THE QUANTUM GROUPS \overline{U}_{q}(sl_{2}) AT THE ROOTS OF UNITY, SELF-DUALITY AND ASCENT

LARS KADISON

Abstract. We compute that \overline{U}_{q}(sl_{2}) is self-dual if q is a fourth root of unity. We prove the converse statement by using Radford’s formula for the fourth power of the antipode. We end with a discussion of the β-Frobenius extension structure of \overline{U}_{i}(sl_{2}) over the Sweedler 4-dimensional Hopf subalgebra, and remark that the triangular structure at λ = 0 ascends for certain general reasons.

1. Introduction

This paper continues some computations performed in [6]. We will prove that the finite dimensional quantum enveloping algebra \overline{U}_{q}(sl_{2}), where q is the fourth root of unity i = \sqrt{−1}, is self-dual as a Hopf algebra. More generally, we establish that

Theorem 1.1. The Hopf algebra \overline{U}_{q}(sl_{2}) at the primitive n’th root of unity q is self-dual if and only if n = 4.

We prove this in Sections 2 and 3 below. In Section 2, we make some general observations on the integrals of \overline{U}_{q}(sl_{2}), unimodularity, the Nakayama automorphism S^{2} where S is the antipode, Radford’s formula for S^{4} and counimodularity. In Section 3, we continue the computations in [6] by computing the Hopf algebra structure of the dual \overline{U}_{i}^{∗} of \overline{U}_{i} := \overline{U}_{i}(sl_{2}) and showing it is isomorphic as Hopf algebras to \overline{U}_{i}.

We conclude with Section 4 discussing our results in light of the structure given in [2] for a β-Frobenius algebra extension formed by a Hopf subalgebra pair, such as the e^{3}-dimensional algebra \overline{U}_{q} := \overline{U}_{q}(sl_{2}) over Taft’s e^{2}-dimensional Hopf algebras [19]. A property P of unital algebras is said to ascend along a special type of algebra extension if a subalgebra with P implies the overalgebra has P in such an extension: we have a similar definition for descent. Although the Taft algebras are self-dual, non-symmetric and non-unimodular, we see then that none of these properties automatically ascend along β-Frobenius extensions.

1991 Mathematics Subject Classification. 16W35,17B37,81R50.

We continue our computations for the fourth root of unity and provide a β-Frobenius structure for the 8-dimensional Hopf algebra \overline{U}_{i} extending Sweedler’s 4-dimensional Hopf algebra. Indepedent of the rest of the section, we observe finally that properties of Hopf algebras like quasi-cocommutativity or being braided or triangular ascend for certain “well-placed” Hopf subalgebras like Sweedler’s algebra within \overline{U}_{i}.

2. General Observations on U_{q} and its quotient

All algebras in this paper are unital and over the complex numbers C. Let q be a complex number different from 0, 1, −1. The algebra U_{q} := U_{q}(sl_{2}) is the free polynomial algebra in noncommuting variables K, E, F, where K has inverse K^{−1}, and subject to the relations,

(1)  KE = q^{2}EK
(2)  KF = q^{−2}F K
(3)  EF − F E = \frac{K − K^{−1}}{q − q^{−1}}

[7]. The additional structure of a Hopf algebra on U_{q} is given by the following values of its comultiplication Δ : U_{q} → U_{q} ⊗ U_{q}, counit ϵ : U_{q} → C and antipode S : U_{q} → U_{q} [16]:

(4)  Δ(K) = K ⊗ K
(5)  Δ(E) = E ⊗ 1 + K ⊗ E
(6)  Δ(F) = F ⊗ K^{−1} + 1 ⊗ F ,
(7)  ϵ(K) = 1, ϵ(E) = 0, ϵ(F) = 0,
(8)  S(K) = K^{−1}, S(E) = −K^{−1}E, S(F) = −F K

Recall that Δ and ϵ are algebra homomorphisms, while the antipode S is an algebra anti-homomorphism: S(ab) = S(b)S(a). The antipode S above has an inverse \overline{S} given by

(9)  \overline{S}(K) = K^{−1}, \overline{S}(E) = −EK^{−1}, \overline{S}(F) = −KF

If q is a primitive root of unity of order d, one defines e to be d if odd, or \frac{d}{2} if even. Then K^{e}, E^{e} and F^{e} belong to the center of U_{q} [5]. One defines the finite dimensional Hopf algebra \overline{U}_{q} as the quotient of U_{q} by the ideal I generated by E^{e}, F^{e} and K^{e} − 1: I is a Hopf ideal [5]. The algebra \overline{U}_{q} (called the Frobenius-Lusztig kernel of U_{q}(sl_{2})) is e^{3}-dimensional with basis {E^{ℓ}F^{j}K^{k}|0 ≤ ℓ, j, k ≤ e − 1} [5].

We make a series of small observations about the algebras U_{q} and \overline{U}_{q} needed later.

QUANTUM GROUPS AT ROOTS OF UNITY, SELF-DUALITY AND ASCENT 3

The Hopf algebra automorphism S^{2} is important here and we record below the well-known fact that S^{2} is an inner automorphism:

Lemma 2.1. S^{2}(a) = K^{−1}aK for all a ∈ U_{q} or \overline{U}_{q}.

Next we note a fact about the Cartan automorphism α on H = U_{q} or \overline{U}_{q}, given by α(K) = K^{−1}, α(E) = F, α(F) = E, which helps account for the differing Hopf structures on U_{q} in the literature (cf. [5, 6]).

Lemma 2.2. The Cartan automorphism α on H = U_{q} or \overline{U}_{q} induces a Hopf algebra isomorphism of H with its co-opposite algebra H^{cop}.

Proof. We note from Equations 4 - 6 that (α ⊗ α)Δ = Δ^{op}α where Δ^{op} := τ ∘ Δ, and τ is the twist on H ⊗ H given by τ(a ⊗ b) := b ⊗ a. Moreover, ϵ ∘ α = ϵ. It follows that α : H ⟶^{≅} H^{cop} as Hopf algebras and that \overline{S} ∘ α = α ∘ S [15, Theorem 1.5d], since \overline{S} is the antipode of H^{cop}.

Since S induces a Hopf algebra isomorphism H ≅ H^{op, cop}, it follows from the lemma that H is Hopf isomorphic to its opposite algebra H^{op}.

Recall that a finite dimensional Hopf algebra H has a right (respectively left) integral t ∈ H with defining property, ta = ϵ(a)t (respectively at = ϵ(a)t) for all a ∈ H [9]. Since the space of right integrals is one-dimensional [9], there is an augmentation m ∈ H^{∗} called the right modular function or right distinguished group-like element in the dual Hopf algebra H^{∗} (for dual Hopf algebra: [5]), such that at = m(a)t for all a ∈ H. The left modular function is defined correspondingly and equals m^{−1} := m ∘ S. If the space of left integrals coincides with the right integrals, H is unimodular, i.e. m = ϵ. The integral for \overline{U}_{q} below resembles somewhat the left and right integrals for Taft’s Hopf algebras [15].

Proposition 2.3. The Hopf algebras \overline{U}_{q} are unimodular.

Proof. From relations 1 and 2, we see that for any integer powers ℓ, j we have K^{ℓ}E^{j}F^{j} = E^{j}F^{j}K^{ℓ}. It follows that

t := E^{e−1}F^{e−1}(1 + K + ⋯ + K^{e−1}) = (1 + K + ⋯ + K^{e−1})E^{e−1}F^{e−1}. (10)

We compute that Kt = t = ϵ(K)t and similarly Et = 0 = F t from E^{e} = 0 = F^{e}. Thus t is a left integral, and similarly we see that t is a right integral.

A right integral t provides a nice pair of bases for a finite dimensional Hopf algebra H as follows: compute Δ(t) = ∑_{(t)} t_{(1)}⊗t_{(2)} in Sweedler’s notation, then \{\overline{S}(t_{(2)})\}, \{t_{(1)}\} are dual bases for H, with definining property that there is a linear functional ϕ ∈ H^{∗} such that

(11)  a = ∑_{(t)} ϕ(a\overline{S}(t_{(2)}))t_{(1)}
(12)  a = ∑_{(t)} \overline{S}(t_{(2)})ϕ(t_{(1)}a)

for all a ∈ H. We note that ⟨a, b⟩ := ϕ(ab) is a nondegenerate associative bilinear form on H. If this bilinear form is symmetric for some ϕ and associated dual bases, H is a symmetric algebra with many nice properties from the point of view of representation theory, such as a symmetric Cartan matrix [20]. Equivalently, the algebra H is symmetric if the (Nakayama) algebra automorphism η of H defined by ϕ(xa) = ϕ(η(a)x) for all a, x ∈ H is an inner automorphism [8].

In terms of the dual bases and the associated ϕ ∈ H^{∗} satisfying ϕ(t) = 1, the Nakayama automorphism for H is given by

η(a) = S^{2}(a) ↼ m^{−1} (13)

[2], where H^{∗} acts on the bimodule H from left and right by f ⇀ a := ∑_{(a)} a_{(1)}f(a_{(2)}) and a ↼ f := ∑_{(a)} f(a_{(1)})a_{(2)} for all f ∈ H^{∗}, a ∈ H. In case H = \overline{U}_{q}, m = ϵ by Proposition 2.3, whence η equals the Hopf algebra automorphism S^{2}. Putting this together with Lemma 2.1, we arrive at the next observation:

Proposition 2.4. The \overline{U}_{q} are symmetric algebras.

For d odd, the last two propositions also follow from putting together [5, Theorem IX.6.6] with Equation 13 and theorems in [1, 13].

We will need a few more general remarks for a finite dimensional Hopf algebra H. H is said to be counimodular if its dual Hopf algebra H^{∗} is unimodular. Of course, a self-dual H is unimodular if and only if it is counimodular. If H is not counimodular, there is a right distinguished group-like element b ∈ H, different from 1, which is the modular function for H^{∗} in H^{∗∗} ≅ H, as defined above. Radford [12] has the following formula for the fourth power of the antipode: for all a ∈ H

S^{4}(a) = b(m^{−1} ⇀ a ↼ m)b^{−1}. (14)

Thus S^{4} = Id_{H} if H is unimodular and counimodular.

3. Computations for U_{q}(sl_{2})^{∗} at the fourth root of unity

In this section we compute the structure of the 8-dimensional algebra \overline{U}_{i}^{∗} := Hom_{C}(\overline{U}_{i}, C) and show that it is Hopf algebra isomorphic to \overline{U}_{i}.

With q = i, the relations 1– 8 and those of the Hopf ideal I become the following defining relations of the Hopf algebra \overline{U}_{i}:

(15)EK = −KE  F K = −KF  EF = F E
(16)  K^{2} = 1  E^{2} = 0  F^{2} = 0
Δ(K) = K ⊗ K Δ(E) = E ⊗ 1 + K ⊗ E Δ(F) = F ⊗ K + 1 ⊗ F
(17) ϵ(K) = 1  ϵ(E) = 0  ϵ(F) = 0
(18) S(K) = K  S(E) = −KE  S(F) = −F K

We base our computations on those in [6]^{1} for the Hopf algebra \overline{U}_{i}, relative to the basis,

\begin{matrix}e_{0} := \frac{1+K}{2} & e_{1} := \frac{1−K}{2} \\ E_{0} := e_{0}E & E_{1} := e_{1}E \\ F_{0} := e_{0}F & F_{1} := e_{1}F \\ C_{0} := e_{0}EF & C_{1} := e_{1}EF. \end{matrix}

We note the identities e_{j}^{2} = e_{j}, e_{0}e_{1} = 0, F_{j}^{2} = E_{j}^{2} = C_{j}^{2} = 0, e_{0}E = Ee_{1}, etc. (j = 0, 1).

We denote the dual (ordered) basis in \overline{U}_{i}^{∗} of the basis above, where ê_{0}(e_{0}) = 1, e_{0}(x) = 0 for x another basis element, etc., by

{ê_{0}, ê_{1}, Ê_{0}, Ê_{1}, F̂_{0}, F̂_{1}, Ĉ_{0}, Ĉ_{1}}.

We compute the multiplication table using the formula, (χρ)(a) = ∑_{(a)} χ(a_{(1)})ρ(a_{(2)}) for multiplication in \overline{U}_{i}^{∗} of χ, ρ ∈ \overline{U}_{i} and the values of Δ on the basis for \overline{U}_{i} just above [6].

ê_{0}  ê_{1}  Ê_{0}  Ê_{1} F̂_{0} F̂_{1} Ĉ_{0}  Ĉ_{1}
ê_{0}  ê_{0}  ê_{1}  Ê_{0}  Ê_{1} F̂_{0} F̂_{1} Ĉ_{0}  Ĉ_{1}
ê_{1}  ê_{1}  ê_{0} −Ê_{1} −Ê_{0} F̂_{1} F̂_{0} −Ĉ_{1} −Ĉ_{0}
Ê_{0}  Ê_{0} Ê_{1} 0  0 Ĉ_{0} Ĉ_{1} 0  0
Ê_{1}  Ê_{1} Ê_{0} 0  0 Ĉ_{1} Ĉ_{0} 0  0
F̂_{0}  F̂_{0} −F̂_{1} −Ĉ_{0} Ĉ_{1} 0 0 0  0
F̂_{1}  F̂_{1} −F̂_{0} Ĉ_{1} −Ĉ_{0} 0 0 0  0
Ĉ_{0}  Ĉ_{0} −Ĉ_{1} 0  0 0 0 0  0
Ĉ_{1}  Ĉ_{1} −Ĉ_{0} 0  0 0 0 0  0

1The preprint [6] is available as a postscript file from the internet address www.cpt.univ-mrs.fr

From the multiplication table, we note that the linear transformation ψ : \overline{U}_{i}^{∗} → \overline{U}_{i}, given by

(19)  ψ(ê_{0}) = 1 ψ(ê_{1}) = K ψ(Ê_{0}) = E
(20)  ψ(Ê_{1}) = EK ψ(F̂_{0}) = −F K ψ(F̂_{1}) = F
(21)  ψ(Ĉ_{0}) = −EF K ψ(Ĉ_{1}) = EF,

is an algebra isomorphism.

We next record the Hopf structure of \overline{U}_{i}^{∗} below. We first compute its comultiplication Δ^{∗} from the formula Δ^{∗}(ρ)(a ⊗ b) = ρ(ab) for all ρ ∈ \overline{U}_{i}^{∗}, a, b ∈ \overline{U}_{i}.

Δ^{∗}(ê_{0}) = ê_{0} ⊗ ê_{0}  Δ^{∗}(ê_{1}) = ê_{1} ⊗ ê_{1}
Δ^{∗}(Ê_{0}) = Ê_{0} ⊗ ê_{1} + ê_{0} ⊗ Ê_{0}  Δ^{∗}(Ê_{1}) = ê_{1} ⊗ Ê_{1} + Ê_{1} ⊗ ê_{0}
Δ^{∗}(F̂_{0}) = F̂_{0} ⊗ ê_{1} + ê_{0} ⊗ F̂_{0}  Δ^{∗}(F̂_{1}) = ê_{1} ⊗ F̂_{1} + F̂_{1} ⊗ ê_{0}

Δ^{∗}(Ĉ_{0}) = Ĉ_{0} ⊗ ê_{0} + ê_{0} ⊗ Ĉ_{0} + F̂_{0} ⊗ Ê_{1} + Ê_{0} ⊗ F̂_{1} Δ^{∗}(Ĉ_{1}) = Ĉ_{1} ⊗ ê_{1} + ê_{1} ⊗ Ĉ_{1} + F̂_{1} ⊗ Ê_{0} + Ê_{1} ⊗ F̂_{0} (22)

We note that (ψ ⊗ψ)∘Δ^{∗} = Δ^{op} ∘ψ. To show that ψ : \overline{U}_{i}^{∗} ⟶^{≅} \overline{U}_{i}^{cop} is a Hopf algebra isomorphism, it suffices to show that ϵ∘ψ = ϵ^{∗} where ϵ^{∗} is defined on \overline{U}_{i}^{∗} by ϵ^{∗}(ρ) := ρ(1) [15]. This follows from a comparison with the formulas for ϵ^{∗} below for j = 0, 1:

ϵ^{∗}(ê_{j}) = 1 ϵ^{∗}(Ê_{j}) = 0 = ϵ^{∗}(F̂_{j}) = ϵ^{∗}(Ĉ_{j}). (23)

Our record of the Hopf structure of \overline{U}_{i}^{∗} is not complete without the antipode S^{∗}(ρ) = ρ ∘ S. However, it follows automatically from the fact that ψ is a bialgebra isomorphism that ψ ∘ S^{∗} = \overline{S} ∘ ψ [15].

\begin{matrix}S^{∗}(ê_{0}) = ê_{0} & S^{∗}(ê_{1}) = ê_{1} & S^{∗}(Ê_{0}) = −Ê_{1} & S^{∗}(Ê_{1}) = Ê_{0} \\ S^{∗}(F̂_{0}) = F̂_{1} & S^{∗}(F̂_{1}) = −F̂_{0} & S^{∗}(Ĉ_{0}) = Ĉ_{0} & S^{∗}(Ĉ_{1}) = Ĉ_{1} \end{matrix} (24)

We can now prove:

Theorem 3.1. The Hopf algebras \overline{U}_{q}(sl_{2}) at the primitive n’th root of unity q are self-dual if and only if n = 4.

Proof. If q = ±i, we have shown in this section that \overline{U}_{q}^{∗} ≅ \overline{U}_{q}^{cop}. But by Lemma 2.2, the Cartan automorphism induces \overline{U}_{q} ≅ \overline{U}_{q}^{cop}. Whence \overline{U}_{q} ≅ \overline{U}_{q}^{∗} and \overline{U}_{q} is self-dual.

Now suppose q is not a fourth root of unity and \overline{U}_{q} is self-dual. By Proposition 2.3 \overline{U}_{q} is unimodular, therefore counimodular. Then by Radford’s formula 14, S^{4} = Id. But from Lemma 2.1

F = S^{4}(F) = K^{−2}F K^{2} = q^{4}F

which implies that q^{4} = 1, a contradiction.

Sommerhäuser [17] observes another proof for odd n that \overline{U}_{q} is not selfdual, which is implied by formulas for the dimensions of the radicals of \overline{U}_{q} and its dual.

4. Observations on Hopf subalgebras of \overline{U}_{q}

In \overline{U}_{q} is the Hopf algebra T_{q} generated by F and K. If e is again the order d of the root of unity q if d is odd, \frac{d}{2} if d is even, then T_{q} is determined by the relations:

(25)  F^{e} = 0 K^{e} = 1 F K^{−1} = q^{−2}K^{−1}F
(26) Δ(K) = K ⊗ K Δ(F) = F ⊗ K^{−1} + 1 ⊗ F
(27) ϵ(K) = 1, S(K) = K^{−1}  ϵ(F) = 0, S(F) = −F K

On the one hand, this is clearly a Hopf subalgebra of \overline{U_{q}} (i.e, Δ(T_{q}) ⊆ T_{q} ⊗ T_{q} and S(T_{q}) ⊆ T_{q}). On the other hand, these are the defining relations and Hopf structure on the Taft algebras (cf. [19, 15]) since q^{−2} is a primitive e’th root of unity.

The Taft algebra T_{q} is not unimodular, having right integral

t = F (1 + K + ⋯ + K^{e−1}) (28)

and non-proportional left integral (1 + K + ⋯ + K^{e−1})F. Therefore, T_{q} is not a symmetric algebra by [4, Corollary 3.3]. But T_{q} is self-dual with isomorphism T_{q} ⟶^{≅} T_{q}^{∗} given by K^{−1} ↦ G where G is the algebra map defined by G(K^{−1}) = q^{2}, G(F) = 0, and F ↦ X where X is the linear map defined by X(K^{−ℓ}F^{j}) = δ_{j,1} for 0 ≤ ℓ, j ≤ e − 1 [15].

Suppose H is a finite dimensional Hopf algebra with Hopf subalgebra T. Then it turns out that H is free as a right or left T -module [11] with special dual bases and right T -module homomorphism Φ : H → T: in the special case that T is the unit subalgebra C1, we recover our discussion in Section 2 (cf. Equation 12) [2]. As a left T -module map Φ is twisted by a new algebra automorphism β : T → T which is given in terms of the Nakayama automorphisms of T and H, respectively, by

β = η_{T} ∘ η_{H}^{−1}. (29)

There are formulas for the dual bases and homomorphism Φ given in [2] as follows. If t is a right integral for H, n a right integral for T, and ϕ ∈ H^{∗} satisfies Equations 12, then

Φ(a) := ∑_{(a)} ϕ(a_{(1)}\overline{S}(n))a_{(2)}, (30)

for every a ∈ H. By freeness of H as a right and left T -module, there exists Λ ∈ H such that

t = Λ̂n. (31)

Let

Λ := η_{H}(\overline{S}(Λ̂)). (32)

Then \{\overline{S}(Λ_{(2)})\}, \{Λ_{(1)}\} are dual bases for the right and left-twisted T -module H as follows: for all a ∈ H,

(33)  a = ∑_{(Λ)} \overline{S}(Λ_{(2)})Φ(Λ_{(1)}a)
(34)  a = ∑_{(Λ)} β^{−1}[Φ(a\overline{S}(Λ_{(2)}))]Λ_{(1)}.

The advantage of these equations comes about when H extends a wellknown algebra T, or vice versa. For example, from Equation 31 we deduce that Hopf subalgebras of a semisimple Hopf algebra are themselves semisimple, since ϵ(t) = ϵ(Λ̂)ϵ(n) ≠ 0 ⇒ ϵ(n) ≠ 0 [10]. However, we add a word of caution next.

For a β-Frobenius extension T ⊆ H of algebras, induction of T - modules, M ↦ M ⊗_{T} H, is naturally isomorphic to co-induction of T -modules, M ↦ Hom_{T}(H, M), up to a category equivalence determined by the automorphism β of T [3]. Certain properties such as self-injectivity of the subalgebra T will ascend to H [3]. However, we now note a number of properties that will not ascend, including selfduality for Hopf algebras (since T_{q} is self-dual, but \overline{U}_{q} is not self-dual for q ≠ ±i) and non-symmetry for Hopf algebras (since T_{q} is not a symmetric algebra, but U_{q} is).

At the fourth root of unity (cf. Section 3), we have the Hopf subalgebra pair \overline{U}_{i} ⊃ T_{i} where T_{i} is the well-known 4-dimensional Sweedler algebra with right integral n = F (1 + K) [18]. Since t = EF (1 + K), Λ̂ = E and Λ = S(E) = −KE.^{2} Then

∑_{(Λ)} \overline{S}(Λ_{(2)}) ⊗ Λ_{(1)} = K ⊗ EK + E ⊗ 1.

Furthermore, for all a ∈ \overline{U}_{i},

Φ(a) = −ϕ(a(1 + K)F)

2In passing, we should note that t provides an easy counterexample to the third Kaplansky conjecture that there are no nonzero central nilpotent elements in Hopf algebras over a field of characteristic zero, since t is a 2-sided integral and t^{2} = 0

where ϕ has dual bases for \overline{U}_{i}, \{\overline{S}(t_{(2)})\}, \{t_{(1)}\} as in Equation 12. We compute these ordered bases for \overline{U}_{i} and then the associated map ϕ:

(35)  \{\overline{S}(t_{(2)})\} = \{e_{0}, −e_{1}, C_{0}, −C_{1}, F_{1}, −F_{0}, E_{1}, −E_{0}\}
(36)  {t_{(1)}} = {C_{0}, C_{1}, e_{0}, e_{1}, E_{0}, E_{1}, F_{0}, F_{1}}.

It follows from Equation 12 that ϕ(C_{j}) = (−1)^{j}, ϕ(e_{j}) = ϕ(F_{j}) = ϕ(E_{j}) = 0 for j = 0, 1. Thus Φ has values

\begin{matrix}a & 1 & F & K & KF & E & EF & KEF & EK \\ Φ(a) & 0 & 0 & 0 & 0 & 1 & F & −KF & K \end{matrix}

Finally, the automorphism β is given by

\begin{matrix}x & 1 & F & K & KF \\ β(x) & 1 & F & −K & −KF \end{matrix}

We note that β(x) is not of the form uxu^{−1} for any u ∈ \overline{U}_{i} by applying ϵ to this equation and setting x = K. If it were of this form, one could choose new Φ and dual bases (by multiplying appropriately by u and its inverse) and get rid of the twist β in Equation 34.

Finally we note that \overline{U}_{i} is a ribbon algebra like T_{i} (with braided category of modules [5]) for reasons that generalize. For T_{i} has a oneparameter family of universal R-matrices R_{λ} [13], which at λ = 0 is

R := R_{0} = \frac{1}{2}(1 ⊗ 1 + 1 ⊗ K + K ⊗ 1 − K ⊗ K). (37)

Then Δ^{op}(a)R = RΔ(a) for a = K, F. The Cartan automorphism α fixes K, transposes E and F and satisfies Δ^{op} ∘ α = (α ⊗ α) ∘ Δ, equivalently Δ ∘ α = (α ⊗ α) ∘ Δ^{op}. Since R^{−1} = R = τ(R), it follows that

Δ^{op}(E)R = Δ^{op}(α(F))(α ⊗ α)(R) =
(α ⊗ α)(Δ(F)R) = (α ⊗ α)(RΔ^{op}(F)
= RΔ(E)

It follows by multiplicativity of Δ and Δ^{op} that \overline{U}_{i} is quasi-cocommutative (almost cocommutative) with same universal R-matrix R. That \overline{U}_{i} is braided (indeed, triangular) follows at once from R satisfying (Δ ⊗ 1)(R) = R_{13}R_{23} and (1 ⊗ Δ)(R) = R_{13}R_{12} [13].

Recall that a braided Hopf algebra H is a ribbon algebra if there is a central element θ ∈ H such that Δ(θ) = (τ(R)R)^{−1}(θ ⊗ θ), ϵ(θ) = 1, and S(θ) = θ [14]. It follows from taking θ = 1 that \overline{U}_{i} is a ribbon algebra.

The preceding generalizes to theorems of the following kind, providing properties that ascend for certain “well-placed” Hopf subalgebras:

Proposition 4.1. Suppose H is a Hopf algebra with braided Hopf subalgebra T and Hopf algebra automorphism α : H → H fixing the universal R-matrix R. If H is generated as an algebra by T and α(T), then H is braided.

Proof. The conclusion follows, as a variation on the theme above, by noting

Δ^{op}(α(x))R = (α ⊗ α)(Δ^{op}(x)R) = (α ⊗ α)(RΔ(x)) = RΔ(α(x)),

for every x ∈ T, since (α ⊗ α)(R) = R.

Finally it should be noted that \overline{U}_{i} is the Radford biproduct of the Hopf subalgebra generated by K and a zero-dimensional local Gorenstein algebra G := C⟨E, F | EF = F E, E^{2} = 0 = F^{2}⟩. G is the algebra of the isolated singularity at (0, 0) of the variety defined by x^{3} + y^{3}. G is equivalent to \overline{U}_{i} as algebras in a slightly weaker sense than Morita equivalence [3], for the bimodule projection G ⊕ GK → G has very pleasant dual bases given by 1 ⊗_{G} 1 + K ⊗_{G} K, which has etale-like properties and no twisting by an automorphism.

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Chalmers University of Technology/Göteborg University, S-412 96 Göteborg, Sweden

E-mail address: kadison@math.ntnu.no