A COMBINATORIAL APPROACH TO QUANTIFICATION OF LIE ALGEBRAS

V.K. KHARCHENKO

Abstract. We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinite series defined by the Serre relations and prove that the set of PBW-generators for each of these enveloping algebras coincides with the Lalonde–Ram basis of the ground Lie algebra with a skew commutator in place of the Lie operation. The similar statement is valid for Hall–Shirshov basis of any Lie algebra defined by one relation, but it is not so in general case.

1.Introduction

Quantum universal enveloping algebras appeared in the famous papers by Drinfeld [14] and Jimbo [17]. Since then a great deal of articles and number of monographs were devoted to their investigation. All of these researches are mainly concerned with a particular quantification of Lie algebras of the classical series. This is accounted for first by the fact that these Lie algebras have applications and visual interpretations in physical speculations, and then by the fact that a general, and commonly accepted as standard, notion of a quantum universal enveloping algebra is not elaborated yet (see a detailed discussion in [1, 31]).

In the present paper we propose a combinatorial solution of this problem by means of the quantum (Lie) operation concept [21, 23, 24]. In line with the main idea of our approach, the skew primitive elements must play the same role in quantum enveloping algebras as the primitive elements do in the classical case. By the Friedrichs criteria [12, 15, 30, 32, 33], the primitive elements form the ground Lie algebra in the classical case. For this reason we consider the space spanned by the skew primitive elements and equipped with the quantum operations as a quantum analogue of a Lie algebra.

In the second section we adduce the main notions and consider some examples. These examples, in particular, show that the Drinfeld–Jimbo enveloping algebra as well as its modifications are quantum enveloping algebras in our sense.

Research at MSRI is supported in part by NSF grant DMS-9701755. Supported in part by CONACyT México, grant 32130-E.

In the third section with the help of the Heyneman–Radford theorem we introduce a notion of a combinatorial rank of a Hopf algebra generated by skew primitive semiinvariants. Then we define the quantum enveloping algebra of an arbitrary rank that slightly generalises the definitions given in the preceding section.

The basis construction problem for the quantum enveloping algebras is considered in the fourth section. We indicate two main methods for the construction of PBWgenerators. One of them modifies the Hall–Shirshov basis construction process by means of replacing the Lie operation with a skew commutator. The set of the PBWgenerators defined in this way, the values of hard super-letters, plays the same role as the basis of the ground Lie algebra does in the PBW theorem. At first glance it would seem reasonable to consider the k[G]-module generated by the values of hard super-letters as a quantum Lie algebra. However, this extremely important module falls far short of being uniquely defined. It essentially depends on the ordering of the main generators, their degrees, and it is almost never antipode stable. Also we have to note the following important fact. Our definition of the hard super-letter is not constructive and, of course, it cannot be constructive in general. The basis construction problem includs the word problem for Lie algebras defined by generators and relations, while the latter one has no general algorithmic solution (see [4, 7]).

The second method is connected with the Kashiwara crystallisation idea [19, 20] (see also a development in [11, 25]). M. Kashiwara has considered the main parameter q of the Drinfeld–Jimbo enveloping algebra as a temperature of some physical medium. When the temperature tend to zero, the medium crystallises. The PBWgenerators must crystallise as well. In our case under this process no one limit quantum enveloping algebra appears since the existence conditions normally include equalities of the form ∏ p_{ij} = 1 (see [23]). Nevertheless if we equate all quantification parameters to zero, the hard super-letters would form a new set of PBW-generators for the given quantum universal enveloping algebra. To put this another way, the PBW-basis defined by the super-letters admits the Kashiwara crystallisation.

In the fifth section we bring a way to construct a Groebner–Shirshov relations system for a quantum enveloping algebra. This system is related to the main skew primitive generators, and, according to the Diamond Lemma (see [3, 5, 37]), it determines the crystal basis. The usefulness of the Groebner–Shirshov systems depends upon the fact that such a system not only defines a basis of an associative algebra, but it also provides a simple diminishing algorithm for expansion of elements on this basis (see, for example [2]).

In the sixth section we adapt a well known method of triangular splitting to the quantification with constants. The original method appeared in studies of simple finite dimensional Lie algebras. Then it has been extended into the field of quantum algebra in a lot of publications (see, for example [8, 29, 38]). By means of this method the investigation of the Drinfeld–Jimbo enveloping algebra amounts to a consideration of its positive and negative homogeneous components, quantum Borel sub-algebras.

In the seventh section we consider more thoroughly the quantum universal enveloping algebras of nilpotent algebras of the series A_{n}, B_{n}, C_{n}, D_{n} defined by the Serre relations. We adduce first lists of all hard super-letters in the explicit form, then Groebner–Shirshov relations systems, and next spaces L(U_{P} (𝔤)) spanned by the skew primitive elements (i.e. the Lie algebra quantifications 𝔤_{P} proper). In all cases the lists of hard super-letters (but the hard super-letters themselves) turn out to be independent of the quantification parameters. This means that the PBW-generators result from the Hall–Shirshov basis of the ground Lie algebra by replacing the Lie operation with the skew commutator. The same is valid for the Groebner–Shirshov relations systems. Note that the Hall–Shirshov bases, under the name standard Lyndon bases, for the classical Lie series were constructed by P. Lalonde and A. Ram [26], while the Groebner–Shirshov systems of Lie relations were found by L.A. Bokut’ and A.A. Klein [6].

Furthermore, in all cases 𝔤_{P} as a quantum Lie algebra (in our sense) proves to be very simple in structure. Either it is a coloured Lie super-algebra (provided that the main parameter p_{11} equals 1), or values of all non-unary quantum operations equal zero on 𝔤_{P} . In particular, if char(k) = 0 and p_{11}^{t} ≠ 1 then the partial quantum operations may be defined on 𝔤_{P} , but all of them have zero values. Thus, in this case we have a reason to consider U_{P} (𝔤) as an algebra of ‘commutative’ quantum polynomials, since the universal enveloping algebra of a Lie algebra with zero bracket is the algebra of ordinary commutative polynomials. From this standpoint the Drinfeld–Jimbo enveloping algebra is a ‘quantum’ Weyl algebra of (skew) differential operators. Immediately afterwards a number of interesting questions appears. What is the structure of other algebras of ‘commutative’ quantum polynomials? Under what conditions are the quantum universal enveloping algebras of homogeneous components of other Kac–Moody algebras defined by the Gabber–Kac relations [16] the algebras of ‘commutative’ quantum polynomials? When do the PBW-generators result from a basis of the ground Lie algebra by means of replacing the Lie operation with the skew commutator? These and other questions we briefly discuss in the last section.

It is as well to bear in mind that the combinatorial approach is not free from flaws: the quantum universal enveloping algebra essentially depends on a combinatorial representation of the ground Lie algebra, i.e. a close connection with the abstract category of Lie algebras is lost.

2. Quantum enveloping algebras

Recall that a variable x is called a quantum variable if an element g_{x} of a fixed Abelian group G and a character χ^{x} ∈ G^{∗} are associated with it. A noncommutative polynomial in quantum variables is called a quantum operation if all of its values in all Hopf algebras are skew primitive provided that every variable x has a value x = a such that

Δ(a) = a ⊗ 1 + g_{x} ⊗ a, g^{−1}ag = χ^{x}(g)a, g ∈ G.  (1)

Let x_{1}, …, x_{n} be a set of quantum variables. For each word u in x_{1}, …, x_{n} we denote by g_{u} an element of G that appears from u by replacing of all x_{i} with g_{x_{i}}. In the same way we denote by χ^{u} a character that appears from u by replacing of all x_{i} with χ^{x_{i}}. Thus on the free algebra k⟨x_{1}, …, x_{n}⟩ a grading by the group G × G^{∗} is defined. For each pair of homogeneous elements u, v we fix the denotations p_{uv} = χ^{u}(g_{v}) = p(u, v).

The quantum operation can be defined equivalently as a G × 1-homogeneous polynomial that has only primitive values in all braided bigraded Hopf algebras provided that all quantum variables have primitive homogeneous values g_{a} = g_{x}, χ^{a} = χ^{x} (see [21, Sect. 1–4]).

Recall that a constitution of a word u is a sequence of non-negative integers (m_{1}, m_{2}, …, m_{n}) such that u is of degree m_{1} in x_{1}, deg_{1}(u) = m_{1}; of degree m_{2} in x_{2}, deg_{2}(u) = m_{2}; and so on. Since the group G is abelian, all constitution homogeneous polynomials are homogeneous with respect to the grading. Let us define a bilinear skew commutator on the set of graded homogeneous noncommutative polynomials by the formula

[u, v] = uv − p_{uv}vu.  (2)

These brackets satisfy the following Jacobi and skew differential identities:

[[u, v], w] = [u, [v, w]] + p_{wv}^{−1}[[u, w], v] + (p_{vw} − p_{wv}^{−1})[u, w] · v;  (3)
[[u, v], w] = [u, [v, w]] + p_{vw}[[u, w], v] + p_{uv}(p_{vw}p_{wv} − 1)v · [u, w];  (4)
[u, v · w] = [u, v] · w + p_{uv}v · [u, w]; [u · v, w] = p_{vw}[u, w] · v + u · [v, w], (5)

where by the dot we denote the usual multiplication. It is easy to see that the following conditional restricted identities are valid as well

[u, v^{n}] = [… [[u, v], v] …, v]; [v^{n}, u] = [v, [… [v, u] …]],  (6)

provided that p_{vv} is a primitive t-th root of unit, and n = t or n = tl^{k} in the case of characteristic l > 0.

Suppose that a Lie algebra 𝔤 is defined by the generators x_{1}, …, x_{n} and the relations f_{i} = 0. Let us convert the generators into quantum variables. For this associate to them elements of G × G^{∗} in arbitrary way. Let P = ||p_{ij}||, p_{ij} = χ^{x_{i}}(g_{x_{j}}) be the quantification matrix.

Definition 2.1. A braided quantum enveloping algebra is a braided bigraded Hopf algebra U_{P}^{b} (𝔤) defined by the variables x_{1}, …, x_{n} and the relations f_{i} = 0, where the Lie operation is replaced with (2), provided that in this way f_{i} are converted into the quantum operations f_{i}^{∗}. The coproduct and the commutation relations in the tensor product are defined by

Δ(x_{i}) = x_{i}\underline{⊗}1 + 1\underline{⊗}x_{i},  (7)
(x_{i}\underline{⊗}x_{j}) · (x_{k}\underline{⊗}x_{m}) = (χ^{x_{k}}(g_{x_{j}}))^{−1}x_{i}x_{k}\underline{⊗}x_{j}x_{m}.  (8)

Definition 2.2. A simple quantification or a quantum universal enveloping algebra of 𝔤 is an algebra U_{P} (𝔤) that is isomorphic to the skew group algebra

U_{P} (𝔤) = U_{P}^{b} (𝔤) ∗ G,  (9)

where the group action and the coproduct are defined by

g^{−1}x_{i}g = χ^{x_{i}}(g)x_{i}, Δ(x_{i}) = x_{i} ⊗ 1 + g_{x_{i}} ⊗ x_{i}, Δ(g) = g ⊗ g.  (10)

Definition 2.3. A quantification with constants is a simple quantification where additionally some generators x_{i} associated to the trivial character are replaced with the constants α_{i}(1 − g_{x_{i}}).

The formulae (10) and (7) correctly define the coproduct since by definition of the quantum operation Δ(f_{i}^{∗}) = f_{i}^{∗} ⊗ 1 + g ⊗ f_{i}^{∗} in the case of ordinary Hopf algebras and Δ(f_{i}^{∗}) = f_{i}^{∗}\underline{⊗}1 + 1\underline{⊗}f_{i}^{∗} in the braided case.

We have to note that the defined quantifications essentially depend on the combinatorial representation of the Lie algebra. For example, an additional relation [x_{1}, x_{1}] = 0 does not change the Lie algebra. At the same time if χ^{x_{1}}(g_{1}) = −1 then this relation admits the quantification and yields a nontrivial relation for the quantum enveloping algebra, 2x_{1}^{2} = 0.

Example 1. Suppose that the Lie algebra is defined by a system of constitution homogeneous relations. If the characters χ^{i} are such that p_{ij}p_{ji} = 1 for all i, j then the skew commutator itself is a quantum operation. Therefore on replacing the Lie operation all relations become quantum operations as well. This means that the braided enveloping algebra is the universal enveloping algebra U(𝔤^{col}) of the coloured Lie super-algebra which is defined by the same relations as the given Lie algebra is. The simple quantification appears as the Radford biproduct U(𝔤^{col}) ⋆ k[G] or, equivalently, as the universal G-enveloping algebra of the coloured Lie super-algebra 𝔤^{col} (see [35] or [21, Example 1.9]).

Example 2. Suppose that the Lie algebra 𝔤 is defined by the generators x_{1}, …, x_{n} and the system of nil relations

x_{j}(adx_{i})^{n_{ij}} = 0, 1 ≤ i ≠ j ≤ n.  (11)

Usually instead of the matrix of degrees (without the main diagonal) ||n_{ij}|| the matrix A = ||a_{ij}||, a_{ij} = 1 − n_{ij} is considered. The Coxeter graph Γ(A) is associated to every such a matrix. This graph has the vertices 1, …, n, where the vertex i is connected by a_{ij}a_{ji} edges with the vertex j.

If a_{ij} = 0 then the relation x_{j}adx_{i} = 0 is in the list (11), and the relation x_{i}(adx_{j})^{n_{ji}} = 0 is a consequence of it. The skew commutator [x_{j}, x_{i}] is a quantum operation if and only if p_{ij}p_{ji} = 1. Under this condition we have [x_{i}, x_{j}] = −p_{ij}[x_{j}, x_{i}]. Therefore both in the given Lie algebra and in its quantification one may replace the relation x_{i}(adx_{j})^{n_{ji}} = 0 with x_{i}adx_{j} = 0. In other words, without loss of generality, we may suppose that a_{ij} = 0 ↔ a_{ji} = 0. By the Gabber-Kac theorem [16] we get that the algebra 𝔤 is the positive homogeneous component 𝔤_{1}^{+} of a Kac-Moody algebra 𝔤_{1}.

Theorem 6.1 [21] describes the conditions for a homogeneous polynomial in two variables which is linear in one of them to be a quantum operation. From this theorem we have the following corollary.

Corollary 2.4. If n_{ij} is a simple number or unit and in the former case p_{ii} is not a primitive n_{ij}-th root of unit, then the relation (11) admits a quantification if and only if p_{ij}p_{ji} = p_{ii}^{a_{ij}}.

Theorem 6.1 [21] provides no essential restrictions on the non-diagonal parameters p_{ij} : if the matrix P correctly defines a quantification of (11) then for every set Z = {z_{ij}|z_{ij}z_{ji} = z_{ii} = 1} the following matrix does as well:

P_{Z} = {p_{ij}z_{ij}|p_{ij} ∈ P, z_{ij} ∈ Z}.  (12)

Example 3. Let G be freely generated by g_{1}, … g_{n} and A be a generalised Cartan matrix symmetrised by d_{1}, …, d_{n}, while the characters are defined by p_{ij} = q^{−d_{i}a_{ij}}. In this case the simple quantification is the positive component of the Drinfeld–Jimbo enveloping algebra together with the group-like elements, U_{P} (𝔤) = U_{q}^{+}(𝔤) ∗ G. By means of an arbitrary deformation (12) one may define a ‘colouring’ of U_{q}^{+}(𝔤) ∗ G.

The braided enveloping algebra equals U_{q}^{+}(𝔤) where the coproduct and braiding are defined by (7) and (8) with the coefficient q^{d_{k}a_{kj}}. The formula (12) correctly defines its ‘colouring’ as well.

Example 4. If in the example above we complete the set of quantum variables by the new ones x_{1}^{−}, …, x_{n}^{−}; z_{1}, …, z_{n} such that

χ^{x^{−}} = (χ^{x})^{−1}, g_{x^{−}} = g_{x}, χ^{z} = id, g_{z_{i}} = g_{i}^{2},  (13)

then by [21, Theorem 6.1] the Gabber–Kac relations (2), (3), and [e_{i}, f_{j}] = δ_{ij}h_{i} (see [16, Theorem 2]) under the identification e_{i} = x_{i}, f_{i} = x_{i}^{−}, h_{i} = z_{i} admit the quantification with constants z_{i} = ε_{i} (1 − g_{i}^{2}). (Unformally we may consider the obtained quantification as one of the Kac–Moody algebra identifying g_{i} with q^{h_{i}}, where the rest of the Kac–Moody algebra relations, [h_{i}, e_{j}] = a_{ij}e_{i}, [h_{i}, f_{j}] = −a_{ij}f_{j}, is quantified to the G-action: g_{j}^{−1}x_{i}^{±}g_{j} = q^{∓d_{ij}a_{ij}}x_{i}^{±}.) This quantification coincides with the Drinfeld–Jimbo one under a suitable choice of x_{i}, x_{i}^{−}, and ε_{i} depending up the particular definition of U_{q}(𝔤) :

\begin{matrix}[28] & x_{i} = E_{i}, g_{i} = K_{i}, x_{i}^{−} = F_{i}K_{i}, p_{ij} = v^{−d_{i}a_{ij}}, ε_{i} = (v^{−d_{i}} − v^{d_{i}})^{−1}; \\ [29] & x_{i} = E_{i}, g_{i} = K̃_{i}, x_{i}^{−} = F_{i}K̃_{i}, p_{iμ} = v^{−⟨μ,i′⟩}, ε_{i} = (v_{i}^{−1} − v_{i})^{−1}; \\ [19]Δ_{+} & x_{i} = e_{i}, g_{i} = t_{i}, x_{i}^{−} = t_{i}f_{i}, p_{ij} = q_{j}^{−⟨h_{j},α_{i}⟩}, ε_{i} = (q_{i} − q_{i}^{3})^{−1}; \\ [19]Δ_{−} & x_{i} = f_{i}, g_{i} = t_{i}, x_{i}^{−} = e_{i}t_{i}, p_{ij} = q_{j}^{⟨h_{j},α_{i}⟩}, ε_{i} = (q_{i}^{−1} − q_{i})^{−1}; \\ [34] & x_{i} = E_{i}K_{i}, g_{i} = K_{i}^{2}, x_{i}^{−} = F_{i}K_{i}, p_{ij} = q^{−2d_{i}a_{ij}}, ε_{i} = (1 − q^{4d_{i}})^{−1}. \end{matrix}

By (13) the brackets [x_{i}, x_{j}^{−}] are quantum operations only if p_{ij} = p_{ji}. So in this case the ‘colourings’ (12) may be only black-white, z_{ij} = ±1.

In the perfect analogy the Kang quantification [18] of the generalised Kac-Moody algebras [9] is a quantification in our sense as well.

3.Combinatorial rank

By the above definitions the quantum enveloping algebras (with or without constants) are character Hopf algebras (see [21, Definition 1.2]). In this section by means of a combinatorial rank notion we identify the quantum enveloping algebras in the class of character Hopf algebras.

Let H be a character Hopf algebra generated by a_{1}, …, a_{n} :

Δ(a_{i}) = a_{i} ⊗ 1 + g_{a_{i}} ⊗ a_{i}, g^{−1}a_{i}g = χ^{a_{i}}(g)a_{i}, g ∈ G.  (14)

Let us associate a quantum variable x_{i} with the parameters (χ^{a_{i}}, g_{a_{i}}) to a_{i}. Denote by G⟨X⟩ the free enveloping algebra defined by the quantum variables x_{1} …, x_{n}. (see [21, Sec. 3] under denotation H⟨X⟩).

The map x_{i} → a_{i} has an extension to a homomorphism of Hopf algebras φ : G⟨X⟩ → H. Denote by I the kernel of this homomorphism. If I ≠ 0 then by the Heyneman–Radford theorem (see, for example [34, pages 65–71]), the Hopf ideal I has a non-zero skew primitive element. Let I_{1} be an ideal generated by all skew primitive elements of I. Clearly I_{1} is a Hopf ideal as well. Now consider the Hopf ideal I/I_{1} of the quotient Hopf algebra G⟨X⟩/I_{1}. This ideal also has non-zero skew primitive elements (provided I_{1} ≠ I). Denote by I_{2}/I_{1} the ideal generated by all skew primitive elements of I/I_{1}, where I_{2} is its preimage with respect to the projection G⟨X⟩ → G⟨X⟩/I_{1}. Continuing the process we will find a strictly increasing, finite or infinite, chain of Hopf ideals of G⟨X⟩ :

0 = I_{0} ⊂ I_{1} ⊂ I_{2} ⊂ … ⊂ I_{n} ⊂ ….  (15)

Definition 3.1. The length of (15) is called a combinatorial rank of H.

By definition, the combinatorial rank of any quantum enveloping algebra (with constants) equals one. In the case of zero characteristic the inverse statement is valid as well.

Theorem 3.2. Each character Hopf algebra of the combinatorial rank 1 over a field of zero characteristic is isomorphic to a quantum enveloping algebra with constants of a Lie algebra.

Proof. By definition, I is generated by skew primitive elements. These elements as noncommutative polynomials are the quantum operations. Consider one of them, say f. Let us decompose f into a sum of homogeneous components f = ∑ f_{i}. All positive components belongs to k⟨X⟩ and they are the quantum operations themselves, while the constant component has the form α(1−g), g ∈ G (see [21, Sec. 3 and Prop. 3.3]). If α ≠ 0 then we introduce a new quantum variable z_{f} with the parameters (id, g). Each f_{i} has a representation through the skew commutator. Indeed, by [21, Theorem 7.5] the complete linearization f_{i}^{lin} of f_{i} has the required representation. By the identification of variables in a suitable way in f_{i}^{lin} we get the required representation for f_{i} multiplied by a natural number, m_{i}f_{i} = f_{i}^{[ ]}.

Now consider a Lie algebra 𝔤 defined by the generators x_{i}, z_{f} and the relations ∑ m_{i}^{−1}f_{i}^{[ ]} + z_{f} = 0, with the Lie multiplication in place of the skew commutator. It is clear that H is the quantification with constants of 𝔤.

In the same way one may introduce the notion of the combinatorial rank for the braided bigraded Hopf algebras. In this case all braided quantum enveloping algebras are of rank 1, and all braided bigraded algebras of rank 1 are the braided quantification of some Lie algebras.

Now we are ready to define a quantification of arbitrary rank. For this in the definitions of the above section it is necessary to change the requirement that all f_{i}^{∗} are quantum operations with the following condition.

The set F splits in a union F = ∪_{j=1}^{n}F_{j} such that F_{1}^{∗} consists of quantum operations; the set F_{2}^{∗} consists of skew primitive elements of G⟨X||F_{1}^{∗}⟩; the set F_{3}^{∗} consists of skew primitive elements of G⟨X||F_{1}^{∗}, F_{2}^{∗}⟩, and so on.

The quantum enveloping algebras of an arbitrary rank are character Hopf algebras also. But it is not clear if any character Hopf algebra is a quantification of some rank of a suitable Lie algebra. It is so if the Hopf algebra is homogeneous and the ground field has a zero characteristic (to appear). Also it is not clear if there exist character Hopf algebras, or braided bigraded Hopf algebras, of infinite combinatorial rank; while it is easy to see that ∪I_{n} = I. Also it is possible to show that F_{1} always contains all relations of a minimal constitution in F. For example, each of (11) is of a minimal constitution in (11). Therefore the quantification of arbitrary rank with the identification g_{i} = exp(h_{i}) of any (generalized) Kac–Moody algebra 𝔤, or its nilpotent component 𝔤^{+}, is always a quantification in the sense of the above section.

4.PBW-generators and crystallisation

The next result yields a PBW basis for the quantum enveloping algebras.

Theorem 4.1. Every character Hopf algebra H has a linearly ordered set of constitution homogeneous elements U = {u_{i} | i ∈ I} such that the set of all products gu_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{m}^{n_{m}}, where g ∈ G, u_{1} < u_{2} < … < u_{m}, 0 ≤ n_{i} < h(i) forms a basis of H. Here if p_{ii} =^{df} p_{u_{i}u_{i}} is not a root of unity then h(i) = ∞; if p_{ii} = 1 then either h(i) = ∞ or h(i) = l is the characteristic of the ground field; if p_{ii} is a primitive t-th root of unity, t ≠ 1, then h(i) = t.

The set U is referred to as a set of PBW-generators of H. This theorem easily follows from [22, Theorem 2]. Let us recall necessary notions.

Let a_{1}, …, a_{n} be a set of skew primitive generators of H, and let x_{i} be the associated quantum variables. Consider the lexicographical ordering of all words in x_{1} > x_{2} > … > x_{n}. A non-empty word u is called standard if vw > wv for each decomposition u = vw with non-empty v, w. The following properties are well known (see, for example [10, 13, 27, 36, 37]).

1s. A word u is standard if and only if it is greater than each of its ends.

2s. Every standard word starts with a maximal letter that it has.

3s. Each word c has a unique representation c = u_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{k}^{n_{k}}, where u_{1} < u_{2} < ⋯ < u_{k} are standard words (the Lyndon theorem).

4s. If u, v are different standard words and u^{n} contains v^{k} as a sub-word, u^{n} = cv^{k}d, then u itself contains v^{k} as a sub-word, u = bv^{k}e.

The set of standard nonassociative words is defined as the smallest set SL that contains all variables x_{i} and satisfies the following properties.

1) If [u] = [[v][w]] ∈ SL then [v], [w] ∈ SL, and v > w are standard.

2) If [u] = [[[v_{1}][v_{2}]][w]] ∈ SL then v_{2} ≤ w.

The following statements are valid as well.

5s. Every standard word has the only alignment of brackets such that the appeared nonassociative word is standard (the Shirshov theorem [36]).

6s. The factors v, w of the nonassociative decomposition [u] = [[v][w]] are the standard words such that u = vw and v has the minimal length ([37]).

Definition 4.2. A super-letter is a polynomial that equals a nonassociative standard word where the brackets mean (2). A super-word is a word in super-letters. By 5s every standard word u defines a super-letter [u].

Let D be a linearly ordered Abelian additive group. Suppose that some positive D-degrees d_{1}, …, d_{n} ∈ D are associated to x_{1}, …, x_{n}. We define the degree of a word to be equal to m_{1}d_{1} + … + m_{n}d_{n} where (m_{1}, …, m_{n}) is the constitution of the word. The order and the degree on the super-letters are defined in the following way:

[u] > [v] ⟺ u > v; D([u]) = D(u).

Definition 4.3. A super-letter [u] is called hard in H provided that its value in H is not a linear combination of values of super-words of the same degree in less than [u] super-letters \underline{and G-super-words of a lesser degree}.

Definition 4.4. We say that a height of a super-letter [u] of degree d equals h = h([u]) if h is the smallest number such that: first p_{uu} is a primitive t-th root of unity and either h = t or h = tl^{r}, where l =char(k); and then the value in H of [u]^{h} is a linear combination of super-words of degree hd in less than [u] super-letters \underline{and G-super-words of a lesser degree}. If there exists no such number then the height equals infinity.

Clearly, if the algebra H is D-homogeneous then one may omit the underlined parts of the above definitions.

Theorem 4.5. ([22, Theorem 2]). The set of all values in H of all G-super-words W in the hard super-letters [u_{i}],

W = g[u_{1}]^{n_{1}}[u_{2}]^{n_{2}} ⋯ [u_{m}]^{n_{m}},  (16)

where g ∈ G, u_{1} < u_{2} < … < u_{m}, n_{i} < h([u_{i}]) is a basis of H.

In order to find the set of PBW-generators it is necessary first to include in U the values of all hard super-letters, then for each hard super-letter [u] of a finite height, h([u]) = tl^{k}, to add the values of [u]^{t}, [u]^{tl}, … [u]^{tl^{(k−1)}}, and next for each hard superletter of infinite height such that p_{uu} is a primitive t-th root of unity to add the value of [u]^{t}.

Obviously the set of PBW-generators plays the same role as the basis of the Lie algebra in the PBW theorem does. Nevertheless the k[G]-bimodule generated by the PBW-generators is not uniquely defined. It depends on the ordering of the main generators, the D-degree, and under the action of antipode it transforms to a different bimodule of PBW-generators k[G]S(U).

Another way to construct PBW-generators is connected with the M.Kashiwara crystallisation idea [19, 20]. M.Kashiwara considered the main parameter of the Drinfeld–Jimbo enveloping algebra as the temperature of some physical medium.

When the temperature tends to zero the medium crystallises. By this means the ‘crystal’ bases must appear. If we replace p_{ij} with zero then [u, v] turns into uv, while [u] turns into u.

Lemma 4.6. (Bases Crystallisation). Under the above crystallisation the set of PBWgenerators constructed in Theorem 4.5 turns into another set of PBW-generators.

Proof. See [22, Corollary 1].

Lemma 4.7. (Super-letters Crystallisation). A super-letter [u] is hard in H if and only if the value of u is not a linear combination of lesser words of the same degree \underline{and G-words of a lesser degree}.

Proof. See [22, Corollary 2].

Lemma 4.8. Let B be a set of the super-letters containing x_{1}, …, x_{n}. If each pair [u], [v] ∈ B, u > v satisfies one of the following conditions

1) [[u][v]] is not a standard nonassociative word;

2) the super-letter [[u][v]] is not hard in H;

3) [[u][v]] ∈ B, then the set B includes all hard in H super-letters.

Proof. Let [w] be a hard super-letter of minimal degree such that [w] ∉ B. Then [w] = [[u][v]], u > v where [u], [v] are hard super-letters. Indeed, if [u] is not hard then by Lemma 4.7 we have u = ∑ α_{i}u_{i} + S, where u_{i} < u and D(u_{i}) = D(u), D(S) < D(u). We have uv = ∑ α_{i}u_{i}v + Sv, where u_{i}v < uv. Therefore by Lemma 4.7, the super-letter [w] = [uv] can not be hard in H. Contradiction. Similarly, if [v] is not hard then v = ∑ α_{i}v_{i} + S, v_{i} < v, D(v_{i}) = D(v), D(S) < D(v). Therefore uv = ∑ α_{i}uv_{i} + uS, uv_{i} < uv, and again [w] can not be hard.

Thus, according to the choice of [w], we get [u], [v] ∈ B. Since this pair satisfies neither condition 1) nor 2), the condition 3), [uv] ∈ B, holds. □

Lemma 4.9. If T ∈ H is a skew primitive element then

T = [u]^{h} + ∑ α_{i}W_{i} + ∑ β_{j}g_{j}W′_{j}

where [u] is a hard super-letter, W_{i} are basis super-words in super-letters less than [u], D(W_{i}) = hD([u]), D(W′_{j}) < hD([u]). Here if p_{uu} is not a root of unity then h = 1; if p_{uu} is a primitive t-th root of unity then h = 1, or h = t, or h = tl^{k}, where l is the characteristic.

Proof. Consider an expansion of T in terms of the basis (16)

T = αgU + ∑_{i=1}^{k} γ_{i}g_{i}W_{i} + W′, α ≠ 0,  (17)

where gU, g_{i}W_{i} are different basis elements of maximal degree, and U is one of the biggest words among U, W_{i} with respect to the lexicographic ordering of words in the super-letters. On basis expansion of tensors, the element Δ(T) − T ⊗ 1 − g_{t} ⊗ T has only one tensor of the form gU ⊗ … and this tensor equals gU ⊗ α(g − 1). Therefore g = 1 and one may apply [22, Lemma 13]. □

5. Groebner–Shirshov relations systems

Let x_{1}, …, x_{n} be variables that have positive degrees d_{1}, …, d_{n} ∈ D. Recall that a Hall ordering of words in x_{1}, …, x_{n} is an order when the words are compared firstly by the degree and then words of the same degree are compared by means of the lexicographic ordering . Consider a set of relations

w_{i} = f_{i}, i ∈ I,  (18)

where w_{i} is a word and f_{i} is a linear combination of Hall lesser words. The system (18) is said to be closed under compositions or a Groebner–Shirshov relations system if first none of w_{i} contains w_{j}, i ≠ j ∈ I as a sub-word, and then for each pair of words w_{k}, w_{j} such that some non-empty terminal of w_{k} coincides with an onset of w_{j}, that is w_{k} = w′_{k}v, w_{j} = vw′_{j}, the difference (a composition) f_{k}w′_{j} − w′_{k}f_{j} can be reduced to zero in the free algebra by means of a sequence of one sided substitutions w_{i} → f_{i}, i ∈ I.

Lemma 5.1. (Diamond Lemma [3, 5, 37]). If the system (18) is closed under compositions then the words that have none of w_{i} as sub-words form a basis of the algebra H defined by (18).

If none of the words w_{i} has sub-words w_{j}, j ≠ i, then the converse statement is valid as well. Indeed, any composition by means of substitutions w_{i} → f_{i} can be reduced to a linear combination of words that have no sub-words w_{i}. Since f_{i}w′_{j} − w′_{i}f_{j} = (f_{i} − w_{i})w′_{j} − w′_{i}(f_{j} − w_{j}), this linear combination equals zero in H. Therefore all the coefficients have to be zero.

Since Bases Crystallisation Lemma provides the basis that consists of words, the above note gives a way to construct the Groebner–Shirshov relations system for any quantum enveloping algebra.

Let H be a character Hopf algebra generated by skew primitive semi-invariants a_{1}, …, a_{n} (or a braided bigraded Hopf algebra generated by graiding homogeneous primitive elements a_{1}, …, a_{n}) and let x_{1}, …, x_{n} be the related quantum variables. A non-hard in H super-letter [w] is referred to as a minimal one if first w has no proper standard sub-words that define non-hard super-letters, and then w has no sub-words u^{h}, where [u] is a hard super-letter of the height h.

By the Super-letters Crystallisation Lemma, for every minimal non-hard in H super-letter [w] we may write a relation in H

w = ∑ α_{i}w_{i} + ∑ β_{j}g_{j}w_{j},  (19)

where w_{j}, w_{i} < w in the Hall sense, D(w_{i}) = D(w), D(w_{j}) < D(w). In the same way if [u] is a hard in H super-letter of a finite height h then

u^{h} = ∑ α_{i}u_{i} + ∑ β_{j}g_{j}u_{j},  (20)

where u_{j}, u_{i} < u^{h} in the Hall sense, D(u_{i}) = hD(u), D(u_{j}) < hD(u). The relations (14) and the group operation provide the relations

x_{i}g = χ^{x_{i}}(g)gx_{i}, g_{1}g_{2} = g_{3}.  (21)

Theorem 5.2. The set of relations (19), (20), and (21) forms a Groebner–Shirshov system that defines H. The basis determined by this system in Diamond Lemma coincides with the crystal basis.

Proof. The property 4s implies that none of the left hand sides of (19), (20), (21) contains another one as a sub-word. Therefore by the Bases Crystallisation Lemma it is sufficient to show that the set of all words c determined in the Diamond Lemma coincides with the crystal basis. By 3s we have c = u_{1}^{n_{1}}u_{2}^{n_{2}} ⋯ u_{k}^{n_{k}}, where u_{1} < … < u_{k} is a sequence of standard words. Every word u_{i} define a hard super-letter [u_{i}] since in the opposite case u_{i}, and therefore c, contains a sub-word w that defines a minimal non-hard super-letter [w]. In the same way n_{i} does not exceed the height of [u_{i}]. □

Lemma 5.3. In terms of Lemma 4.8 the set of all super-letters [[u][v]] that satisfy the condition 2) contains all minimal non-hard super-letters, but non-hard generators x_{i}.

Proof. If [w] is a minimal non-hard super-letter then [w] = [[u][v]], where [u], [v] are hard super-letters. By Lemma 4.8 we have [u], [v] ∈ B, while [[u][v]] neither satisfies 1) nor 3). □

6.Quantification with constants

By means of the Diamond Lemma in some instances the investigation of a quantification with constants can be reduced to one of a simple quantification.

Let H_{1} = ⟨x_{1}, …, x_{k}||F_{1}⟩ be a character Hopf algebra defined by the quantum variables x_{1}, …, x_{k} and the grading homogeneous relations {f = 0 : f ∈ F_{1}}, while H_{2} = ⟨x_{k+1}, …, x_{n}||F_{2}⟩ is a character Hopf algebra defined by the quantum variables x_{k+1}, …, x_{n} and the grading homogeneous relations {h = 0 : h ∈ F_{2}}. Consider the algebra H = ⟨x_{1}, …, x_{n}||F_{1}, F_{2}, F_{3}⟩, where F_{3} is the following system of relations with constants

[x_{i}, x_{j}] = α_{ij}(1 − g_{i}g_{j}), 1 ≤ i ≤ k < j ≤ n.  (22)

If the conditions below are met then the character Hopf algebra structure on H is uniquely determined:

p_{ij}p_{ji} = 1, 1 ≤ i ≤ k < j ≤ n; χ^{x_{i}}χ^{x_{j}} ≠ 1 ⟹ α_{ij} = 0.  (23)

Indeed, in this case the difference w_{ij} between the left and right hand sides of (22) is a skew primitive semi-invariant of the free enveloping algebra G⟨x_{1}, …, x_{n}⟩. Consider the ideals of relations I_{1} =id(F_{1}) and I_{2} =id(F_{2}) of H_{1} and H_{2} respectively. They are, in the present context, Hopf ideals of G⟨x_{1}, …, x_{k}⟩ and G⟨x_{k+1}, …, x_{n}⟩, respectively. Therefore V = I_{1} + I_{2}+ ∑kw_{ij} is an antipode stable coideal of G⟨X⟩. Consequently the ideal generated by V is a Hopf ideal. It remains to note that this ideal is generated in G⟨X⟩ by w_{ij} and F_{1}, F_{2}.

Lemma 6.1. Every hard in H super-letter belongs to either H_{1} or H_{2}, and it is hard in the related algebra.

Proof. If a standard word contains at least one of the letters x_{i}, i ≤ k then it has to start with one of them (see s2). If this word contains a letter x_{j}, j > k then it has a sub word of the form x_{i}x_{j}, i ≤ k < j. Therefore by Lemma 4.7 and relations (22) this word defines a non-hard super-letter. □

The converse statement is not universally true. In order to formulate the necessary and sufficient conditions let us define partial skew derivatives:

(x_{j})′_{i} = (x_{i})′_{j} = α_{ij}(1 − g_{i}g_{j}), i ≤ k < j;
(v · w)′_{i} = (v)′_{i} · w + p(x_{i}, v)v · (w)′_{i}, i ≤ k, v, w ∈ G⟨x_{k+1}, …, x_{n}⟩;
(u · v)′_{j} = p(v, x_{j})(u)′_{j} · v + u · (v)′_{j}, j > k, u, v ∈ G⟨x_{1}, …, x_{k}⟩.  (24)

Lemma 6.2. All hard in H_{1} or H_{2} super-letters are hard in H if and only if (h)′_{i} = 0 in H_{2} for all i ≤ k, h ∈ F_{2} and (f)′_{j} = 0 in H_{1} for all j > k, f ∈ F_{1}. If these conditions are met then

H ≅ H_{2} ⊗_{k[G]} H_{1}  (25)

as k[G]-bimodules and the space generated by the skew primitive elements of H equals the sum of these spaces for H_{1} and H_{2}.

Proof. By (5) and (24) the following equalities are valid in H :

0 = [x_{i}, h] = (h)′_{i}; 0 = [f, x_{j}] = (f)′_{j}, i ≤ k < j.  (26)

If all hard in H_{1} or H_{2} super-letters are hard in H then H_{1}, H_{2} are sub-algebras of H. So (26) proves the necessity of the lemma conditions.

Conversely. Let us consider an algebra R defined by the generators g ∈ G, x_{1}, …, x_{n} and the relations (21), (22). Evidently this system is closed under the compositions. Therefore by Diamond Lemma the set of words gvw forms a basis of R where g ∈ G; v is a word in x_{j}, j > k; and w is a word in x_{i}, i ≤ k. In other words R as a bimodule over k[G] has a decomposition

R = G⟨x_{k+1}, …, x_{n}⟩ ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩.  (27)

Let us show that the two sided ideal of R generated by F_{2} coincides with the right ideal I_{2}R = I_{2} ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩. It will suffice to show that I_{2}R admits left multiplication by x_{i}, i ≤ k. If v is a word in x_{k+1}, …, x_{k}, h ∈ F_{2}, r ∈ R then x_{i}vhr = [x_{i}, vh]r + p(x_{i}, vh)vhx_{i}r. The second term belongs to I_{2}R, while the first one can be rewritten by (5): [x_{i}, v]h + p(x_{i}, v)v[x_{i}, h]. Both of these addends belong to I_{2}R since [x_{i}, v] = (v)′_{i} ∈ G⟨x_{k+1}, …, x_{n}⟩ and [x_{i}, h] = (h)′_{i} ∈ I_{2}.

Furthermore, consider a quotient algebra R_{1} = R/I_{2}R :

R_{1} = (G⟨x_{k+1}, …, x_{n}⟩ ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩)/(I_{2} ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩) =
H_{2} ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩,

where the equality means the natural isomorphism of k[G]-bimodules.

Along similar lines, the left ideal R_{1}I_{1} = H_{2} ⊗_{k[G]} I_{1} of this quotient algebra coincides with the two sided ideal generated by F_{1}. Therefore

H = R_{1}/R_{1}I_{1} = H_{2} ⊗_{k[G]} G⟨x_{1}, …, x_{k}⟩/H_{2} ⊗_{k[G]} I_{1} = H_{2} ⊗_{k[G]} H_{1}.

Thus the monotonous restricted G-words in hard in H_{1} or H_{2} super-letters form a basis of H. This, in particular, proves the first statement.

Now let T = ∑ α_{t}g_{t}V_{t}W_{t} be the basis decomposition of a skew primitive element, g_{t} ∈ G, V_{t} ∈ H_{2}, W_{t} ∈ H_{1}, α_{t} ≠ 0. We have to show that for each t one of the super-words V_{t} or W_{t} is empty. Suppose that it is not so. Among the addends with non-empty V_{t}, W_{t} we choose the largest one in the Hall sense, say g_{s}V_{s}W_{s}. Under the basis decomposition of Δ(T)−T ⊗1−g(T)⊗T the term α_{s}g_{s}g(V_{s})W_{s} ⊗g_{s}V_{s} appears and cannot be cancelled with other. Indeed, since the coproduct is homogeneous (see [22, Lemma 9]) and since under the basis decomposition the super-words are decreased (see [22, Lemma 7]) the product α_{s}(g_{s} ⊗ g_{s})Δ(V_{s})Δ(W_{s}) has the only term of the above type. By the same reasons α_{t}(g_{t} ⊗ g_{t})Δ(V_{t})Δ(W_{t}) has a term of the above type only if V_{t} ≥ V_{s} and W_{t} ≥ W_{s} with respect to the Hall ordering of the set of all super-words. However, by the choise of s, we have D(V_{s}W_{s}) ≥ D(V_{t}W_{t}). Hence D(V_{t}) = D(V_{s}) and D(W_{t}) = D(W_{s}). In particular V_{t} is not a proper onset of V_{s}. Therefore V_{t} = V_{s} since otherwise the inequality V_{t} > V_{s} yields a contradiction V_{t}W_{t} > V_{s}W_{s}. The inequality W_{t} > W_{s} get the same contradiction. Therefore V_{t} = V_{s} and W_{t} = W_{s}, in which case g_{t}g(V_{t})W_{t} ⊗ g_{t}V_{t} = g_{s}g(V_{s})W_{s} ⊗ g_{s}V_{s}. Thus g_{t} = g_{s} and t = s. □

7.Quantification of the classical series

In this section we apply the above general results to the infinite series A_{n}, B_{n}, C_{n}, D_{n} of nilpotent Lie algebras defined by the Serre relations (11). Let 𝔤 be any such Lie algebra.

Lemma 7.1. If a standard word u has no sub words of the type

x_{i}^{s}x_{j}x_{i}^{m}, where s + m = 1 − a_{ij}  (28)

then [u] is a hard in U_{P} (𝔤) super-letter.

Proof. Let R be defined by the generators x_{1}, …, x_{n} and the relations

x_{i}^{s}x_{j}x_{i}^{m} = 0, where s + m = 1 − a_{ij}.  (29)

Clearly (29) implies (11) with the skew commutator in place of the Lie operation. Therefore R is a homomorphic image of U_{P} (𝔤). The system (29) is closed under compositions since a composition of monomial relations always has the form 0 = 0.

Let u have no sub-words (28). If [u] is not hard then, by the Super-letters Crystallisation Lemma, u is a linear combination of lesser words in U_{P} (𝔤). Therefore u is a linear combination of lesser words in R as well. This contradicts the fact that u belongs to the Groebner–Shirshov basis of R, since every word either belongs to this basis or equals zero in R. □

Theorem A_{n}. Suppose that 𝔤 is of the type A_{n}, and p_{ii} ≠ −1. Denote by B the set of the super-letters given below:

[u_{km}] =^{df} [x_{k}x_{k+1} … x_{m}], 1 ≤ k ≤ m ≤ n.  (30)

The following statements are valid.

1. The values of [u_{km}] in U_{P} (𝔤) form a PBW-generators set.

2. Each of the super-letters (30) has infinite height in U_{P} (𝔤).

3. The values of all non-hard in U_{P} (𝔤) super-letters equal zero.

4. The following relations with (21) form the Groebner–Shirshov relations system that determines the crystal basis of U_{P} (𝔤) :

\begin{matrix}[u_{0}] =^{df} [x_{k}x_{m}] = 0, & 1 ≤ k < m − 1 < n; \\ [u_{1}] =^{df} [x_{k}x_{k+1} … x_{m}x_{k+1}] = 0, & 1 ≤ k < m ≤ n; \\ [u_{2}] =^{df} [x_{k}x_{k+1} … x_{m}x_{k}x_{k+1} … x_{m+1}] = 0, & 1 ≤ k ≤ m < n. \end{matrix}  (31)

5. If p_{11} ≠ 1 then the generators x_{i}, the constants 1−g, g ∈ G, and, in the case that p_{11} is a primitive t-th root of 1, the elements x_{i}^{t}, x_{i}^{tl^{k}} form a basis of 𝔤_{P} = L(U_{P} (𝔤)). Here l is the characteristic of the ground field.

6. If p_{11} = 1 then the elements (30) and, in the case l > 0, their l^{k}-th powers, together with 1 − g, g ∈ G form a basis of 𝔤_{P} .

By Corollary 2.4 the relations (11) with a Cartan matrix A of type A_{n} admit a quantification if and only if

p_{ii} = p_{11}, p_{ii+1}p_{i+1i} = p_{11}^{−1}; p_{ij}p_{ji} = 1, i − j > 1.  (32)

In this case the quantified relations (11) take up the form

x_{i}x_{i+1}^{2} = p_{ii+1}(1 + p_{i+1i+1})x_{i+1}x_{i}x_{i+1} − p_{ii+1}^{2}p_{i+1i+1}x_{i+1}^{2}x_{i},  (33)
x_{i}^{2}x_{i+1} = p_{ii+1}(1 + p_{ii})x_{i}x_{i+1}x_{i} − p_{ii+1}^{2}p_{ii}x_{i+1}x_{i}^{2},  (34)
x_{i}x_{j} = p_{ij}x_{j}x_{i}, i − j > 1.  (35)

Let us introduce a congruence u ≡_{k} v on G⟨X⟩. This congruence means that the value of u − v in U_{P}^{b} (𝔤) belongs to the subspace generated by values of all words with the initial letters x_{i}, i ≥ k. Clearly, this congruence admits right multiplication by arbitrary polynomials as well as left multiplication by the independent of x_{k−1} ones (see (35)). For example, by (33) and (34) we have

x_{i}x_{i+1}^{2} ≡_{i+1} 0; x_{i}x_{i+1}x_{i} ≡_{i+1} αx_{i}^{2}x_{i+1}, α ≠ 0.  (36)

Lemma 7.2. If y = x_{i}, m + 1 ≠ i > k or y = x_{i}^{2}, m + 1 = i > k then

u_{km}y ≡_{k+1} 0.  (37)

Proof. Let y = x_{m+1}^{2}, m + 1 > k. By (36) and (35) we have that u_{km}y = u_{k m−1}\underline{x_{m}x_{m+1}^{2}} ≡_{m+1} 0. If y = x_{i} and m+1 ≠ i > k then we get u_{km}y = αu_{k i−1}\underline{x_{i}x_{i+1}x_{i}} u_{i+2m} ≡_{i+1} β\underline{u_{k i−1}x_{i}^{2}}u_{i+1m} ≡_{k+1} 0 by the above case. □

Lemma 7.3. The brackets in [u_{km}] are left-ordered, [u_{km}] = [x_{k}[u_{k+1m}]].

Proof. The statement immediately follows from the properties 6s and 2s. □

Lemma 7.4. If a nonassociative word [[u_{km}][u_{rs}]] is standard then k = m ≤ r; or r = k + 1, m ≥ s; or r = k, m < s.

Proof. By definition, u_{km} > u_{rs} if and only if either k < r; or k = r, m < s. If k = m then u_{km} = x_{k} and m ≤ r. If k ≠ m then [u_{km}] = [x_{k}[u_{k+1m}]]. Therefore u_{k+1m} ≤ u_{rs}, i.e. either k+1 > r; or k+1 = r and m ≥ s. The former case contradicts k < r while the latter one does k = r. Thus only the possibilities set in the lemma remain. □

Lemma 7.5. If [w] = [[u_{km}][u_{rs}]], n ≥ 1 is a standard nonassociative word then the constitution of [w]^{h} does not equal the constitution of any super-word in less than [w] super-letters from B.

Proof. The inequalities at the last column of the following tableaux are valid for all [u] ∈ B that are less than the super-letters located on the same row, where as above deg_{i}(u) means the degree of u in x_{i}.

[x_{k}u_{k+1s}]  deg_{k}(u) ≤ deg_{s+1}(u);
[x_{k}u_{rs}],  k ≤ r ≠ k + 1 deg_{k}(u) ≤ deg_{k+1}(u);
[u_{km}u_{k+1s}], m ≥ s  deg_{k}(u) ≤ deg_{m+1}(u);
[u_{km}u_{ks}], m < s  deg_{k}(u) ≤ deg_{m+1}(u).

(38)

If all super-letters of a super-word U satisfy one of these inequalities then U does as well. Clearly, no one of the super-letters in the first column satisfies the degree inequality on the same row. Finally, by Lemma 7.4 the first column contains all standard nonassociative words of the type [[u_{km}][u_{rs}]]. □

Lemma 7.6. If p_{11} ≠ 1 then the values of [u_{km}]^{h}, k < m, h ≥ 1 are not skew primitive, in particular they are non-zero.

Proof. The sub-algebra generated by x_{2}, … x_{n} is defined by the Cartan matrix of the type A_{n−1}. This allows us to use induction on n. If n = 1 then the lemma is correct in the sense that [u_{km}]^{h} = x_{1}^{h} ≠ 0.

Let n > 1. If k > 1 then we may use the inductive supposition directly. Consider the decomposition Δ([u_{1m}]) = ∑ u^{(1)} ⊗ u^{(2)}. Since

[u_{1m}] = x_{1}[u_{2m}] − p(x_{1}, u_{2m})[u_{2m}]x_{1},  (39)

we have

Δ([u_{1m}]) = (x_{1} ⊗ 1 + g_{1} ⊗ x_{1})Δ([u_{2m}])−
p(x_{1}, u_{2m})Δ([u_{2m}])(x_{1} ⊗ 1 + g_{1} ⊗ x_{1}).  (40)

Therefore the sum of all tensors u^{(1)} ⊗ u^{(2)} with deg_{1}(u^{(2)}) = 1, deg_{k}(u^{(2)}) = 0, k > 1 has the form εg_{1}[u_{2m}] ⊗ x_{1}, where ε = 1 − p(x_{1}, u_{2m})p(u_{2m}, x_{1}) since [u_{2m}]g_{1} = p(u_{2m}, x_{1})g_{1}[u_{2m}]. By (32) we have p_{ij}p_{ji} = 1 for i − 1 > j. Therefore ε = 1 − p_{12}p_{21} = 1 − p_{11}^{−1} ≠ 0.

This implies that in the decomposition Δ([u_{1m}]^{h}) = ∑ v^{(1)} ⊗ v^{(2)} the sum of all tensors v^{(1)} ⊗ v^{(2)} with deg_{1}(v^{(2)}) = h, deg_{k}(v^{(2)}) = 0, k > 1 equals ε^{h}[u_{2m}]^{h} ⊗ x_{1}^{h}. Thus [u_{1m}]^{h} is not skew primitive in U_{P} (𝔤). □

Proof of Theorem A_{n}. Let us show firstly that B satisfies the conditions of Lemma 4.8. By the Super-letter Crystallisation Lemma [w] = [[u_{km}][u_{rs}]] is non-hard if the value of u_{km}u_{rs} is a linear combination of lesser words. For k = m, r = k + 1 we have [w] = [u_{ks}] ∈ B. If k = m, r > k + 1 then the word x_{k}u_{rs} can be diminished by (34) or (35). If k ≠ m then by Lemma 7.4 the word u_{km}u_{rs} has a sub-word of the type u_{1} or u_{2}. Thus we need show only that the values in U_{P} (𝔤) of u_{1} and u_{2} are linear combinations of lesser words.

The word u_{1} has such a representation by Lemma 7.2. Consider the word u_{2}. Let us show by downward induction on k that

u_{km}u_{k m+1} ≡_{k+1} γu_{k m+1}u_{km}, γ ≠ 0.  (41)

If k = m then one may use (34) with i = k. Let k < m. Let us transpose the second letter x_{k} of u_{2} as far to the left as possible by (35). We get

u_{2} = α\underline{x_{k}x_{k+1}x_{k}}x_{k+2} ⋯ x_{m}x_{k+1} ⋯ x_{m+1}, α ≠ 0.

By (34) we have

u_{2} ≡_{k+1} βx_{k}^{2}(x_{k+1}x_{k+2} ⋯ x_{m}x_{k+1} ⋯ x_{m+1}), β ≠ 0.

Let us apply the inductive supposition to the word in the parentheses. Since x_{i}, i > k + 1 commutes with x_{k}^{2} according to the formulae (35), we get

u_{2} ≡_{k+1} γ\underline{x_{k}^{2}x_{k+1}}x_{k+2} ⋯ x_{m+1}x_{k+1} ⋯ x_{m}.

Now it remains to replace the underlined sub-word according to (34) and then to transpose the second letter x_{k} to its former position by (35).

Note that for the diminishing of u_{1}, u_{2} we did not use, and we could not use, the relation [x_{n−1}x_{n}^{2}] = 0 since deg_{n}(u_{1}) ≤ 1, deg_{n}(u_{2}) ≤ 1.

Thus B satisfies the conditions of Lemma 4.8. Since none of [u_{km}] has sub-words (28), Lemmas 7.1 and 4.8 show that the first statement is correct.

If [u_{km}] has a finite height h then the value of the polynomial [u_{km}]^{h} in U_{P} (𝔤) is a linear combination of words in hard super-letters that are less than [u_{km}]. However by Lemma 7.5 this linear combination is trivial, [u_{km}]^{h} = 0, since the defining relations are homogeneous. By Lemma 7.6 the second statement is correct for p_{11} ≠ 1.

Similarly consider the skew primitive elements. Since both the defining relations and the coproduct are homogeneous, all the homogeneous components of a skew primitive element are skew primitive itself. Therefore it remains to describe all skew primitive elements homogeneous in each x_{i}. Let T be such an element. By Lemma 4.9 we have

T = [u]^{h} + ∑ α_{i}W_{i},

where [u] is a hard super-letter, u = u_{km}, and W_{i} are super-words in less than [u] super-letters from B. By the homogeneity all W_{i} have the same constitution as [u_{km}]^{h} does. However by Lemma 7.5 there exist no such super-words. This means that the only case possible is T = [u_{km}]^{h}. Thus, by Lemma 7.6 the fifth statement is valid as well.

If p_{11} = 1 then p_{ij}p_{ji} = p_{ii} = 1 for all i, j. So we are under the conditions of Example 1, that is U_{P}^{b} (𝔤) is the universal enveloping algebra of the colour Lie algebra 𝔤^{col}. Further, [u_{km}] ∈ 𝔤^{col} and [u_{km}] are linearly independent in 𝔤^{col} since they are hard super-letters and no one of them can be a linear combination of the lesser ones. Let us complete B to a homogeneous basis B′ of 𝔤^{col}. Then by the PBW theorem for the colour Lie algebras the products b_{1}^{n_{1}} ⋯ b_{k}^{n_{k}}, b_{1} < … < b_{k} form a basis of U(𝔤^{col}) = U_{P}^{b} (𝔤). However, the monotonous restricted words in B form a basis of U_{P}^{b} (𝔤) also. Thus B′ = B and all hard super-letters have the infinite height.

In particular, we get that the second statement is valid in complete extent. Moreover, if p_{11} = 1 then p(u_{km}, u_{km}) = 1, thus for l = 0 all homogeneous skew primitive elements became exhausted by [u_{km}], while for l > 0 the powers [u_{km}]^{l^{k}} are added to them (of course, here l ≠ 2 since −1 ≠ p_{ii} = 1).

So we have proved all statements, but the third and fourth ones. These statements will follow Theorem 5.2 and Lemma 5.3 if we prove that all non-hard super-letters [[u_{km}][u_{rs}]] equal zero in U_{P} (𝔤). By the homogeneous definition, [[u_{km}][u_{rs}]] is a linear combination of super-words in lesser hard super-letters. However, by Lemma 7.5, there exist no such super-words of the same constitution. Therefore, by the homogeneity, the above linear combination equals zero. □

Theorem B_{n}. Let 𝔤 be of the type B_{n}, and p_{ii} ≠ −1, 1 ≤ i < n, p_{nn}^{[3]} ≠ 0. Denote by B the set of the super-letters given below:

\begin{matrix}[u_{km}] =^{df} [x_{k}x_{k+1} … x_{m}], & 1 ≤ k ≤ m ≤ n; \\ [w_{km}] =^{df} [x_{k}x_{k+1} … x_{n} · x_{n} … x_{m}], & 1 ≤ k < m ≤ n. \end{matrix}  (42)

The following statements are valid.

1. The values of (42) in U_{P} (𝔤) form the PBW-generators set.

2. Every super-letter [u] ∈ B has infinite height in U_{P} (𝔤).

3. The relations (21) with the following ones form a Groebner–Shirshov system that determines the crystal basis of U_{P} (𝔤).

\begin{matrix}[u_{0}] =^{df} [x_{k}x_{m}] = 0, & 1 ≤ k < m − 1 < n; \\ [u_{1}] =^{df} [u_{km}x_{k+1}] = 0, & 1 ≤ k < m ≤ n, k ≠ n − 1; \\ [u_{2}] =^{df} [u_{km}u_{k m+1}] = 0, & 1 ≤ k ≤ m < n; \\ [u_{3}] =^{df} [w_{km}x_{k+1}] = 0, & 1 ≤ k < m ≤ n, k ≠ m − 2; \\ [u_{4}] =^{df} [w_{kk+1}x_{k+2}] = 0, & 1 ≤ k < n − 1; \\ [u_{5}] =^{df} [w_{km}w_{k m−1}] = 0, & 1 ≤ k < m − 1 ≤ n − 1; \\ [u_{6}] =^{df} [u_{kn}^{2}x_{n}] = 0, & 1 ≤ k < n. \end{matrix}  (43)

4. If p_{11} ≠ 1 then the generators x_{i} and their powers x_{i}^{t}, x_{i}^{tl^{k}}, such that p_{ii} is a primitive t-th root of 1, together with the constants 1 − g, g ∈ G form a basis of 𝔤_{P} = L(U_{P} (𝔤)). Here l is the characteristic of the ground field.

5. If p_{nn} = p_{11} = 1 then the elements (42) and, for l > 0, their l^{k}-th powers, together with 1−g, g ∈ G form a basis of 𝔤_{P} . If p_{nn} = −p_{11} = −1 then [u_{kn}]^{2}, [u_{kn}]^{2l^{k}} are added to them.

Recall that in the case B_{n} the algebra U_{P}^{b} (𝔤) is defined by (33), (34), (35) where in (33) the last relation, i = n − 1, is replaced with

x_{n−1}x_{n}^{3} = p_{n−1n}p_{nn}^{[3]}x_{n}x_{n−1}x_{n}^{2} − p_{n−1n}^{2}p_{nn}p_{nn}^{[3]}x_{n}^{2}x_{n−1}x_{n} + p_{n−1n}^{3}p_{nn}^{3}x_{n}^{3}x_{n−1}. (44)

By Corollary 2.4 we get the existence conditions

p_{ii} = p_{11}, p_{ii+1}p_{i+1i} = p_{11}^{−1} = p_{nn}^{−2}, 1 ≤ i ≤ n − 1; p_{ij}p_{ji} = 1, i − j > 1. (45)

The relations (33) and (44) show that

x_{i}x_{i+1}^{2} ≡_{i+1} 0, i < n − 1; x_{n−1}x_{n}^{3} ≡_{n} 0,  (46)

while the relations (34) imply

x_{i}x_{i+1}x_{i} ≡_{i+1} αx_{i}^{2}x_{i+1}, α ≠ 0.  (47)

By means of these relations and (35), (44) we have

x_{n−2}x_{n−1}\underline{x_{n}^{2}x_{n−1}x_{n}} ≡_{n−1} 0.  (48)

Lemma 7.7. The brackets in [w_{km}] are set by the recurrence formulae:

\begin{matrix}[w_{km}] = [x_{k}[w_{k+1m}]], & if 1 ≤ k < m − 1 < n; \\ [w_{kk+1}] = [[w_{kk+2}]x_{k+1}], & if 1 ≤ k < n. \end{matrix}  (49)

Here by the definition w_{k n+1} = u_{kn}.

Proof. It is enough to use the property 6s and then 1s and 2s. □

Lemma 7.8. The nonassociative word [[w_{km}][w_{rs}]] is standard only in the following two cases: 1) s ≥ m > k + 1 = r; 2) s < m, r = k.

Proof. If [[w_{km}][w_{rs}]] is standard then w_{km} > w_{rs} and by (49) either w_{k+1} ≤ w_{rs}, or m = k + 1 and x_{k+1} ≤ w_{rs}. The inequality w_{km} > w_{rs} is correct only in two cases: k < r or k = r, m > s. We get four possibilities: 1) k < r, k < m − 1, w_{k+1m} ≤ w_{rs};

2) k < r, m = k + 1, x_{k+1} ≤ w_{rs}; 3) k = r, m > s, k < m − 1, w_{k+1m} ≤ w_{rs};

4) k = r, m > s, m = k +1, x_{k+1} ≤ w_{rs}. Only the first and third ones are consistent since in the second case x_{k+1} ≤ w_{rs} implies k + 1 > r, while in the fourth case r < s and k = r < s < m = k + 1. If now we decode w_{k+1m} ≤ w_{rs} in the first and third cases, we get the two possibilities mentioned in the lemma. □

Lemma 7.9. The nonassociative word [[u_{km}][w_{rs}]] is standard only in the following two cases: 1) k = r; 2) k = m < r.

Proof. The inequality u_{km} > w_{rs} means k ≤ r. Since [u_{km}] = [x_{k}[u_{k+1m}]], for k ≠ m we get u_{k+1m} ≤ w_{rs}, so k + 1 > r and k = r. If k = m ≠ r then x_{m} > w_{rs} and m < r. □

Lemma 7.10. The nonassociative word [[w_{km}][u_{rs}]] is standard only in the following two cases: 1) r = k + 1 < m; 2) r = k + 1 = m = s.

Proof. The inequality w_{km} > u_{rs} implies r > k. If k < m − 1 then by the first formula (49) we have w_{k+1m} ≤ u_{rs} that is equivalent to k + 1 ≥ r. Therefore r = k + 1 < m. If k = m − 1 then by the second formula (49) we get x_{k+1} ≤ u_{rs}, i.e. either k + 1 > r or k + 1 = r = s. The former case contradicts r > k while the latter one is mentioned in the lemma. □

Lemma 7.11. If [u], [v] ∈ B then one of the statements below is correct. 1) [[u][v]] is not a standard nonassociative word;

2) uv contains a sub-word of one of the types u_{0}, u_{1}, u_{2}, u_{3}, u_{4}, u_{5}, u_{6};

3) [[u][v]] ∈ B.

Proof. The proof results from Lemmas 7.4, 7.8, 7.9, 7.10. □

Lemma 7.12. If a super-word W equals one of the super-letters [u_{1}]–[u_{6}] or [u_{km}]^{h}, [w_{km}]^{h}, h ≥ 1 then its constitution does not equal the constitution of any super-word in less than W super-letters from B.

Proof. The proof is akin to Lemma 7.5 with the following tableaux:

[u_{km}], [u_{km}x_{k+1}], [u_{km}u_{k m+1}] deg_{k}(u) ≤ deg_{m+1}(u);
[w_{km}], [w_{km}x_{k+1}], [w_{km}w_{k m−1}] 2deg_{k}(u) ≤ deg_{m−1}(u);
[w_{kk+1}x_{k+2}]  deg_{k}(u) = 0;
[u_{kn}^{2}x_{n}]  deg_{k}(u) ≤ deg_{n}(u).

(50)

□

Lemma 7.13. If y = x_{i}, m − 1 ≠ i > k or y = x_{i}^{2}, m − 1 = i > k then

w_{km}y ≡_{k+1} 0.  (51)

Proof. If i < m − 1 then by means of (35) it is possible to permute y to the left beyond x_{n}^{2} and use Lemma 7.2 with m′ = n − 1. If y = x_{i}^{2}, m − 1 = i > k then by the above case, i < m − 1, we get

w_{km}y = w_{k m+1}\underline{x_{m}x_{m−1}^{2}} = \underline{w_{k m+1}x_{m−1}}(αx_{m}x_{m−1} + βx_{m−1}x_{m}) ≡_{k+1} 0, (52)

where for m = n by definition w_{k n+1} = u_{kn}, and u_{kn}x_{n−1} ≡_{n−1} 0.

If y = x_{i}, i = m > k then for m = n one may use the second equality (46). For m < n we have w_{km}y = w_{k m+1}y_{1} where y_{1} = x_{m}^{2}. Therefore for k < n − 1 we may use (52) with m + 1 in place of m. For k = n − 1 we have w_{km}x_{n} = x_{n−1}x_{n}^{3} ≡_{n} 0.

Finally, if y = x_{i}, i > m > k then by (35) we have w_{km}y = αw_{ki+1}\underline{x_{i}x_{i−1}x_{i}} · v. For i = n one may use (48), while for i < n, changing the underlined word according to (33), we may use the above considered cases: m′ − 1 = i′, where m′ = i + 1, i′ = i; and i′ < m′ − 1, where m′ = i + 1, i′ = i − 1. □

Another interesting relation appears if we multiply (44) by x_{n−1} from the left and subtract (34) with i = n − 1 multiplied from the right by x_{n}^{2} :

x_{n−1}x_{n}x_{n−1}x_{n}^{2} ≡_{n} αx_{n−1}x_{n}^{2}x_{n−1}x_{n},  (53)

in which case α = p_{n−1n}p_{nn}^{[3]} ≠ 0.

Lemma 7.14. For k < s < m ≤ n the following relation is valid.

w_{km}w_{ks} ≡_{k+1} εw_{ks}w_{km}, ε ≠ 0.  (54)

Proof. Let us use downward induction on k. For this we first transpose the second letter x_{k} of w_{km}w_{ks} as far to the left as possible by means of (35), and then change the onset x_{k}x_{k+1}x_{k} according to (47). We get

w_{km}w_{ks} ≡_{k+1} αx_{k}^{2}(w_{k+1m}w_{k+1s}), α ≠ 0.  (55)

For k + 1 < s we apply the inductive supposition to the word in the parentheses and then by (47) and (35) transpose x_{k} to its former position.

The case k+1 = s, the basis of the induction on k, we prove by downward induction on s.

Let k + 1 = s = n − 1. Then m = n. Let us show firstly that

\underline{x_{n−1}x_{n}^{2}x_{n−1}x_{n}}x_{n}x_{n−1} ≡_{n} αx_{n−1}x_{n}^{2}x_{n−1}^{2}x_{n}^{2} + βx_{n−1}x_{n}x_{n−1}^{2}x_{n}^{3}, α ≠ 0. (56)

For this in the left hand side we transpose the first letter x_{n} by means of (53) to the penultimate position, and then replace the ending x_{n}^{3}x_{n−1} by (44). We get a linear combination of three words. One of them equals the second word of (56), while two other have the following forms.

x_{n−1}x_{n}\underline{x_{n−1}x_{n}x_{n−1}}x_{n}^{2}, \underline{x_{n−1}x_{n}x_{n−1}x_{n}^{2}}x_{n−1}x_{n}.

The former word by (34) transforms into the form (56). The latter one, after the application of (53) and the replacing of x_{n−1}x_{n}x_{n−1} by (34), will have an additional term \underline{x_{n−1}x_{n}^{3}}x_{n−1}^{2}x_{n} to which it is possible to apply (46). The direct calculation of the coefficients shows that α = p_{n−1n}p_{nn} ≠ 0.

Now let us multiply (56) by x_{n−2}^{2} from the left and use (34) with i = n − 2. We get that w_{n−2 n}w_{n−2 n−1} with respect to ≡_{n−1} equals

γx_{n−2}x_{n−1}x_{n}^{2}\underline{x_{n−2}x_{n−1}^{2}}x_{n}^{2} + δx_{n−2}x_{n−1}x_{n}\underline{x_{n−2}x_{n−1}^{2}}x_{n}^{3}, γ ≠ 0.  (57)

Let us apply (46) and then (47) and (46) to the second word. We get that this word with respect to ≡_{n−1} equals zero. The first word after application of (34) takes up the form

εw_{n−2 n−1}w_{n−2 n} + ε′\underline{w_{n−2 n}x_{n−1}^{2}}x_{n−2}x_{n}^{2}, ε ≠ 0.

Thus, by Lemma 7.13, the basis of the induction on s is proved.

Let us carry out the inductive step. Let k + 1 = s < n − 1. If m > s + 1 = k + 2 then by the inductive supposition on s we may write

w_{km}w_{ks} = (w_{km}w_{kk+2})x_{k+1} ≡_{k+1} αw_{kk+2}w_{km}x_{k+1} =
βw_{kk+2}\underline{x_{k}x_{k+1}x_{k+2}x_{k+1}}w_{k+3 m}.  (58)

Taking into account (51) we may neglect the words starting with x_{k+1}^{2}, x_{k+2} while transforming the underlined part:

x_{k}\underline{x_{k+1}x_{k+2}x_{k+1}} ≡ γ\underline{x_{k}x_{k+1}^{2}}x_{k+2} ≡ δx_{k+1}x_{k}x_{k+1}x_{k+2}.  (59)

In this way (58) is transformed into (54).

If m = s + 1 = k + 2 < n then the relation (55) takes up the form

w_{km}w_{ks} ≡_{k+1} αx_{k}^{2}(w_{k+1k+2}w_{k+1k+3})x_{k+2}x_{k+1}.

Let us apply the inductive supposition with k′ = k + 1, s′ = k + 2, m′ = k + 3 to the word in the parentheses. We get

w_{km}w_{ks} ≡_{k+1} αε^{−1}x_{k}^{2}w_{k+1k+3}w_{k+1k+3}\underline{x_{k+2}^{2}x_{k+1}},

or after an evident replacement

w_{km}w_{ks} ≡_{k+1} γx_{k}^{2}w_{k+1k+3}w_{k+1k+2} · x_{k+1}x_{k+2} + δx_{k}^{2}w_{k+1k+3}^{2}x_{k+1}x_{k+2}^{2}.

In both terms we may transpose one letter x_{k} to its former position by means of (47) and (35). We get

w_{km}w_{ks} ≡_{k+1} γ′\underline{w_{kk+3}w_{kk+1}}x_{k+2} + δ′w_{kk+3}^{2}x_{k+1}x_{k+2}^{2}.  (60)

It is possible to apply (54) with m′ = k + 3, s′ = k + 1 to the first term since the case m > s + 1 is completely considered. Therefore it is enough to show that the second term equals zero with respect to ≡_{k+1} . When we transpose the third letter x_{k+1} as far to the left as possible we get the word

w_{kk+3}\underline{x_{k}x_{k+1}x_{k+2}x_{k+1}}w_{k+3 k+3}x_{k+2}^{2}.  (61)

Taking into account (51) we may neglect the words starting with x_{k+1} while transforming the underlined part:

x_{k}\underline{x_{k+1}x_{k+2}x_{k+1}} ≡ x_{k+2}\underline{x_{k}x_{k+1}^{2}} ≡ x_{k+2}x_{k+1}x_{k}x_{k+1}.  (62)

Therefore the word (61) equals w_{kk+1}\underline{w_{kk+3}x_{k+2}^{2}} with respect to ≡_{k+1} and it remains only to apply Lemma 7.13 twice. □

Lemma 7.15. The set B satisfies the conditions of Lemma 4.8.

Proof. By Lemmas 7.11 and 4.7 it is sufficient to show that in U_{P}^{b} (𝔤) all words of the form u_{0}, …, u_{6} are linear combinations of lesser ones. The words u_{0} are diminished by (35). The words u_{1}, u_{2} have been presented in this way, without using [x_{n−1}x_{n}^{2}] = 0, in the proof of the above theorem. The relation (51) shows that u_{3} ≡_{k+1} 0, u_{4} ≡_{k+1} 0. Lemma 7.14 with s = m − 1 yields the necessary representation for u_{5}.

Let us prove by downward induction on k that

u_{6} =^{df} u_{kn}^{2}x_{n} ≡_{k+1} εu_{kn}x_{n}u_{kn}, ε ≠ 0.

For k = n − 1 this equality takes up the form (53). Let k < n − 1. Let us transpose the second letter x_{k} of u_{kn}^{2}x_{n} as far to the left as possible by means of (35) and then apply (33). We get

u_{kn}^{2}x_{n} ≡_{k+1} αx_{k}^{2}(u_{k+1n}^{2}x_{n}), α ≠ 0.

We may apply the inductive supposition to the term in the parentheses and then by (33), (35) transpose one of x_{k}’s to its former position. □

Lemma 7.16. If p_{11} ≠ 1 then the values of polynomials [v]^{h}, where [v] ∈ B, v ≠ x_{i} h ≥ 1 are not skew primitive, in particular, they are non-zero.

Proof. Note that for n > 2 the sub-algebra generated by x_{2}, … x_{n} is defined by the Cartan matrix of the type B_{n−1}. This allows us to carry out the induction on n with additional supposition that the statements 1 and 2 of Theorem B_{n} are valid for lesser values of n. It is convenient formally consider the sub-algebras ⟨x_{i}⟩ as algebras of the type B_{1}. In this case for n = 1 the lemma and the statements 1 and 2 are correct in the evident way. If v starts with x_{k} ≠ x_{1} then we may directly use the inductive supposition. If v = u_{1m}, one may literally repeat the arguments of Lemma 7.6 starting at the formula (39). Let v = w_{1m}. If m > 2 then by Lemma 7.7 we have w_{1m} = [x_{1}[w_{2m}]]. This provides a possibility to repeat the same arguments of Lemma 7.6 with w in place of u.

Consider the last case v = w_{12}. By Lemma 7.7 we have

[w_{12}] = [w_{13}]x_{2} − p(w_{13}, x_{2})x_{2}[w_{13}],  (63)
[w_{13}] = x_{1}[w_{23}] − p(x_{1}, w_{23})[w_{23}]x_{1}.  (64)

Applying the coproduct first to (64) then to (63) we may find the sum Σ of all tensors w^{(1)} ⊗ w^{(2)} of Δ([w_{12}]) with deg_{1}(w^{(2)}) = 1, deg_{k}(w^{(2)}) = 0, k > 1 (in much the same way as (40)):

Σ = (εg_{1}[w_{23}] ⊗ x_{1})(x_{2} ⊗ 1) − p(w_{13}, x_{2})(x_{2} ⊗ 1)(εg_{1}[w_{23}] ⊗ x_{1}) =
εg_{1}([w_{23}]x_{2} − p(w_{13}, x_{2})p(x_{2}, x_{1})x_{2}[w_{23}]) ⊗ x_{1}.  (65)

For n > 2, taking into account first the bicharacter property of p, then the equality [x_{2}[w_{23}]] = x_{2}[w_{23}]− p(x_{2}, w_{23})[w_{23}]x_{2}, and next the following relations p_{ij}p_{ji} = 1, i − j > 1; p_{11}^{−1} = p_{12}p_{21} = p_{22}^{−1} = p_{23}p_{32}, we may write

Σ = εg_{1}(−p(w_{13}, x_{2})p_{21}[x_{2}w_{23}] + (1 − p_{11}^{−1})[w_{23}] · x_{2}) ⊗ x_{1}.  (66)

Consider the left hand side of this tensor on applying the inductive supposition. Note that x_{2}w_{23} is a standard word and [x_{2}w_{23}] equals [x_{2}[w_{23}]]. This super-letter is non-hard in U_{P} (𝔤) since x_{2}w_{23} contains the sub-word x_{2}^{2}x_{3}. Thus [x_{2}w_{23}] is a linear combination of monotonous non-decreasing super-words in lesser super-letters. Among these super-words there is no [w_{23}] · x_{2} since x_{2} > x_{2}w_{23}. On the other hand, [w_{23}] · x_{2} is a monotonous non-decreasing super-word and hence its value in U_{P} (𝔤) is a basis element. Therefore for n > 2 the left hand side W of Σ is non-zero.

For n = 2, by the definition w_{23} = x_{2}, w_{13} = x_{1}x_{2}, and the equality (65) takes up the form Σ = εg_{1}(1 − p_{12}p_{22}p_{21})x_{2}^{2} ⊗ x_{1}. Since 1 ≠ p_{11}^{−1} = p_{12}p_{21} = p_{22}^{−2}, we get (1 − p_{12}p_{22}p_{21}) = 1 − p_{22}^{−1} ≠ 0. Therefore in this case Σ ≠ 0 as well.

By [22, Corollary 10] the sub-algebra generated by x_{2}, …, x_{n} has no zero divisors. In particular W^{h} ≠ 0 and Σ^{h} ≠ 0 in any case.

It remains to note that for n > 1 the sum of all tensors w^{(1)} ⊗ w^{(2)} of Δ([w_{12}]^{h}) such that deg_{1}(w^{(2)}) = h, deg_{k}(w^{(2)}) = 0, k > 1 equals Σ^{h}, hence [w_{12}]^{h} can not be skew-primitive. □

Proof of Theorem B_{n}. Since none of u_{km}, w_{km} contains sub-words (28), Lemmas 7.15, 7.1, 4.8 imply the first statement.

If [v] ∈ B is of finite height then by Lemma 7.12 and the homogeneous version of Definition 4.4 we have [v]^{h} = 0. For p_{11} ≠ 1 this contradicts Lemma 7.16.

Along similar lines, by Lemma 4.9, every skew primitive homogeneous element has the form [v]^{h}. This, together with Lemma 7.16, proves the fourth statement and, for p_{11} ≠ 1, the second one too.

If p_{11} = 1 then by (45) we have p_{nn}^{2} = 1, p_{ii} = 1, i < n. Besides, p_{ij}p_{ji} = 1 for all i, j. This means that the skew commutator is a quantum operation. Hence all elements of B are skew primitive. In the case p_{nn} = 1 these elements span a colour Lie algebra, while in the case p_{nn} = −1 they span a colour Lie super-algebra. Now as in Theorem A_{n}, we may use the P BW -theorem for the colour Lie super-algebras.

The third statement will follow Theorem 5.2 and Lemmas 5.3, 7.11 if we prove that all super-letters (43) are zero in U_{P} (𝔤). We have already proved that these super-letters are non-hard. Therefore it remains to use the homogeneous version of Definition 4.3 and Lemma 7.12. □

Theorem C_{n}. Suppose that 𝔤 is of the type C_{n}, and p_{ii} ≠ −1, 1 ≤ i ≤ n, p_{n−1n−1}^{[3]} ≠ 0. Denote by B the set of the following super-letters:

\begin{matrix}[u_{km}] & =^{df} & [x_{k}x_{k+1} … x_{m}], & 1 ≤ k ≤ m ≤ n; \\ [v_{km}] & =^{df} & [x_{k}x_{k+1} … x_{n} · x_{n−1} … x_{m}], & 1 ≤ k < m < n; \\ [v_{k}] & =^{df} & [u_{k n−1}u_{kn}], & 1 ≤ k < n. \end{matrix}  (67)

The statements given below are valid.

1. The values of the super-letters (67) in U_{P} (𝔤) form the PBW-generators set.

2. Each of these super-letters has the infinite height in U_{P} (𝔤).

3. The following relations with (21) form a Groebner–Shirshov system that determines the crystal basis of U_{P} (𝔤).

\begin{matrix}[u_{0}] & =^{df} & [x_{k}x_{m}] = 0, & 1 ≤ k < m − 1 < n; \\ [u_{1}] & =^{df} & [u_{km}x_{k+1}] = 0, & 1 ≤ k < m ≤ n, (k, m) ≠ (n − 2, n); \\ [u_{2}] & =^{df} & [u_{km}u_{km+1}] = 0, & 1 ≤ k ≤ m < n − 1; \\ [w_{3}] & =^{df} & [v_{km}x_{k+1}] = 0, & 1 ≤ k < m < n, k ≠ m − 2; \\ [w_{4}] & =^{df} & [v_{kk+1}x_{k+2}] = 0, & 1 ≤ k < n − 1; \\ [w_{5}] & =^{df} & [v_{km}v_{km−1}] = 0, & 1 ≤ k < m − 1 ≤ n − 1; \\ [w_{6}] & =^{df} & [u_{k n−1}^{3}x_{n}] = 0, & 1 ≤ k < n. \end{matrix}  (68)

4. If p_{11} ≠ 1 then the generators x_{i} and their powers x_{i}^{t}, x_{i}^{tl^{k}}, such that p_{ii} is a primitive t-th root of 1 together with the constants 1 − g, g ∈ G form a basis of 𝔤_{P} = L(U_{P} (𝔤)). Here l is the characteristic of the ground field.

5. If p_{11} = 1 then the elements (67) and in the case of prime characteristic l theirs l^{k}-th powers, together with the constants 1 − g, g ∈ G form a basis of 𝔤_{P} .

In the case C_{n} the algebra U_{P}^{b} (𝔤) is defined by the same relations (33), (34), (35), where in (34) the last relation, i = n − 1, is replaced with

x_{n−1}^{3}x_{n} = p_{n−1n}p_{n−1n−1}^{[3]}x_{n−1}^{2}x_{n}x_{n−1} +
−p_{n−1n}^{2}p_{n−1n−1}p_{n−1n−1}^{[3]}x_{n−1}x_{n}x_{n−1}^{2} + p_{n−1n}^{3}p_{n−1n−1}^{3}x_{n}x_{n−1}^{3}.  (69)

By Corollary 2.4 we get the existence conditions

p_{ii} = p_{11}, p_{i−1i}p_{ii−1} = p_{11}^{−1}, 1 < i < n,
p_{n−1n}p_{nn−1} = p_{nn}^{−1} = p_{n−1n−1}^{−2}; p_{ij}p_{ji} = 1, i − j > 1.  (70)

Therefore the following relations are correct

x_{i}x_{i+1}^{2} ≡_{i+1} 0,  1 ≤ i < n;  (71)
x_{i}x_{i+1}x_{i} ≡_{i+1} αx_{i}^{2}x_{i+1}, 1 ≤ i < n − 1, α ≠ 0;  (72)
x_{n−1}x_{n}x_{n−1}^{2} ≡_{n} αx_{n−1}^{3}x_{n} + βx_{n−1}^{2}x_{n}x_{n−1}, α, β ≠ 0.  (73)

The left multiplication by x_{n−2} of the last relation implies

x_{n−2}x_{n−1}x_{n}x_{n−1}^{2} ≡_{n−1} 0.  (74)

Lemma 7.17. The brackets in [v_{km}], [v_{k}] are set according to the following recurrence formulae, where by the definition v_{kn} = u_{kn}.

\begin{aligned}[v_{km}] & = & [x_{k}[v_{k+1m}]], & if 1 ≤ k < m − 1 < n − 1; \\ [v_{kk+1}] & = & [[v_{kk+2}]x_{k+1}], & if 1 ≤ k < n − 1; \\ [v_{k}] & = & [[u_{k n−1}][u_{kn}]], & if 1 ≤ k < n. \end{aligned}  (75)

Proof. It is enough to use the properties 6s, 1s and 2s. □

Lemma 7.18. If [u], [v] ∈ B then one of the following statements is valid. 1) [[u][v]] is not a standard nonassociative word;

2) uv contains a sub-word of one of the types u_{0}, u_{1}, u_{2}, w_{3}, w_{4}, w_{5}, w_{6};

3) [[u][v]] ∈ B.

Proof. The first two formulae (75) coincide with (49) up to replacement of v with w provided k + 1 ≠ n > m. Obviously for m < n the inequality v_{km} > v_{rs} is equivalent to w_{km} > w_{rs}, while v_{km} > u_{rs} is equivalent to w_{km} > w_{rs}. Hence Lemmas (7.8), (7.9), (7.10) are still valid under the replacement of w with v :

[[v_{km}][v_{rs}]] is standard ⇔ s ≥ m > k + 1 = r ∨ (s < m&r = k);
[[u_{km}][v_{rs}]] is standard ⇔ k = r ∨ k = m < r;
[[v_{km}][u_{rs}]] is standard ⇔ r = k + 1 < m ∨ r = k + 1 = m = s.

(76)

Further, v_{k} > v_{r} if and only if k < r, and under this condition [[v_{k}][v_{r}]] is not standard since u_{kn} > u_{rn−1}u_{rn}.

In a similar manner v_{k} > u_{rm} is equivalent to k < r, while v_{k} > v_{rm} is equivalent to k ≤ r. Therefore none of the words [[v_{k}][u_{rm}]], [[v_{k}][v_{rm}]] is standard since u_{kn} > u_{rm} and u_{kn} > v_{rm}, respectively.

For the remaining two cases we have only two possibilities

[[u_{km}][v_{r}]] is standard ⇔ r = k ≤ m < n;
[[v_{km}][v_{r}]] is standard ⇔ r = k + 1&k < m − 1.

(77)

The treatment in turn of the eight possibilities (76), (77) proves the lemma. □

Lemma 7.19. If a super-word W equals one of the super-letters (68) or [v]^{h}, [v] ∈ B, h ≥ 1, then its constitution does not equal the constitution of any word in less then W super-letters from B.

Proof. The proof is akin to Lemma 7.5 with the following tableaux:

[u_{km}]^{h}, [u_{km}x_{k+1}], [u_{km}u_{km+1}] deg_{k}(u) ≤ deg_{m+1}(u);
[v_{km}]^{h}, [v_{km}x_{k+1}], [v_{km}v_{km−1}] 2deg_{k}(u) ≤ deg_{m−1}(u);
[v_{kk+1}x_{k+2}]  deg_{k}(u) = 0;
[v_{k}]^{h}  deg_{k}(u) ≤ deg_{n}(u);
[u_{kn−1}^{3}x_{n}]  deg_{k}(u) ≤ 2deg_{n}(u).

(78)

□

Lemma 7.20. If y = x_{i}, m − 1 ≠ i > k or y = x_{i}^{2}, m − 1 = i > k then

v_{km}y ≡_{k+1} 0.  (79)

Proof. For i < m − 1, we may transpose y by means of (35) to the left across x_{n}^{2} and then use Lemma 7.2 with m′ = n − 1.

If y = x_{i}^{2}, m − 1 = i > k then by the above case, i < m − 1, we get

v_{km}y = v_{km+1}\underline{x_{m}x_{m−1}^{2}} = \underline{v_{km+1}x_{m−1}}(αx_{m}x_{m−1} + βx_{m−1}x_{m}) ≡_{k+1} 0,  (80)

where by definition v_{kn} = u_{kn} and u_{kn}x_{n−2} ≡_{n−2} 0, while n − 2 = i > k.

If y = x_{i}, i = m > k then for m = n − 1 we may use the inequality (74), while for m < n − 1 we have v_{km}y = v_{km+1}y_{1} where y_{1} = x_{m}^{2}. Hence we may use (80) replacing m by m + 1.

If y = x_{i}, i > m > k then by (35) we get v_{km}y = αv_{ki+1}\underline{x_{i}x_{i−1}x_{i}} · w. Changing the underlined by (33), we may apply the previously considered cases: m′ − 1 = i′, where m′ = i + 1, i′ = i; and i′ < m′ − 1, where m′ = i + 1, i′ = i − 1. □

If we multiply (69) by x_{n} from the right and subtract (33) with i = n−1 multiplied from the left by x_{n−1}^{2}, then by means of p_{n−1n−1}^{−2} = p_{nn−1}p_{n−1n} = p_{nn}^{−1} we get

x_{n−1}^{2}\underline{x_{n}x_{n−1}x_{n}} ≡_{n} p_{n−1n}(p_{n−1n−1}^{[3]}x_{n−1}x_{n}x_{n−1}^{2}x_{n} − p_{n−1n−1}x_{n−1}^{2}x_{n}^{2}x_{n−1}). (81)

Let us first multiply this relation by x_{n−2}^{2} from the left and then apply (33) to the underlined sub-word. Taking into account the relation x_{n−2}^{2}x_{n−1}^{3} ≡_{n−1} 0, we get that the left hand side of the multiplied (81) equals p_{n−1n}p_{nn}(1 + p_{nn})^{−1}x_{n−2}^{2}x_{n−1}^{2}x_{n}^{2}x_{n−1} up to ≡_{n−1}, i.e. it is proportional to the second term of the right hand side. As a result the relation below with α = p_{n−1n−1}^{−1}(1 + p_{nn}) ≠ 0 is correct.

x_{n−2}^{2}x_{n−1}^{2}x_{n}^{2}x_{n−1} ≡_{n−1} αx_{n−2}^{2}x_{n−1}x_{n}x_{n−1}^{2}x_{n}.  (82)

Lemma 7.21. If k < s < m ≤ n and as above v_{kn} = u_{kn} then

v_{km}v_{ks} ≡_{k+1} εv_{ks}v_{km}, ε ≠ 0,  (83)

Proof. Let us use downward induction on k. For this we first transpose the second letter x_{k} of v_{km}v_{ks} as far to the left as possible by means of (35), and then change the onset x_{k}x_{k+1}x_{k} according to (72). We get

v_{km}v_{ks} ≡_{k+1} αx_{k}^{2}(v_{k+1m}v_{k+1s}), α ≠ 0.  (84)

For k+1 < s we may apply the inductive supposition to the word in the parentheses, and then transpose x_{k} to its former position by (72), (35).

For k + 1 = s we will use downward induction on s.

Let k + 1 = s = n − 1. In this case m = n and (84) becomes:

v_{n−2 n}v_{n−2 n−1} ≡_{n−1} βx_{n−2}^{2}(x_{n−1}\underline{x_{n}x_{n−1}x_{n}}x_{n−1}).

Let us replace the underlined part according to (33). Since x_{n−2}^{2}x_{n−1}x_{n}^{2} ≡_{n} 0, we may continue by (82):

≡_{n−1} β_{1}x_{n−2}^{2}x_{n−1}^{2}x_{n}^{2}x_{n−1} ≡_{n−1} β_{2}\underline{x_{n−2}^{2}x_{n−1}}x_{n}x_{n−1}^{2}x_{n} ≡_{n−1}
β_{3}x_{n−2}x_{n−1}\underline{x_{n−2}x_{n}}x_{n−1}^{2}x_{n} ≡_{n−1} β_{4}x_{n−2}x_{n−1}x_{n}\underline{x_{n−2}x_{n−1}^{2}}x_{n}.

With the help of (33) we get

= εv_{n−2 n−1}v_{n−2 n} + β_{5}x_{n−2}\underline{x_{n−1}x_{n}x_{n−1}^{2}}x_{n−2}x_{n}, ε ≠ 0.

By (73) and (71) we see that the second term equals zero up to ≡_{n−1} .

The inductive step on s coincides the inductive step on s in Lemma 7.14 up to replacing both the citations of Lemma 7.13 with the citations of Lemma 7.20 and w with v. □

Lemma 7.22. The set B satisfies the Lemma 4.8 conditions.

Proof. According to the Super-letter Crystallisation Lemma and Lemma 7.11 it is sufficient to show that words of the form u_{0}, u_{1}, u_{2}, w_{3}, w_{4}, w_{5}, w_{6} are linear combinations of lesser words in U_{P} (𝔤). The words u_{0} are diminished by (35). The words u_{1}, u_{2} have been diminished in Theorem A_{n} since in the case C_{n} the words u_{2} are independent of x_{n}, while u_{1} depends on x_{n} only if u_{1} = x_{n−1}x_{n}^{2}. The relation (79) shows that w_{3} ≡_{k+1} 0, w_{4} ≡_{k+1} 0. Lemma 7.21 with s = m − 1 gives the required representation for u_{5}.

Consider the words w_{6}. For k = n − 1 the relation (69) defines the required decomposition. Let k < n − 1. Since x_{1}, …, x_{n−1} generate a sub-algebra of the type A_{n−1}, the crystal decomposition of u_{kn−2}^{3}x_{n−1} has the form

u_{kn−2}^{3}x_{n−1} = ∑ αu_{m_{1}s_{1}}u_{m_{2}s_{2}} ⋯ u_{m_{t}s_{t}},  (85)

where u_{m_{1}s_{1}} ≤ u_{m_{2}s_{2}} ≤ … ≤ u_{m_{t}s_{t}}, that is m_{1} ≥ m_{2} ≥ … ≥ m_{t} and s_{i} ≥ s_{i+1} if m_{i} = m_{i+1}. In particular, if m_{1} = k then m_{2} = … = m_{t} = k and, due to the homogeneity, t = 3, s_{1} = n − 1, s_{2} = s_{3} = n − 2. Therefore

u_{kn−2}^{3}x_{n−1} ≡_{k+1} εu_{kn−1}u_{kn−2}^{2}.  (86)

Along similar lines, the following relations are valid as well

u_{kn−2}^{3}x_{n−1}^{2} ≡_{k+1} μu_{kn−1}^{2}u_{kn−2}, u_{k n−2}^{2}x_{n−1}^{3} ≡_{k+1} 0.  (87)

Now let us multiply (33) with i = n − 2 by x_{n−1} from the right, and then add to the result the same relation multiplied by p_{n−2 n−1}(1 + p_{n−1n−1})x_{n−1} from the left. We get the following relation with α = p_{n−2 n−1}^{2}p_{n−1n−1}^{[3]} ≠ 0.

x_{n−2}x_{n−1}^{3} = αx_{n−1}^{2}x_{n−2}x_{n−1} + βx_{n−1}^{3}x_{n−2},  (88)

Further, we may write

u_{k n−1}^{3} = β_{1}u_{k n−2}u_{k n−3}\underline{x_{n−1}x_{n−2}x_{n−1}}u_{k n−1}, β_{1} ≠ 0,  (89)

where for k = n − 2 the term u_{k n−3} is absent. Let us apply (33) with i = n − 2 to the underlined word. Since u_{k n−2}u_{k n−3}x_{n−1}^{2} ≡_{n−1} 0, we have got

u_{k n−1}^{3} ≡_{n−1} β_{2}u_{k n−2}^{2}u_{k n−3}\underline{x_{n−1}^{2}x_{n−2}x_{n−1}}.  (90)

Let us apply (88). Taking into account the second of (87) we get

u_{k n−1}^{3} ≡_{k+1} β_{3}u_{k n−2}^{3}x_{n−1}^{3}.  (91)

Let us multiply this relation from the right by x_{n}. By (69) we have

u_{k n−1}^{3}x_{n} ≡_{k+1} α\underline{u_{k n−2}^{3}x_{n−1}}x_{n}x_{n−1}^{2} + β\underline{u_{k n−2}^{3}x_{n−1}^{2}}x_{n}x_{n−1}.  (92)

By means of (86) and (87) we have got

u_{k n−1}^{3}x_{n} ≡_{k+1} α_{1}u_{k n−1}x_{n}u_{k n−2}^{2}x_{n−1}^{2} + β_{1}u_{k n−1}^{2}x_{n}u_{k n−2}x_{n−1},

and both of these words are less than u_{k n−1}^{3}x_{n}. □

Lemma 7.23. If p_{11} ≠ 1 then the values of [v]^{h}, where [v] ∈ B, v ≠ x_{i}, h ≥ 1 are not skew primitive. In particular they are non-zero.

Proof. Note that for n > 3 the algebra generated by x_{2}, … x_{n} is a sub-algebra of the type C_{n−1}. Therefore we may use induction on n with additional supposition that the theorem statements 1 and 2 are valid for the lesser values of n. We will formally consider the sub-algebra generated by x_{n−1}, x_{n} as an algebra of the type C_{2}, and the sub-algebra generated by x_{n} as an algebra of type C_{1}. In this case for n = 1 the present lemma and the statements 1 and 2 are valid in obvious way.

If the first letter x_{k} of v is less than x_{1} then we may use the inductive supposition directly. If v = u_{1m} then one may literally repeat arguments of Lemma 7.6 starting at (39).

If v = v_{1m} and n > 3 then we may repeat arguments of Lemma 7.16 starting at (63) up to replacing w with v. For n = 3 in these arguments the formula (66) assumes the form

Σ = εg_{1}(−p(v_{13}, x_{2})p_{21}[x_{2}^{2}x_{3}] + (1 − p_{11}^{−1})[x_{2}x_{3}] · x_{2}) ⊗ x_{1}.  (93)

Therefore the left component of the tensor Σ is a non-zero linear combination of the basis elements. For n = 2 the set B has no elements v_{1m} at all.

Consider the last case, v = v_{1} = [u_{1n−1}^{2}x_{n}]. Let S_{k} be the sum of all tensors of Δ([u_{kn}]) = ∑ u^{(1)} ⊗ u^{(2)} with deg_{n}(w^{(1)}) = 1, deg_{k}(w^{(1)}) = 0, k < n. Evidently S_{n} = x_{n}⊗1. Let us show by downward induction on k that S_{k} = (1−p_{11}^{−1})g(u_{kn−1})x_{n}⊗ [u_{kn−1}] at k < n. We have

Δ([u_{kn}]) = Δ(x_{k})Δ([u_{k+1n}]) − p(x_{k}, u_{k+1n})Δ([u_{k+1n}])Δ(x_{k}).  (94)

Consequently,

S_{k} = (g_{k} ⊗ x_{k})S_{k+1} − p(x_{k}, u_{k+1n})S_{k+1}(g_{k} ⊗ x_{k}).  (95)

This implies the required formula since by (70) at k < n − 1 we have

p(x_{k}, u_{k+1n})p(x_{n}, x_{k}) = p(x_{k}, u_{k+1n−1}),

while at k = n − 1 we have p(x_{n−1}, x_{n})p(x_{n}, x_{n−1}) = p_{11}^{−1}.

In a similar manner, consider the sum S of all tensors of Δ([u_{kn}^{2}x_{n}]) = ∑ w^{(1)} ⊗w^{(2)} with deg_{n}(w^{(1)}) = 1, deg_{i}(w^{(1)}) = 0, at i < n.

Δ([[u_{1n−1}][u_{1n}]]) = Δ([u_{1n−1}])Δ([u_{1n}]) − p(u_{1n−1}, u_{1n})Δ([u_{1n}])Δ([u_{1n−1}]). (96)

Since we now S_{1}, we may calculate S :

S = (g(u_{1n−1}) ⊗ [u_{1n−1}])S_{1} − p(u_{1n−1}, u_{1n})S_{1}(g(u_{1n−1}) ⊗ [u_{1n−1}]) =
(1 − p_{11}^{−1})g(u_{1n−1}^{2})x_{n} ⊗ (1 − p(u_{1n−1}, u_{1n})p(x_{n}, u_{1n−1}))[u_{1n−1}]^{2}.  (97)

By (70), using the bicharacter property of p, we have

1 − p(u_{1n−1}, u_{1n})p(x_{n}, u_{1n−1}) = 1 − p(u_{1n−1}, u_{1n−1})p_{n−1n}p_{nn−1} =
1 − p_{n−1n−1}p_{n−1n−1}^{−2} = 1 − p_{11}^{−1} ≠ 0.

Because of this, S ≠ 0 and the sum of all tensors w^{(1)} ⊗ w^{(2)} with deg_{n}(w^{(1)}) = h, deg_{k}(w^{(1)}) = 0, k < n of the basis decomposition of Δ([v_{1}]^{h}) equals S^{h} ≠ 0. Therefore [v_{1}]^{h} is not skew primitive. □

Proof of Theorem C_{n}. For the first statement it will suffice to prove that all superletters (67) are hard in U_{P} (𝔤). Since none of u_{km}, v_{km} contains a sub-word (28), Lemma 7.1 implies that [u_{km}], [v_{km}] are hard.

If [v_{k}] is not hard then, by the homogeneous version of Definition 4.3, its value is a polynomial in lesser hard super-letters. In line with Lemmas 7.22 and 4.8, all hard super-letters belong to B. Therefore, by Lemma 7.19, [v_{k}] = 0. Since deg_{n}(v_{k}) = 1 and deg_{n−1}(v_{k}) = 2, the equality [v_{k}] = 0 is valid in the algebra C′ which is defined by all relations of U_{P} (𝔤), but ones of degree greater than 1 in x_{n} and ones of degree greater than 2 in x_{n−1}, that is in the algebra defined by (33), (34) with i < n − 1, and (35). These relations do not reverse the order of x_{n−1} and x_{n} in monomials since none of them has both x_{n−1} and x_{n}. This implies that the sum of all monomials of [v_{k}] = [u_{kn−1}]·[u_{kn}]− p(u_{kn−1}, u_{kn})[u_{kn}]·[u_{kn−1}] in which x_{n} is prefixed to x_{n−1} equals zero in C′, that is [u_{kn}] · [u_{kn−1}] = 0. Especially, this equality is valid in U_{P} (𝔤). Since, by Theorem 4.5, the super-word [u_{kn}] · [u_{kn−1}] is a basis element, the first statement is proved.

If [v] ∈ B is of finite height then, by Lemma 7.19 and the homogeneous version of Definition 4.4, we have [v]^{h} = 0. For p_{11} ≠ 1 this contradicts Lemma 7.23. In a similar manner, according to Lemma 4.9, every skew primitive homogeneous element has the form [v]^{h}. This, together with Lemma 7.23, proves the fourth statement and, for p_{11} ≠ 1, the second one too. If p_{11} = 1 then according to (70) we have p_{ii} = p_{ij}p_{ji} = 1 at all i, j. In particular, the skew commutator is a quantum operation. Hence all elements of B are skew primitive. These elements span a colour Lie algebra. Now, as in Theorem A_{n}, we may use the coloured PBW theorem.

The third statement will follow from Theorem 5.2 and Lemmas 5.3, 7.18 provided we note that all super-letters (68) are zero in U_{P} (𝔤). We have proved already that these super-letters are non-hard. So it remains to use first the homogeneous version of Definition 4.3 and then Lemma 7.26. □

Theorem D_{n}. Let 𝔤 be of the type D_{n}, and p_{ii} ≠ −1, 1 ≤ i ≤ n. Denote by B the set of the following super-letters:

\begin{matrix}[u_{km}] & =^{df} & [x_{k}x_{k+1} … x_{m}], & 1 ≤ k ≤ m < n; \\ [e_{km}] & =^{df} & [x_{k}x_{k+1} … x_{n−2} · x_{n}x_{n−1} … x_{m}], & 1 ≤ k < m ≤ n, \\ [e_{n−1n}] & =^{df} & x_{n}. \end{matrix}  (98)

The statements given below are valid.

1. The values of (98) in U_{P} (𝔤) form the PBW-generators set.

2. Each of the super-letters (98) has infinite height in U_{P} (𝔤).

3. The relations (21) together with the following ones form a Groebner–Shirshov system that determines the crystal basis of U_{P} (𝔤).

\begin{matrix}[u_{0}] & =^{df} & [x_{k}x_{m}] = 0, & 1 ≤ k < m − 1 < n, (k, m) ≠ (n − 2, n); \\ [u_{1}] & =^{df} & [u_{km}x_{k+1}] = 0, & 1 ≤ k < m < n; \\ [u′_{1}] & =^{df} & [x_{n−2}x_{n}^{2}] = 0, \\ [u_{2}] & =^{df} & [u_{km}u_{k m+1}] = 0, & 1 ≤ k ≤ m < n − 1; \\ [v_{3}] & =^{df} & [e_{km}x_{k+1}] = 0, & 1 ≤ k < m ≤ n, n − 1 ≠ k ≠ m − 2; \\ [v_{4}] & =^{df} & [e_{kk+1}x_{k+2}] = 0, & 1 ≤ k < n − 2; \\ [v′_{4}] & =^{df} & [e_{n−3 n−2}x_{n}] = 0, \\ [v_{5}] & =^{df} & [e_{km}e_{k m−1}] = 0, & 1 ≤ k < m − 1 ≤ n − 1; \\ [v_{6}] & =^{df} & [u_{km}e_{kn}] = 0, & 1 ≤ k ≤ m < n, n − 2 ≤ m. \end{matrix} (99)

4. If p_{11} ≠ 1, then the generators x_{i}, their powers x_{i}^{t}, x_{i}^{tl^{k}}, such that p_{ii} is a primitive t-th root of 1, together with the constants 1−g, g ∈ G form a basis of 𝔤_{P} = L(U_{P} (𝔤)). Here l = char(k).

5. If p_{11} = 1, then the elements of B and, for l > 0, their l^{k}-th powers together with the constants 1 − g, g ∈ G form a basis of 𝔤_{P} .

In the case D_{n} the algebra U_{P}^{b} (𝔤) can be defined by the condition that the subalgebras U_{n−1} and U_{n} generated, respectively, by x_{1}, …, x_{n−1} and x_{1}, …, x_{n−2}, x′_{n−1} = x_{n} are quantum universal enveloping algebras of the type A_{n−1}, and by the only additional relation

[x_{n−1}x_{n}] = 0.  (100)

The existence conditions take up the form

p_{ii} = p_{nn} = p_{11}, p_{i+1i}p_{ii+1} = p_{n−2n}p_{nn−2} = p_{11}^{−1}, if 1 ≤ i < n,
p_{n−1n}p_{nn−1} = p_{ij}p_{ji} = 1, if i − j > 1&(i, j) ≠ (n, n − 2).  (101)

Lemma 7.24. The brackets in (98) are set up by the recurrence formulae

\begin{aligned}[e_{km}] & = & [x_{k}[e_{k+1m}]], & if 1 ≤ k < m − 1 < n, k ≠ n − 1; \\ [e_{kk+1}] & = & [[e_{kk+2}]x_{k+1}], & if 1 ≤ k < n − 1. \end{aligned}  (102)

Proof. It is enough to use the properties 6s, 1s, and 2s. □

Lemma 7.25. If [u], [v] ∈ B, then one of the statements below is correct. 1) [[u][v]] is not a standard nonassociative word;

2) uv contains a sub-word of one of the types u_{0}, u_{1}, u′_{1}u_{2}, v_{3}, v_{4}, v′_{4}, v_{5}, v_{6};

3) [[u][v]] ∈ B.

Proof. The formulae (102) coincides with (49) at k ≠ n − 1 up to replacing e by w. The inequality e_{km} > e_{rs} is set up by the same conditions, k < r ∨ (k = r&m < s), as the inequality w_{km} > w_{rs} does. Likewise u_{km} > e_{rs} is set up by the same condition, k ≤ r, as u_{km} > w_{rs} does. Therefore Lemmas 7.8, 7.9, 7.10 remain valid with e in place of w :

[[e_{km}][e_{rs}]] is standard ⇔ s ≥ m > k + 1 = r ∨ (s < m&r = k);
[[u_{km}][e_{rs}]] is standard ⇔ k = r ∨ k = m < r;
[[e_{km}][u_{rs}]] is standard ⇔ r = k + 1 < m ∨ r = k + 1 = m = s.

(103)

By looking over all of these possibilities we get the lemma statement. □

Lemma 7.26. If a super-word W equals one of the super-letters (99) or [v]^{h}, [v] ∈ B, h ≥ 1 then its constitution does not equal the constitution of any super-word in less than W super-letters from B.

Proof. The proof is similar to the one of Lemma 7.5 with the tableaux

[u_{km}]^{h}, [u_{km}x_{k+1}], [u_{km}u_{k m+1}]  deg_{k}(u) ≤ deg_{m+1}(u);
[e_{km}]^{h}, [e_{km}x_{k+1}], [e_{km}e_{k m−1}], m < n 2deg_{k}(u) ≤ deg_{m−1}(u);
[e_{kn}]^{h}, [e_{kn}x_{k+1}], [e_{kn}e_{k n−1}]  deg_{k}(u) ≤ deg_{m−1}(u);
[e_{kk+1}x_{k+2}] deg_{k}(u) = 0;
[e_{n−3n−2}x_{n}] deg_{n−3}(u) = 0;
[u_{k n−2}e_{kn}] deg_{k}(u) ≤ deg_{n−1}(u) + deg_{n}(u);
[u_{k n−1}e_{kn}] deg_{k}(u) ≤ deg_{n}(u).

(104)

Lemma 7.27. If y = x_{i}, m − 1 ≠ i > k or y = x_{i}^{2}, m − 1 = i > k then

e_{km}y ≡_{k+1} 0.  (105)

Proof. If i < m − 1, m ≠ n, or m = n, i < n − 2, then with the help of (35) and (100) it is possible to permute y to the left beyond x_{n} and then to use Lemma 7.2 for U_{n−1}.

If m = n, i = n − 2 then we may use Lemma 7.2 for U_{n}.

If y = x_{i}^{2}, m − 1 = i > k then for m < n by the above case we get

e_{km}y = e_{k m+1}x_{m}x_{m−1}^{2} = \underline{e_{k m+1}x_{m−1}}(αx_{m}x_{m−1} + βx_{m−1}x_{m}) ≡_{k+1} 0. (106)

For m = n we have e_{kn}x_{n−1}^{2} = α\underline{u_{k n−2}x_{n−1}^{2}}x_{n} ≡_{n−1} 0 since the underlined part belongs to U_{n−1}.

If y = x_{i}, i = m > k then for m = n we may use Lemma 7.2 applied to U_{n}; for m = n − 1 we may use the same lemma applied to U_{n−1} provided that beforehand we permute x_{n} with y by (100); for m < n − 1 we may first rewrite e_{km}y = e_{k m+1}y_{1}, where y_{1} = x_{m}^{2}, and then use (106) with m + 1 in place of m.

If y = x_{i}, i > m > k then for i < n we have e_{km}y = αe_{ki+1}\underline{x_{i}x_{i−1}x_{i}} · v. Replacing the underlined word by (33) in U_{n−1}, we may use the previously considered cases: m′ − 1 = i′, where m′ = i + 1, i′ = i; and i′ < m′ − 1, where m′ = i + 1, i′ = i − 1. For i = n, and m = n − 1 we have e_{k n−1}x_{n} = α\underline{u_{k n−2}x_{n}^{2}}x_{n−1} and one may apply Lemma 7.2 to U_{n}. Finally, for i = n and m < n − 1 we get

e_{km}x_{n} = β_{1}u_{k n−2}x_{n}x_{n−1}x_{n−2}x_{n} · v = β_{2}u_{k n−2}x_{n−1}\underline{x_{n}x_{n−1}x_{n}} · v =
β_{3}\underline{u_{k n−2}x_{n−1}x_{n−2}}x_{n}^{2} · v + β_{4}u_{k n−2}\underline{x_{n−1}x_{n}^{2}}x_{n−2} · v.

One may apply first Lemma 7.2 for U_{n−1} to the underlined sub-word of the first term, and then, after (100), Lemma 7.2 for U_{n} to the second term. □

Lemma 7.28. If k < s < m ≤ n then e_{km}e_{ks} ≡_{k+1} εe_{ks}e_{km}, ε ≠ 0.

Proof. Let us carry out downward induction on k. The largest value of k equals n − 2. In this case s = n − 1, m = n and we have

\underline{x_{n−2}x_{n} · x_{n−2}}x_{n}x_{n−1} ≡_{n} x_{n−2}^{2}\underline{x_{n}^{2}x_{n−1}} = α\underline{x_{n−2}^{2}x_{n−1}}x_{n}^{2} ≡_{n−1}
βx_{n−2}x_{n−1}\underline{x_{n−2}x_{n}^{2}} ≡_{n} εx_{n−1}x_{n} · x_{n−1}x_{n−2}x_{n}.  (107)

Let us first transpose the second letter x_{k} of e_{km}e_{ks} as far to the left as possible by (35), and then replace the onset x_{k}x_{k+1}x_{k} by (36). We get

e_{km}e_{ks} ≡_{k+1} αx_{k}^{2}(e_{k+1m}e_{k+1s}), α ≠ 0.  (108)

For k + 1 < s it suffices to apply the inductive supposition to the word in the parentheses and then by (36) and (35) to put x_{k} to the proper place.

For k + 1 = s one may use downward induction on s. The basis of this induction, s = n − 1, has been proved, see (107). For k < n − 3 the inductive step on s coincides with the one of Lemma 7.14 with e in place of w since in this case the active variables x_{k}, x_{k+1} q-commute with x_{n}. If k = n − 3 then in consideration of Lemma 7.14 the variable x_{k+1} = x_{n−2} is transposed across x_{n} twice: in (58) and in the second word of (60).

In (58) with k = n − 3 we have s = n − 2, m = n; and (58) becomes

e_{n−3n}e_{n−3n−2} ≡_{n−2} βe_{n−3n−1}\underline{x_{n−3}x_{n−2}x_{n}x_{n−2}}.  (109)

In view of Lemma 7.27, we may transform the underlined part in U_{n} neglecting the words starting with x_{n−2}^{2} and x_{n} in much the same way as in (59), with x_{n} in place of x_{k+1}. So (109) reduces to the required form.

The second word of (60) with k = n − 3 assumes the form e_{n−3n}^{2}x_{n−2}x_{n−1}^{2} = e_{n−3n}\underline{x_{n−3}x_{n−2}x_{n}x_{n−2}}x_{n−1}^{2}. By Lemma 7.2 applied to U_{n}, the underlined word is a linear combination of words starting with x_{n−2} and x_{n}. However, by Lemma 7.27 both e_{n−3n}x_{n−2} and e_{n−3n}x_{n} equal zero up to ≡_{n−2} . □

Lemma 7.29. The set B satisfies the conditions of Lemma 4.8.

Proof. By Lemmas 7.25 and 4.7 one need show only that in U_{P}^{b} (𝔤) the words (99) are linear combinations of lesser ones. The words v_{6} with m = n − 2, and u_{0}, u_{1}, u′_{1}, u_{2} have the required decomposition since they belong either to U_{n−1} or to U_{n}. Lemma 7.27 shows that v_{3} ≡_{k+1} 0, v_{4} ≡_{k+1} 0, v′_{4} ≡_{k+1} 0. Lemma 7.28 with s = m − 1 yields the required representation for v_{5}. Consider v_{6} with m = n − 1. Let us prove by downward induction on k that

u_{k n−1}e_{kn} ≡_{k+1} εe_{kn}u_{k n−1}, ε ≠ 0.

For k = n−1 this equality assumes the form (100). Let k < n−1. Let us transpose the second letter x_{k} of u_{k n−1}e_{kn} as far to the left as possible in U_{n−1}. After an application of (33) we get

u_{k n−1}e_{kn} ≡_{k+1} αx_{k}^{2}(u_{k+1n−1}e_{k+1n}), α ≠ 0.

It suffices to apply the inductive supposition to the term in the parentheses, and then by (33) and (35) for U_{n} to move x_{k} to the proper place. □

Lemma 7.30. If p_{11} ≠ 1 then the values of [v]^{h}, where [v] ∈ B, v ≠ x_{i}, h ≥ 1 are not skew primitive, in particular they are non-zero.

Proof. One need consider only super-letters that belong neither to U_{n−1} nor to U_{n}. That is [e_{km}] with m < n. We use induction on n.

For n = 3 the algebra of the type D_{3} reduces to the algebra of the type A_{3} with a new ordering of variables x_{2} > x_{1} > x_{3}. Therefore we may use Theorem A_{n}, after the decomposition below of e_{12} in the PBW-basis:

[[x_{1}x_{3}]x_{2}] = −p_{12}p_{32}[x_{2}[x_{1}x_{3}]] + β[x_{1}x_{3}] · x_{2}.

Let n > 3. If k > 1 then the inductive supposition works. For k = 1, m > 2 we have e_{1m} = [x_{1}[e_{2m}]], and one may repeat the arguments of Lemma 7.6 with e in place of u starting at (39). If m = 2 then we may repeat the arguments of Lemma 7.16 with e on place of w starting at (63). □

Proof of Theorem D_{n}. For the first statement it will suffice to prove that all superletters (98) are hard in U_{P}^{b} (𝔤).

Since none of u_{km} contains sub-words (28), [u_{km}] are hard.

Suppose [e_{km}] is non-hard. By Lemmas 7.29 and 4.8 all hard super-letters belong to B. Thus, by Lemma 7.26, we get [e_{km}] = 0. Since deg_{n}(e_{km}) =deg_{n−1}(e_{km}) = 1, the equality [e_{km}] = 0 is also valid in the algebra D′ defined by the same relations as U_{P}^{b} (𝔤) is, but [x_{n−2}x_{n}^{2}] = 0 and [x_{n−2}x_{n−1}^{2}] = 0. Let us equate to zero all monomials in all the defining relations of D′, but [x_{n−1}x_{n}] = 0. Consider the algebra R′ defined by (100) and by the resulting system of monomial relations. It is easy to verify that the mentioned relations system Σ of R′ is closed under the compositions. Since e_{km} contains none of leading words of Σ, the super-letter [e_{km}] is non-zero in R′, and so in D′ too. This contradiction proves the first statement.

If [v]^{h}, [v] ∈ B is of finite height then by Lemma 7.26 and the homogeneous version of Definition 4.4 we have [v]^{h} = 0. For p_{11} ≠ 1 this contradicts Lemma 7.30. In a similar manner, by Lemma 4.9, every skew primitive homogeneous element has the form [v]^{h}. This, together with Lemma 7.30, proves both the fourth statement and the second one with p_{11} ≠ 1.

If p_{11} = 1 then by (101) we have p_{ii} = p_{ij}p_{ji} = 1 for all i, j. This means that the skew commutator itself is a quantum operation. Hence all elements of B are skewprimitive. These elements span a colour Lie super-algebra. Now, as in Theorem A_{n}, one may use the PBW theorem for colour Lie super-algebras.

For the third statement it will suffice to show that all super-letters (99) are zero in U_{P} (𝔤). We have proved already that they are non-hard. Therefore it remains to use the homogeneous version of Definition 4.3 and Lemma 7.26. □

8. Conclusion

We see that in all Theorems A_{n}–D_{n} the lists of hard super-letters are independent of the parameters p_{ij}. This fact signifies that the Lalonde–Ram basis of the ground Lie algebra (see, [26, Figure 1]) with the skew commutator in place of the Lie operation coincides with the set of all hard super-letters. It is very interesting to clarify how general this statement is. On the one hand, this does not hold without exception for all quantum enveloping algebras since in Theorems A_{n}–D_{n} a restriction does exist. If p_{ii} = −1, 1 ≤ i < n, n > 2 then it is easy to see by means of the Diamond Lemma that the sets of hard super-letters are infinite. On the other hand, this is not a specific property of Lie algebras defined by the Serre relations. By the Shirshov theorem [36] any relation can be reduced to a linear combination of standard nonassociative words.

Corollary 8.1. If 𝔤 is defined by the only relation f = 0, where f is a linear combination of standard nonassociative words, then the set of all hard in U_{P} (𝔤) superletters coincides with the Hall–Shirshov basis of 𝔤 with the skew commutator in place of the Lie operation.

Proof. The only relation f^{∗} = 0 forms a Groebner–Shirshov system since, according to 1s, none of onsets of its leading word, say w, coincides with a proper terminal of w. Consequently, a super-letter [u] is hard if and only if u does not contain w as a sub-word. We see that this criteria is independent of p_{ij} as well. □

Furthermore, the third statement of Theorem A_{n} shows that U_{P}^{b} (𝔤) can be defined by the following relations in the PBW-generators X_{u} = [u].

\begin{matrix}[X_{u}, X_{v}] = 0, & u > v, & [[u][v]] ∉ B \\ [X_{u}, X_{v}] = X_{uv}, & & [[u][v]] ∈ B. \end{matrix}  (110)

This is an argument in favour of considering the super-letters PBW-generators k[G]- module as a quantum analogue of a Lie algebra. However in the cases B_{n}, C_{n}, D_{n} the defining relations became more complicated. For example,

\begin{matrix}B_{n} : [[u_{k n−1}][w_{kn}]] = α[u_{kn}]^{2}, & α ≠ 0 if p_{nn} ≠ 1; \\ C_{n} : [[u_{k n−2}][v_{k n−1}]] = α[v_{k}] + β[u_{kn}] · [u_{k n−1}], & β ≠ 0 if p_{11} ≠ 1; \\ D_{n} : [[u_{k n−2}][e_{k n−1}]] = α[e_{kn}] · [u_{k n−1}], & α ≠ 0 if p_{11} ≠ ±1. \end{matrix} (111)

It is far more interesting that for p_{11} ≠ 1 the algebra 𝔤_{P} turns out to be very simple in structure. Only unary quantum operations can be non-zero. Other ones may be defined, but due to the homogeneity their values equal zero. In particular, if p_{11}^{[t]} ≠ 0 then without exception all quantum operations have zero values. This provides reason enough to consider U_{P} (𝔤) = U(𝔤_{P}) as an algebra of ‘commutative’ quantum polynomials. Certainly it is very interesting to elucidate to what extent this statement is still retained for the quantum universal enveloping algebras of homogeneous components of other Kac–Moody algebras defined by the Gabber–Kac relations (11). Also it is interesting to investigate the structure of other ‘commutative’ quantum polynomial algebras. For example, one may note that if a semi-group generated by p_{ij}p_{ji} does not contain 1, then G⟨x_{1}, …, x_{n}⟩ itself is a ‘commutative’ quantum polynomial algebra merely since in this case there exists no non-zero quantum operation at all. In another extreme case when p_{ij}p_{ji} = 1 for all i, j, the ‘commutative’ quantum variables commute by x_{i}x_{j} = p_{ij}x_{j}x_{i}.

In a similar manner, the Drinfeld–Jimbo enveloping algebra can be considered as a ‘quantum’ Weyl algebra of (skew) differential operators (see Sec. 6). The resulting ‘quantum’ Weyl algebra is simple in the following sense.

Corollary 8.2. Let 𝔤 be a simple finite dimensional Lie algebra of the infinite series. If q^{[m]} ≠ 0, m ≥ 2 then every non-zero Hopf ideal I of the Drinfeld–Jimbo enveloping algebra contains all generators x_{i}, x_{i}^{−}.

Proof. By the Heyneman–Radford theorem, the ideal I has a non-zero skew primitive element, say a. According to Lemma 6.2 and Theorems A_{n}–C_{n}, the element a is either a constant, α(1−g), or proportional to one of the elements x_{i}, x_{i}^{−}. In the former case I contains all x_{i} with χ^{i}(g) ≠ 1 since x_{i}a − χ^{i}(g)a_{i}x_{i} = α(1 − χ^{i}(g))x_{i}. Here the equality χ^{i}(g) = 1 can not be valid for all i since χ^{i}(g_{j}) = q^{−d_{i}a_{ij}} (see, Example 4 of Section 2) and the columns of the Cartan matrix are linearly independent. In the latter case (and now in the former one as well) we get [x_{i}, x_{i}^{−}] = ε_{i}(1 − g_{i}^{2}) ∈ I, i.e. as above I contains all elements y = x_{i}^{±} with 1 ≠ χ^{y}(g_{i}^{2}) = q^{±2d_{j}a_{ij}}. Since the Coxeter graph is connected, I contains all x_{i}, x_{i}^{−}. □

Acknowledgements. The author is grateful to Dr. J. A. Montaraz, the director of the FES-C UNAM, Dra. S. Rodríguez-Romo, and A.V. Lara Sagahon for providing facilities for the research and also to Dr. L.A. Bokut’ and Dr. R. Bautista for helpful comments on the subject matter.

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V.K. Kharchenko, UNAM, Calle de 1 mayo s/n, 54769, Edo. de México, México., and Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

  E-mail address: vlad servidor.unam.mx