A QUANTUM OCTONION ALGEBRA

Georgia Benkart José M. Pérez-Izquierdo

To the memory of Alberto Izquierdo

Introduction

Using the representation theory of the quantum group U_{q}(D_{4}) of D_{4} we construct a quantum analogue 𝕆_{q} of the split octonions and study its properties.

A unital algebra over a field with a nondegenerate bilinear form ( |) of maximal Witt index which admits composition,

(x · y|x · y) = (x|x)(y|y),

must be the field, two copies of the field, the split quaternions, or the split octonions. There is a natural q-version of the composition property that the algebra 𝕆_{q} of quantum octonions is shown to satisfy (see Prop. 4.12 below). We also prove that the quantum octonion algebra 𝕆_{q} satisfies the “q-Principle of Local Triality” (Prop. 3.12). Inside the quantum octonions are two nonisomorphic 4-dimensional subalgebras, which are q-deformations of the split quaternions. One of them is unital, and both of them give gl_{2} when considered as algebras under the commutator product [x, y] = x · y − y · x.

By its construction, 𝕆_{q} is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of U_{q}(D_{4}). Associative algebras with a Yang-Baxter operator (or r-algebras) arise in Manin’s work on noncommutative geometry and include such important examples as Weyl and Clifford algebras, quantum groups, and certain universal enveloping algebras (see for example [B]).

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 17A75, 17B37, 81R50.

The first author gratefully acknowledges support from National Science Foundation Grant #DMS–9622447.

The second author is grateful for support from the Programa de Formación del Personal Investigador en el Extranjero and from Pb 94-1311-C03-03, DGICYT.

Both authors acknowledge with gratitude the support and hospitality of the Mathematical Sciences Research Institute, Berkeley. Research at MSRI is supported in part by NSF grant DMS-9701755.

In the process of constructing 𝕆_{q} we define an algebra ℙ_{q}, which is a q analogue of the 8-dimensional para-Hurwitz algebra. The quantum para-Hurwitz algebra ℙ_{q} is shown to satisfy certain identities which become familiar properties of the para-Hurwitz algebra at q = 1. The para-Hurwitz algebra is a (non-unital) 8-dimensional algebra with a non-degenerate associative bilinear form admitting composition. The algebra ℙ_{q} exhibits related properties (see Props. 5.1, 5.3).

While this paper was in preparation, we received a preprint of [Br], which constructs a quantized octonion algebra using the representation theory of U_{q}(sl_{2}). Although both Bremner’s quantized octonions and our quantum octonions reduce to the octonions at q = 1, they are different algebras. The quantized octonion algebra in [Br] is constructed from defining a multiplication on the sum of irreducible U_{q}(sl_{2})-modules of dimensions 1 and 7. As a result, it carries a U_{q}(sl_{2})-module structure and has a unit element. The quantum octonion algebra constructed in this paper using U_{q}(D_{4}) has a unit element only for the special values q = 1, −1.

An advantage to the U_{q}(D_{4}) approach is that allows properties such as the ones mentioned above to be derived from its representation theory. Using the fact that the product in 𝕆_{q} is a U_{q}(D_{4})-module homomorphism, we prove identities for 𝕆_{q} (see Section 4) and as a consequence obtain at q = 1 new “representation theory” proofs for very well-known identities satisfied by the octonions that had been established previously by other methods. The U_{q}(D_{4}) approach also affords connections with fixed points of graph automorphisms - although as we show in Section 7, the fixed point subalgebras of U_{q}(D_{4}) are not the quantum groups U_{q}(G_{2}) and U_{q}(B_{3}), because those quantum groups do not have Hopf algebra embeddings into U_{q}(D_{4}). In the final section of this paper we explore connections with quantum Clifford algebras. In particular, we obtain a U_{q}(D_{4})-isomorphism between the quantum Clifford algebra C_{q}(8) and the endomorphism algebra End(𝕆_{q} ⊕ 𝕆_{q}) and discuss its relation to the work of Ding and Frenkel [DF] (compare also [KPS]).

We take this opportunity to express our appreciation to Alberto Elduque for his interest in our work and to Arun Ram for several helpful discussions.

§1. Preliminaries on Quantum Groups

The quantum group U_{q}(𝔤).

Let 𝔤 be a finite-dimensional complex simple Lie algebra with Cartan subalgebra 𝔥, root system Φ, and simple roots Π = {α_{i} | i = 1, …, r} ⊂ Φ relative to 𝔥. Let 𝔄 = (a_{α,β})_{α,β∈Π} be the corresponding Cartan matrix. Thus, a_{α,β} = 2(β, α)/(α, α), and there exist relatively prime positive integers,

d_{α} = \frac{(α, α)}{2},  (1.1)

so the matrix (d_{α}a_{α,β}) is symmetric. These integers are given explicitly by

A_{r}, D_{r}, E_{r} (r = 6, 7, 8)  d_{α_{i}} = 1 for all 1 ≤ i ≤ r
B_{r}  d_{α_{i}} = 2 for all 1 ≤ i ≤ r − 1, and d_{α_{r}} = 1
C_{r}  d_{α_{i}} = 1 for all 1 ≤ i ≤ r − 1, and d_{α_{r}} = 2
F_{4}  d_{α_{i}} = 2, i = 1, 2, and d_{α_{i}} = 1, i = 3, 4
G_{2}  d_{α_{1}} = 1 and d_{α_{2}} = 3. (1.2)

Let K be a field of characteristic not 2 or 3. Fix q ∈ K such that q is not a root of unity, and such that q^{\frac{1}{2}} ∈ K. For m, n, d ∈ Z_{>0}, let

[m]_{d} = \frac{q^{md} − q^{−md}}{q^{d} − q^{−d}} ,  and set  (1.3)

[m]_{d}! = ∏_{j=1}^{m}[j]_{d}  (1.4)

and [0]_{d}! = 1. Let

\left[{m \atop n}\right]_{d} = \frac{[m]_{d}!}{[n]_{d}![m − n]_{d}!}.  (1.5)

Definition 1.6. The quantum group U = U_{q}(𝔤) is the unital associative algebra over K generated by elements E_{α}, F_{α}, K_{α}, and K_{α}^{−1} (for all α ∈ Π) and subject to the relations,

(Q1) K_{α}K_{α}^{−1} = 1 = K_{α}^{−1}K_{α}, K_{α}K_{β} = K_{β}K_{α}

(Q2) K_{α}E_{β}K_{α}^{−1} = q^{(α,β)}E_{β}

(Q3) K_{α}F_{β}K_{α}^{−1} = q^{−(α,β)}F_{β}

(Q4) E_{α}F_{β} − F_{β}E_{α} = δ_{α,β} \frac{K_{α} − K_{α}^{−1}}{q^{d_{α}} − q^{−d_{α}}}

(Q5) ∑_{s=0}^{1−a_{α,β}}(−1)^{s} \left[{1 − a_{α,β} \atop s}\right]_{d_{α}} E_{α}^{1−a_{α,β}−s}E_{β}E_{α}^{s} = 0

(Q6) ∑_{s=0}^{1−a_{α,β}}(−1)^{s} \left[{1 − a_{α,β} \atop s}\right]_{d_{α}} F_{α}^{1−a_{α,β}−s}F_{β}F_{α}^{s} = 0.

Corresponding to each λ = ∑_{α∈Π} m_{α}α in the root lattice ZΦ there is an element

K_{λ} = ∏_{α∈Π} K_{α}^{m_{α}},

in U_{q}(𝔤), and K_{λ}K_{μ} = K_{λ+μ} for all λ, μ ∈ ZΦ. Using relations (Q2),(Q3), we have

K_{λ}E_{β}K_{λ}^{−1} = q^{(λ,β)}E_{β}  and  K_{λ}F_{β}K_{λ}^{−1} = q^{−(λ,β)}F_{β}  (1.7)

for all λ ∈ ZΦ and β ∈ Π.

The algebra U_{q}(𝔤) has a noncommutative Hopf structure with comultiplication Δ, antipode S, and counit ϵ given by

\begin{matrix}Δ(K_{α}) = K_{α} ⊗ K_{α}, \\ Δ(E_{α}) = E_{α} ⊗ 1 + K_{α} ⊗ E_{α} \\ Δ(F_{α}) = F_{α} ⊗ K_{α}^{−1} + 1 ⊗ F_{α} \\ S(K_{α}) = K_{α}^{−1}, \\ S(E_{α}) = −K_{α}^{−1}E_{α}  S(F_{α}) = −F_{α}K_{α} \\ ϵ(K_{α}) = 1,  ϵ(E_{α}) = ϵ(F_{α}) = 0. \end{matrix}  (1.8)

The opposite comultiplication Δ^{op} has the property that if

Δ(a) = ∑_{a} a_{(1)} ⊗ a_{(2)}, then Δ^{op}(a) = ∑_{a} a_{(2)} ⊗ a_{(1)}.

(We are adopting the commonly used Sweedler notation for components of the comultiplication.) The algebra U with the same multiplication, same identity, and same counit, but with comultiplication given by Δ^{op} and with S^{−1} as its antipode is also a Hopf algebra. The comultiplication Δ^{op} can be regarded as the composition τ ∘Δ, where τ : U ⊗U → U ⊗U is given by τ(a⊗b) = b⊗a. The map ψ : U → U_{q}(𝔤) with E_{α} ↦ F_{α}, F_{α} ↦ E_{α}, K_{α} ↦ K_{α}^{−1} is an algebra automorphism, and a coalgebra anti-automorphism, so U is isomorphic to U_{q}(𝔤) as a Hopf algebra (see [CP, p. 211]).

The defining property of the antipode S in any Hopf algebra is that it is an inverse to the identity map id_{U} with respect to the convolution product, so that for all a ∈ U,

∑_{a} S(a_{(1)})a_{(2)} = ϵ(a)id_{U} = ∑_{a} a_{(1)}S(a_{(2)}).  (1.9)

Representations.

For any Hopf algebra U the comultiplication operator allows us to define a Umodule structure on the tensor product V ⊗ W of two U-modules V and W, where

a(v ⊗ w) = ∑_{a} a_{(1)}v ⊗ a_{(2)}w.

The counit ϵ affords a representation of U on the one-dimensional module given by a1 = ϵ(a)1. If V is a finite-dimensional U-module, then so is its dual space V^{∗} where the U-action is given by the antipode mapping,

(af)(v) = f(S(a)v),  (1.10)

for all a ∈ U, f ∈ V^{∗}, and v ∈ V.

A finite-dimensional module M for U = U_{q}(𝔤) is the sum of weight spaces, M = ⨁_{ν,χ} M_{ν,χ}, where M_{ν,χ} = {x ∈ M | K_{α}x = χ(α)q^{(ν,α)}x for all α ∈ Π}, and χ : ZΦ → {±1} is a group homomorphism. All the modules considered in this paper will be of type I, that is χ(α) = 1 for all α ∈ Π, so we will drop χ from the notation and from our considerations.

Each finite-dimensional irreducible U-module M has a highest weight λ, which is a dominant integral weight for 𝔤 relative to Π, and a unique (up to scalar multiple) maximal vector v^{+} so that E_{α}v^{+} = 0 and K_{α}v^{+} = q^{(λ,α)}v^{+} for all α ∈ Π. In particular, the trivial U-module, K1, has highest weight 0. We denote by L(λ) the irreducible U-module with highest weight λ (and χ ≡ 1.)

When V is a finite-dimensional U-module, we can suppose {x_{ν,i}} is a basis of V of weight vectors with i = 1, …, dim V_{ν}. Let {x_{ν,i}^{∗}} be the dual basis in V^{∗} so that x_{μ,i}^{∗}(x_{ν,j}) = δ_{μ,ν}δ_{i,j}. Then

\begin{aligned}(K_{α}x_{μ,i}^{∗})(x_{ν,j}) & = x_{μ,i}^{∗}(S(K_{α})x_{ν,j}) = x_{μ,i}^{∗}(K_{α}^{−1}x_{ν,j}) \\ & = q^{−(ν,α)}x_{μ,i}^{∗}(x_{ν,j}) = q^{−(ν,α)}δ_{μ,ν}δ_{i,j} = q^{−(μ,α)}δ_{μ,ν}δ_{i,j}, \end{aligned} (1.11)

from which we see that x_{μ,i}^{∗} has weight −μ.

Suppose M and N are U-modules for a quantum group U. Define a U-module structure on Hom_{K}(M, N) by

(aγ)(m) = ∑_{a} a_{(1)}γ(S(a_{(2)})m).  (1.12)

(This just means that the natural map N ⊗ M^{∗} → Hom_{K}(M, N) is a U-module homomorphism.)

Note that if γ ∈ Hom_{U}(M, N), then

(aγ)(m) = ∑_{a} a_{(1)}γ(S(a_{(2)})m) = ∑_{a} a_{(1)}S(a_{(2)})γ(m) = ϵ(a)γ(m)

so that aγ = ϵ(a)γ. That says

Hom_{U}(M, N) ⊆ Hom_{K}(M, N)^{U} = {γ ∈ Hom_{K}(M, N) | aγ = ϵ(a)γ}.

Conversely, if γ belongs to the invariants Hom_{K}(M, N)^{U}, then we have

0 = ϵ(K_{α})γ = γ = K_{α} ∘ γ ∘ K_{α}^{−1}, and hence,
0 = K_{α} ∘ γ − γ ∘ K_{α}
0 = ϵ(E_{α})γ = E_{α}γ = E_{α} ∘ γ − K_{α} ∘ γ ∘ K_{α}^{−1} ∘ E_{α}, which implies
0 = E_{α} ∘ γ − γ ∘ E_{α}
0 = ϵ(F_{α})γ = F_{α}γ = F_{α} ∘ γ ∘ K_{α} − γ ∘ F_{α} ∘ K_{α} which implies
0 = F_{α} ∘ γ − γ ∘ F_{α}

because of the relations in (1.8). Since these elements generate U_{q}(𝔤), we see that γ ∈ Hom_{K}(M, N)^{U} implies γ ∈ Hom_{U}(M, N), so the two spaces are equal:

Hom_{U}(M, N) = Hom_{K}(M, N)^{U}.  (1.13)

Now if M = N = L(λ), a finite-dimensional irreducible U-module of highest weight λ, then any f ∈ Hom_{U}(M, M) must map the highest weight vector v^{+} to a multiple of itself. Since v^{+} generates M as a U-module, it follows that f is a scalar multiple of the identity. Thus, Hom_{U}(M, M) = Kid_{M}. So the space of invariants Hom_{K}(M, M)^{U}, or equivalently (M ⊗ M^{∗})^{U}, is one-dimensional.

R-matrices.

The quantum group U = U_{q}(𝔤) has a triangular decomposition U = U^{−}U^{0}U^{+}, where U^{+} (resp. U^{−}) is the subalgebra generated by the E_{α}, (resp. F_{α}), and U^{0} is the subalgebra generated by the K_{α}^{±1}, for all α ∈ Π. Then U^{+} = ∑_{μ} U_{μ}^{+} where U_{μ}^{+} = {a ∈ U^{+} | K_{α}aK_{α}^{−1} = q^{(μ,α)}a}. It is easy to see using the automorphism ψ above, that U^{−} has the decomposition U^{−} = ∑_{μ} U_{−μ}^{−}, and U_{μ}^{+} ≠ (0) if and only if U_{−μ}^{−} ≠ (0). There is a nondegenerate bilinear form ( , ) on U (see [Jn, Lemma 6.16, Prop. 6.21]) which satisfies

(1.14)

(a) (b, a) = q^{(2ρ,μ−ν)}(a, b) for all a ∈ U_{−ν}^{−}U^{0}U_{μ}^{+} and b ∈ U_{−μ}^{−}U^{0}U_{ν}^{+}, where ρ is the half-sum of the positive roots of 𝔤.

(b) (F_{α}, E_{β}) = −δ_{α,β}(q_{α} − q_{α}^{−1})^{−1}.

Choose a basis a_{1}^{μ}, a_{2}^{μ}, …, a_{r(μ)}^{μ} of U_{μ}^{+} and a corresponding dual basis b_{1}^{μ}, b_{2}^{μ}, …, b_{r(μ)}^{μ} of U_{−μ}^{−} so that (b_{i}^{μ}, a_{j}^{μ}) = δ_{i,j}. Set

Θ_{μ} = ∑_{i=1}^{r(μ)} b_{i}^{μ} ⊗ a_{i}^{μ}.

In particular, Θ_{α} = −(q_{α} − q_{α}^{−1})F_{α} ⊗ E_{α} for all α ∈ Π.

Suppose π : U_{q}(𝔤) → End(V) and π′ : U_{q}(𝔤) → End(V′) are two finitedimensional U_{q}(𝔤)-modules. Since the set of weights of V is finite, there are only finitely many μ ∈ ZΦ which are differences of weights of V. Therefore, all but finitely many Θ_{μ} act as zero on V and V′, and Θ_{V,V′} = (π ⊗π′)(Θ) for Θ = ∑_{μ} Θ_{μ} is a well-defined mapping on V ⊗ V′. By suitably ordering the basis of V ⊗ V′, we see that each Θ_{μ} for μ ≠ 0 acting on V ⊗ V′ has a strictly upper triangular matrix relative to that basis. Since Θ_{0} = 1 ⊗ 1, the transformation Θ_{V,V′} is unipotent.

Consider the map f = f_{V,V′}, which is defined by f_{V,V′} : V ⊗ V′ → V ⊗ V′, f(v ⊗ w) = q^{−(λ,μ)}v ⊗ w for all v ∈ V_{λ} and w ∈ V′_{μ}.

Proposition 1.15. (Compare [Jn, Thm. 7.3].) Let V, V′ be finite-dimensional U_{q}(𝔤)-modules, and let σ = σ_{V′,V} be defined by

\begin{matrix}σ : V′ ⊗ V & → V ⊗ V′ \\ w ⊗ v & ↦ v ⊗ w. \end{matrix}

Then the mapping Ř_{V′,V} : V′ ⊗ V → V ⊗ V′ given by Ř_{V′,V} = Θ ∘ f ∘ σ is a U_{q}(𝔤)-module isomorphism.

There is an alternate expression for Θ_{μ} which is more explicit and can be best understood using the braid group action. The braid group B_{𝔤} associated to 𝔤 has generators T_{α}, α ∈ Π, and defining relations

(T_{α}T_{β})^{m_{α,β}} = (T_{β}T_{α})^{m_{α,β}}

where m_{α,β} = 2, 3, 4, 6 if a_{α,β}a_{β,α} = 0, 1, 2, 3 respectively. It acts by algebra automorphisms on U_{q}(𝔤) according to the following rules (see [Jn, Sec. 8.14]):

\begin{matrix}T_{α}(K_{β}) = K_{s_{α}β},  T_{α}(E_{α}) = −F_{α}K_{α},  T_{α}(F_{α}) = −K_{α}^{−1}E_{α} \\ T_{α}(E_{β}) = ∑_{t=0}^{−a_{α,β}}(−1)^{t}q_{α}^{−t}(E_{α})^{(−a_{α,β}−t)}E_{β}(E_{α})^{(t)} α ≠ β \\ T_{α}(F_{β}) = ∑_{t=0}^{−a_{α,β}}(−1)^{t}q_{α}^{t}(F_{α})^{(t)}F_{β}(F_{α})^{(−a_{α,β}−t)} α ≠ β, \end{matrix}  (1.16)

where s_{α} is the simple reflection corresponding to α in the Weyl group W of 𝔤, and

(E_{α})^{(n)} = \frac{E_{α}^{n}}{[n]_{d_{α}}!}, (F_{α})^{(n)} = \frac{F_{α}^{n}}{[n]_{d_{α}}!},

(the factorials are as defined in (1.4)).

Let s_{i} = s_{α_{i}} be the reflection in the hyperplane perpendicular to the simple root α_{i} and fix a reduced decomposition w_{0} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{m}} of the longest element of W. Every positive root occurs precisely once in the sequence

β_{1} = α_{i_{1}}, β_{2} = s_{i_{1}}(α_{i_{2}}), …, β_{m} = s_{i_{1}}s_{i_{2}} ⋯ s_{i_{m−1}}(α_{i_{m}}).  (1.17)

The elements

\begin{aligned}E_{β_{t}} & = T_{α_{i_{1}}} T_{α_{i_{2}}} ⋯ T_{α_{i_{t−1}}} (E_{α_{i_{t}}}) \\ F_{β_{t}} & = T_{α_{i_{1}}} T_{α_{i_{2}}} ⋯ T_{α_{i_{t−1}}} (F_{α_{i_{t}}}) \end{aligned}  (1.18)

for t = 1, …, m, are called the positive (resp. negative) root vectors of U_{q}(𝔤). For any m-tuple \underline{ℓ} = (ℓ_{1}, …, ℓ_{m}) of nonnegative integers let

\begin{aligned}E\underline{ℓ} & = E_{β_{m}}^{ℓ_{m}}E_{β_{m−1}}^{ℓ_{m−1}} ⋯ E_{β_{1}}^{ℓ_{1}} \\ F \underline{ℓ} & = F_{β_{m}}^{ℓ_{m}}F_{β_{m−1}}^{ℓ_{m−1}} ⋯ F_{β_{1}}^{ℓ_{1}}. \end{aligned}

Then the elements E\underline{ℓ} (resp. F \underline{ℓ}) determine a basis of U^{+} (resp. U^{−}). (See for example, [Jn, Thm. 8.24].)

Consider for t = 1, …, m the sum

Θ^{[t]} = ∑_{j≥0}(−1)^{j}q_{α}^{−j(j−1)/2} \frac{(q_{α} − q_{α}^{−1})^{j}}{[j]_{d_{α}}!} F_{β_{t}}^{j} ⊗ E_{β_{t}}^{j}

where α = α_{i_{t}} and E_{β_{t}} and F_{β_{t}} are as in (1.18). This lies in the direct product of all the U_{−μ}^{−} ⊗ U_{μ}^{+}. Then (as discussed in [Jn, Sec. 8.30]), Θ_{μ} is the (U_{μ}^{−} ⊗ U_{μ}^{+})- component of the product

Θ^{[m]}Θ^{[m−1]} ⋯ Θ^{[2]}Θ^{[1]}.

That component involves only finitely many summands.

§2. 8-dimensional Representations of U_{q}(D_{4})

In this section we specialize to the case that 𝔤 is a finite-dimensional simple complex Lie algebra of type D_{4}. We can identify 𝔤 with the special orthogonal Lie algebra so(8). The set of roots of 𝔤 is given by {±ϵ_{i} ± ϵ_{j} | 1 ≤ i ≠ j ≤ 4}, where ϵ_{i}, i = 1, …, 4, is an orthonormal basis of R^{4}, and the simple roots may be taken to be α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{3} = ϵ_{3} − ϵ_{4}, α_{4} = ϵ_{3} + ϵ_{4}. The corresponding fundamental weights have the following expressions:

\begin{aligned}ω_{1} & = α_{1} + α_{2} + (1/2)α_{3} + (1/2)α_{4} = ϵ_{1}, \\ ω_{2} & = α_{1} + 2α_{2} + α_{3} + α_{4} = ϵ_{1} + ϵ_{2} \\ ω_{3} & = (1/2)α_{1} + α_{2} + α_{3} + (1/2)α_{4} = (1/2)(ϵ_{1} + ϵ_{2} + ϵ_{3} − ϵ_{4}) \\ ω_{4} & = (1/2)α_{1} + α_{2} + (1/2)α_{3} + α_{4} = (1/2)(ϵ_{1} + ϵ_{2} + ϵ_{3} + ϵ_{4}), \end{aligned}  (2.1)

and they satisfy (ω_{i}, α_{j}) = δ_{i,j} for all i, j = 1, …, 4.

The Lie algebra 𝔤 has three 8-dimensional representations - its natural representation and two spin representations with highest weights ω_{1}, ω_{3}, and ω_{4} respectively, and so does its quantum counterpart U_{q}(D_{4}). (Note we are writing U_{q}(D_{4}) rather than U_{q}(𝔤) to emphasize the role that D_{4} is playing here.) In particular, for the natural representation V of U_{q}(D_{4}), there is a basis {x_{μ} | μ = ±ϵ_{i}, i = 1, …, 4} (compare [Jn, Chap. 5]) such that

K_{α}x_{μ} = q^{(μ,α)}x_{μ},

E_{α}x_{μ} = \{ \begin{matrix}0 if (μ, α) ≠ −1 \\ x_{μ+α} if (μ, α) = −1 \end{matrix}

F_{α}x_{μ} = \{ \begin{matrix}0 if (μ, α) ≠ 1 \\ x_{μ−α} if (μ, α) = 1. \end{matrix}

(2.2)

For D_{4}, the group of graph automorphisms of its Dynkin diagram can be identified with the symmetric group S_{3} of permutations on {1, 2, 3, 4} fixing 2. Thus, ϕ ∈ S_{3} permutes the simple roots ϕ(α_{i}) = α_{ϕi} and fixes α_{2}. Each graph automorphism ϕ induces an automorphism, which we also denote by ϕ, on U_{q}(D_{4}) defined by

E_{α_{i}} ↦ E_{α_{ϕi}}  F_{α_{i}} ↦ F_{α_{ϕi}}  K_{α_{i}} ↦ K_{α_{ϕi}}.

Corresponding to ϕ there is new representation, π_{ϕ} : U_{q}(D_{4}) → End(V) of U_{q}(D_{4}) on its natural representation V such that

π_{ϕ}(a)x = ϕ(a)x

for all a ∈ U_{q}(D_{4}) and x ∈ V. We denote the U_{q}(D_{4})-module with this action by V_{ϕ}. Then we have

π_{ϕ}(K_{α})x_{μ} = K_{ϕα}x_{μ} = q^{(μ,ϕα)}x_{μ} = q^{(ϕ^{−1}μ,α)}x_{μ}

π_{ϕ}(E_{α})x_{μ} = E_{ϕα}x_{μ} = \{ \begin{matrix}0 if (ϕ^{−1}μ, α) = (μ, ϕα) ≠ −1 \\ x_{μ+ϕα} if (ϕ^{−1}μ, α) = (μ, ϕα) = −1 \end{matrix}

π_{ϕ}(F_{α})x_{μ} = F_{ϕα}x_{μ} = \{ \begin{matrix}0 if (ϕ^{−1}μ, α) = (μ, ϕα) ≠ 1 \\ x_{μ−ϕα} if (ϕ^{−1}μ, α) = (μ, ϕα) = 1. \end{matrix}

(2.3)

From these formulas the following is apparent:

Proposition 2.4. V_{ϕ} is an irreducible U_{q}(D_{4})-module with highest weight ϕ^{−1}ω_{1}. In particular, we have the following highest weights:

(i) ω_{1} for ϕ = id or (3 4),

(ii) ω_{3} for ϕ = (1 3) or (1 4 3),

(iii) ω_{4} for ϕ = (1 4) or (1 3 4).

The operators E_{α}, F_{α} map elements in the basis {x_{μ}} to other basis elements or 0, and each vector x_{μ} is a weight vector of weight ϕ^{−1}μ. For the natural representation V, the basis elements are x_{±ϵ_{i}}, i = 1, …, 4, which we abbreviate as x_{±i}. We can view the module V_{ϕ} as displayed below, where the index i on an arrow indicates that F_{α_{i}} maps the higher vector down to the lower vector and E_{α_{i}} maps the lower vector up to the higher one. The action of all F_{α_{j}}’s and E_{α_{j}}’s not shown is 0.

\begin{matrix}V_{ϕ} \\ x_{1} & ∘ \\ ↓ ϕ^{−1}1 \\ x_{2} & ∘ \\ ↓ 2 \\ x_{3} & ∘ \\ ϕ^{−1}3 ↙ & ↘ ϕ^{−1}4 \\ x_{4} ∘ & ∘ x_{−4} \\ ϕ^{−1}4 ↘ & ↙ ϕ^{−1}3 \\ x_{−3} & ∘ \\ ↓ 2 \\ x_{−2} & ∘ \\ ↓ ϕ^{−1}1 \\ x_{−1} & ∘ \end{matrix}  (2.5)

In particular, taking the identity, θ = (1 4 3), and θ^{2} = (1 3 4) we have the following pictures for the natural representation V, and the spin representations V_{−} and V_{+}:

\begin{aligned}& V \\ & x_{1}∘ \\ & ↓ 1 \\ & x_{2}∘ \\ & ↓ 2 \\ & x_{3}∘ \\ 3 ↙ & & ↘ 4 \\ x_{4} ∘ & & ∘ x_{−4} \\ 4 ↘ & & ↙ 3 \\ & x_{−3}∘ \\ & ↓ 2 \\ & x_{−2}∘ \\ & ↓ 1 \\ & x_{−1}∘ \end{aligned}  \begin{aligned}& V_{−} \\ & x_{1}∘ \\ & ↓ 3 \\ & x_{2}∘ \\ & ↓ 2 \\ & x_{3}∘ \\ 4 ↙ & & ↘ 1 \\ x_{4} ∘ & & ∘ x_{−4} \\ 1 ↘ & & ↙ 4 \\ & x_{−3}∘ \\ & ↓ 2 \\ & x_{−2}∘ \\ & ↓ 3 \\ & x_{−1}∘ \end{aligned}  \begin{aligned}& V_{+} \\ & x_{1}∘ \\ & ↓ 4 \\ & x_{2}∘ \\ & ↓ 2 \\ & x_{3}∘ \\ 1 ↙ & & ↘ 3 \\ x_{4} ∘ & & ∘ x_{−4} \\ 3 ↘ & & ↙ 1 \\ & x_{−3}∘ \\ & ↓ 2 \\ & x_{−2}∘ \\ & ↓ 4 \\ & x_{−1}∘ \end{aligned} (2.6)

There is a unique (up to scalar multiple) U_{q}(D_{4})-module homomorphism ∗ : V_{−} ⊗ V_{+} → V, which we can compute directly using the expressions for the comultiplication in (1.8). We display the result of this computation in Table 2.7 below. To make the table more symmetric, we scale the mapping by q^{\frac{3}{2}} .

Table 2.7 ; \begin{matrix}∗ & x_{1} & x_{2} & x_{3} & x_{−4} & x_{4} & x_{−3} & x_{−2} & x_{−1} \\ x_{1} & 0 & 0 & 0 & q^{− \frac{3}{2}} x_{1} & 0 & q^{− \frac{3}{2}} x_{2} & q^{− \frac{3}{2}} x_{3} & q^{− \frac{3}{2}} x_{−4} \\ x_{2} & 0 & 0 & −q^{− \frac{1}{2}} x_{1} & 0 & −q^{− \frac{1}{2}} x_{2} & 0 & q^{− \frac{3}{2}} x_{4} & q^{− \frac{3}{2}} x_{−3} \\ x_{3} & 0 & q^{\frac{1}{2}} x_{1} & 0 & 0 & −q^{− \frac{1}{2}} x_{3} & −q^{− \frac{1}{2}} x_{4} & 0 & q^{− \frac{3}{2}} x_{−2} \\ x_{4} & −q^{\frac{3}{2}} x_{1} & 0 & 0 & 0 & −q^{− \frac{1}{2}} x_{−4} & −q^{− \frac{1}{2}} x_{−3} & −q^{− \frac{1}{2}} x_{−2} & 0 \\ x_{−4} & 0 & q^{\frac{1}{2}} x_{2} & q^{\frac{1}{2}} x_{3} & q^{\frac{1}{2}} x_{4} & 0 & 0 & 0 & q^{− \frac{3}{2}} x_{−1} \\ x_{−3} & −q^{\frac{3}{2}} x_{2} & 0 & q^{\frac{1}{2}} x_{−4} & q^{\frac{1}{2}} x_{−3} & 0 & 0 & −q^{− \frac{1}{2}} x_{−1} & 0 \\ x_{−2} & −q^{\frac{3}{2}} x_{3} & −q^{\frac{3}{2}} x_{−4} & 0 & q^{\frac{1}{2}} x_{−2} & 0 & q^{\frac{1}{2}} x_{−1} & 0 & 0 \\ x_{−1} & −q^{\frac{3}{2}} x_{4} & −q^{\frac{3}{2}} x_{−3} & −q^{\frac{3}{2}} x_{−2} & 0 & −q^{\frac{3}{2}} x_{−1} & 0 & 0 & 0 \end{matrix}

It can be seen readily from (1.8) that the relations

\begin{matrix}(ϕ ⊗ ϕ)Δ(a) = Δ(ϕ(a)) \\ ϕ(S(a)) = S(ϕ(a)) \end{matrix}  (2.8)

hold for all a ∈ U_{q}(D_{4}) and all graph automorphisms ϕ. Therefore, by identifying V_{−} with V_{θ} and V_{+} with V_{θ^{2}} we obtain from

a(x ∗ y) = ∑_{a} θ(a_{(1)})x ∗ θ^{2}(a_{(2)})y

that the following relations hold:

\begin{matrix}π_{θ}(a)(x ∗ y) = ∑_{a} θ^{2}(a_{(1)})x ∗ a_{(2)}y \\ π_{θ^{2}}(a)(x ∗ y) = ∑_{a} a_{(1)}x ∗ θ(a_{(2)})y. \end{matrix}  (2.9)

They allow us to conclude

Proposition 2.10. The product ∗ : V_{−} ⊗ V_{+} → V displayed in Table 2.7 above is a U_{q}(D_{4})-module homomorphism, and it gives U_{q}(D_{4})-module homomorphisms,

\begin{matrix}∗ : V_{+} ⊗ V → V_{−} \\ ∗ : V ⊗ V_{−} → V_{+}. \end{matrix}

Suppose now that ζ = (1 3) and η = (1 4). Specializing (2.5) with ϕ equal to these permutations gives:

\begin{aligned}& V_{ζ} \\ & x_{1}∘ \\ & ↓ 3 \\ & x_{2}∘ \\ & ↓ 2 \\ & x_{3}∘ \\ 1 ↙ & & ↘ 4 \\ x_{4} ∘ & & ∘ x_{−4} \\ 4 ↘ & & ↙ 1 \\ & x_{−3}∘ \\ & ↓ 2 \\ & x_{−2}∘ \\ & ↓ 3 \\ & x_{−1}∘ \end{aligned}  \begin{aligned}& V_{η} \\ & x_{1}∘ \\ & ↓ 4 \\ & x_{2}∘ \\ & ↓ 2 \\ & x_{3}∘ \\ 3 ↙ & & ↘ 1 \\ x_{4} ∘ & & ∘ x_{−4} \\ 1 ↘ & & ↙ 3 \\ & x_{−3}∘ \\ & ↓ 2 \\ & x_{−2}∘ \\ & ↓ 4 \\ & x_{−1}∘ \end{aligned}  (2.11)

As these diagrams indicate, the transformation specified by

\begin{matrix}ȷ : V & → V \\ x_{i} & ↦ −x_{i}  i ≠ ±4 \\ x_{4} & ↦ −x_{−4} \\ x_{−4} & ↦ −x_{4} \end{matrix}  (2.12)

determines explicit U_{q}(D_{4})-isomorphisms,

ȷ : V_{−} → V_{ζ}  ȷ : V_{+} → V_{η}.  (2.13)

This together with Proposition 2.10 enables to conclude the following:

Proposition 2.14. The map · : V_{ζ} ⊗ V_{η} → V given by x ⊗ y ↦ x · y = ȷ(x) ∗ ȷ(y) is a U_{q}(D_{4})-module homomorphism.

Eigenvalues of Ř_{V,V}.

Suppose now V = L(ω_{1}), the natural representation of U_{q}(D_{4}). The module V^{⊗2} is completely reducible: V^{⊗2} = L(2ω_{1}) ⊕ L(ω_{2}) ⊕ L(0). By determining the eigenvalues of Ř_{V,V} on the corresponding maximal vectors, we obtain the eigenvalues of Ř_{V,V} on the summands.

First, on the maximal vector x_{1} ⊗ x_{1} of the summand L(2ω_{1}), Ř_{V,V} (x_{1} ⊗ x_{1}) = q^{−(ϵ_{1},ϵ_{1})}x_{1} ⊗ x_{1} = q^{−1}(x_{1} ⊗ x_{1}). The vector x_{2} ⊗ x_{1} − q^{−1}x_{1} ⊗ x_{2} is a maximal vector for L(ω_{2}). Then

Ř_{V,V} (x_{2} ⊗ x_{1} − q^{−1}x_{1} ⊗ x_{2}) = q^{−(ϵ_{1},ϵ_{2})}Θ(x_{1} ⊗ x_{2} − q^{−1}x_{2} ⊗ x_{1})
= x_{1} ⊗ x_{2} − q^{−1}x_{2} ⊗ x_{1} − (q − q^{−1})x_{2} ⊗ x_{1}
= x_{1} ⊗ x_{2} − qx_{2} ⊗ x_{1} = −q(x_{2} ⊗ x_{1} − q^{−1}x_{1} ⊗ x_{2}),

so that Ř_{V,V} acts as −q on L(ω_{2}). We will argue in the remark following Proposition 2.19 below that Ř_{V,V} acts on L(0) as q^{7}. Thus, the eigenvalues of Ř_{V,V} are given by

\begin{matrix}q^{−1} & L(2ω_{1}) \\ −q & L(ω_{2}) \\ q^{7} & L(0), \end{matrix}  (2.15)

(compare [D], [R]). Setting S^{2}(V) = L(2ω_{1}) ⊕ L(0) we see that Ř_{V,V} + qid_{V} maps V^{⊗2} onto S^{2}(V), the quantum symmetric space. There are analogous decompositions for the two spin representations of U_{q}(D_{4}) obtained by replacing ω_{1} by ω_{3} and ω_{4}.

For each graph automorphism ϕ, we consider Ř = Ř_{V_{ϕ},V_{ϕ}} on V_{ϕ} ⊗ V_{ϕ}. Note the bilinear form ( , ) on U_{q}(D_{4}) in [Jn, (6.12)] is (ϕ ⊗ ϕ)-invariant, so

Θ_{ϕ(μ)} = ∑_{i} ϕ(b_{i}^{μ}) ⊗ ϕ(a_{i}^{μ}) = (ϕ ⊗ ϕ)(Θ_{μ}).

Now

\begin{aligned}Ř(v ⊗ w) & = (Θ ∘ f ∘ σ)(v ⊗ w) = q^{−(ϕ^{−1}λ,ϕ^{−1}ν)}Θ(w ⊗ v) \\ & = q^{−(λ,ν)}Θ(w ⊗ v) = Ř_{V,V} (v ⊗ w), \end{aligned}  (2.16)

whenever w is in V_{λ} and v ∈ V_{ν}. Consequently, the actions of the R-matrices agree. It follows that

S^{2}(V) = S^{2}(V_{ϕ})  (2.17)

for all graph automorphisms ϕ.

A “test” element of S^{2}(V).

In the next two chapters we will compute various U_{q}(D_{4})-module homomorphisms and prove that they are equal by evaluating them on a particular element of S^{2}(V). A good “test” element for these calculations is x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}, but first we need to demonstrate that it does in fact belong to S^{2}(V). Since Ř_{V,V} +qid_{V} maps V^{⊗2} onto S^{2}(V) for V = L(ω_{1}), the natural representation, it suffices to show that the image of x_{1} ⊗ x_{−1} under Ř_{V,V} + qid_{V} is x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}. Using the fact that E_{γ}x_{1} = 0 for all positive roots γ we obtain

\begin{aligned}Ř_{V,V} (x_{1} ⊗ x_{−1}) & = (Θ ∘ f ∘ σ)(x_{1} ⊗ x_{−1}) \\ & = qΘ(x_{−1} ⊗ x_{1}) = q(x_{−1} ⊗ x_{1}). \end{aligned}

Consequently,

(Ř_{V,V} + qid_{V})(x_{1} ⊗ x_{−1}) = qx_{−1} ⊗ x_{1} + qx_{1} ⊗ x_{−1},

which implies the desired conclusion

x_{−1} ⊗ x_{1} + x_{1} ⊗ x_{−1} ∈ S^{2}(V).

Bilinear Forms.

For the natural representation V of U = U_{q}(D_{4}), the dual space V^{∗} has weights ±ϵ_{i}, i = 1, …, 4 by (1.11). The vector x_{−1}^{∗} is a maximal vector since it corresponds to the unique dominant weight, ϵ_{1}, and V^{∗} is isomorphic to V. We have seen in (1.13) that the space of invariants (V ⊗V^{∗})^{U} is one-dimensional. Since V ≅ V^{∗}, we have that the space of invariants (V ⊗V)^{U} is one-dimensional. Thus, in V ⊗V there is a unique copy of the trivial module, and hence a unique (up to scalar multiple) U_{q}(D_{4})-module homomorphism

( |) : V ⊗ V → K.

The condition of ( |) being a U_{q}(D_{4})-module homomorphism is equivalent to

(ax|y) = (x|S(a)y)  (2.18)

for all a ∈ U and x, y ∈ V (see [Jn, p.122]). There is an analogous bilinear form giving the unique (up to multiples) U_{q}(D_{4})-module homomorphism, V_{ϕ} ⊗ V_{ϕ} → K. All these bilinear forms can be regarded as U_{q}(D_{4})-module maps from the symmetric tensors S^{2}(V_{ϕ}) → K, since the trivial module is an irreducible summand of S^{2}(V_{ϕ}) in each case.

Now for a ∈ U_{q}(D_{4}),

\begin{aligned}∑_{a}(π_{ϕ}(a_{(1)})x|π_{ϕ}(a_{(2)})y) & = ∑_{a}(ϕ(a_{(1)})x|ϕ(a_{(2)})y) \\ & = ∑_{a}(x|S(ϕ(a_{(1)}))ϕ(a_{(2)})y) \\ & = ∑_{a}(x|ϕ(S(a_{(1)})a_{(2)})y \\ & = (x|ϕ( ∑_{a} S(a_{(1)})a_{(2)}))y \\ & = (x|ϕ(ϵ(a)1)y) = ϵ(a)(x|y). \end{aligned}  (2.19)

(We have used in the third line the fact that ϕ and S commute (2.8).) Hence, by the calculation in (2.19), and by direct computation of ( |) on V ⊗ V, we have

Proposition 2.20. The U_{q}(D_{4})-module homomorphism: ( |) : V ⊗ V → K determines a U_{q}(D_{4})-module homomorphism: ( |) : V_{ϕ} ⊗ V_{ϕ} → K for any graph automorphism ϕ. It is given by

(x_{i}|x_{j}) = \{ \begin{matrix}0 if j ≠ −i \\ (−1)^{i+1} \frac{q^{−(4−i)}}{q^{3} + q^{−3}} if i > 0 and j = −i \\ (−1)^{i+1} \frac{q^{i+4}}{q^{3} + q^{−3}} if i < 0 and j = −i. \end{matrix}

We are free to scale the bilinear form ( |) on V_{ϕ} by any nonzero element in K. In writing down the values of (x_{i}|x_{−i}) in Proposition 2.20, we have chosen a particular scaling with an eye towards results (such as Proposition 4.2) to follow. It is easy to see using the basis elements x_{i} that the following holds.

Proposition 2.21. The transformation ȷ is an isometry, ( (x|y) = (ȷ(x)|ȷ(y)) for all x, y ∈ V), relative to the bilinear form in Proposition 2.20.

Remark. Observe that Ř_{V,V} ∘ ( |) = ξ( |), where ξ is the eigenvalue of Ř_{V,V} on L(0). Since Ř_{V,V} (x_{1} ⊗ x_{−1}) = q(x_{−1} ⊗ x_{1}), as our calculations for the “test element” show, we have ξ(x_{1}|x_{−1}) = q(x_{−1}|x_{1}). Therefore, ξq^{−3} = qq^{3} or ξ = q^{7}, as asserted in (2.15).

For D_{4}, the half-sum of the positive roots is given by ρ = 3α_{1} + 5α_{2} + 3α_{3} + 3α_{4}, so K_{2ρ}^{−1} = K_{α_{1}}^{−6}K_{α_{2}}^{−10}K_{α_{3}}^{−6}K_{α_{4}}^{−6}. Therefore, on the natural representation V, K_{2ρ}^{−1} has a diagonal matrix relative to the basis {x_{1}, x_{2}, x_{3}, x_{4}, x_{−4}, x_{−3}, x_{−2}, x_{−1}} with corresponding diagonal entries given by {q^{−6}, q^{−4}, q^{−2}, 1, 1, q^{2}, q^{4}, q^{6}}. It is easy to verify directly that

(x|y) = (y|K_{2ρ}^{−1}x).  (2.22)

§3. Octonions and Quantum Octonions

Suppose ζ = (1 3), η = (1 4), and V_{ϕ} is the representation of U_{q}(D_{4}) coming from the graph automorphism ϕ. Then

dim_{K} Hom_{U_{q}(D_{4})}(S^{2}(V_{ζ}) ⊗ S^{2}(V_{η}), K) = 1,  (3.1)

and we can realize these homomorphisms explicitly as multiples of the mapping

\begin{aligned}S^{2}(V_{ζ}) & ⊗ S^{2}(V_{η}) ⊂ V_{ζ} ⊗ V_{ζ} ⊗ V_{η} ⊗ V_{η} → K \\ & ∑_{k,ℓ} u_{k} ⊗ v_{k} ⊗ w_{ℓ} ⊗ y_{ℓ} ↦ ∑_{k,ℓ}(u_{k}|v_{k})(w_{ℓ}|y_{ℓ}). \end{aligned}  (3.2)

On the other hand, the mapping

S^{2}(V_{ζ}) ⊗ S^{2}(V_{η}) ⊂ V_{ζ} ⊗ V_{ζ} ⊗ V_{η} ⊗ V_{η} → V_{ζ} ⊗ V_{η} ⊗ V_{ζ} ⊗ V_{η} → V ⊗ V → K (3.3)

∑_{k,ℓ} u_{k} ⊗ v_{k} ⊗ w_{ℓ} ⊗ y_{ℓ} ↦ ∑_{k,ℓ} u_{k} ⊗ Ř_{V_{ζ},V_{η}}(v_{k} ⊗ w_{ℓ}) ⊗ y_{ℓ}

↦ ∑_{k,ℓ} ∑_{i} u_{k} · w_{ℓ}^{(i)} ⊗ v_{k}^{(i)} · y_{ℓ} ↦ ∑_{k,ℓ} ∑_{i}(u_{k} · w_{ℓ}^{(i)}|v_{k}^{(i)} · y_{ℓ}),

is a U_{q}(D_{4})-module homomorphism, (where “·” is the product in Proposition 2.14 and Ř_{V_{ζ},V_{η}}(v_{k} ⊗ w_{ℓ}) = ∑_{i} w_{ℓ}^{(i)} ⊗ v_{k}^{(i)} is the braiding morphism). Thus, it must be a multiple of the mapping in (3.2).

We now consider what would happen if q = 1 and compute the multiple in that case. Here the braiding morphism may be taken to be v ⊗ w ↦ w ⊗ v since that is a module map in the q = 1 case. We evaluate the mappings at e ⊗ e ⊗ e ⊗ e where e = x_{4} − x_{−4}. Now by Table 2.7, we have

\begin{aligned}e^{2} & = ȷ(e) ∗ ȷ(e) = (x_{4} − x_{−4}) ∗ (x_{4} − x_{−4}) \\ & = q^{\frac{1}{2}} x_{4} − q^{− \frac{1}{2}} x_{−4}, \end{aligned}  (3.4)

which shows that at q = 1 that e^{2} = e. By Proposition 2.19,

(e|e) = −(x_{4}|x_{−4}) − (x_{−4}|x_{4}) = \frac{2}{q^{3} + q^{−3}} ,  (3.5)

which gives (e|e)_{1} = 1 at q = 1. (Here and throughout the paper we use ( |)_{1} to denote the bilinear form at q = 1.) Therefore, (e|e)_{1}(e|e)_{1} = 1 and (e^{2}|e^{2})_{1} = (e|e)_{1} = 1. Thus, the two maps in (3.2) and (3.3) are equal at q = 1. By specializing ∑_{k} u_{k} ⊗ v_{k} = x ⊗ x and ∑_{ℓ} w_{ℓ} ⊗ y_{ℓ} = y ⊗ y (which are symmetric elements at q = 1), we have as a consequence,

(x · y|x · y)_{1} = (x|x)_{1}(y|y)_{1}.  (3.6)

Using the multiplication table (Table 2.7) for x∗y and the formula x·y = ȷ(x)∗ȷ(y), we can verify that e = x_{4} −x_{−4} is the unit element relative to the product “·” when q = 1. (For further discussion, see the beginning of Section 5.) Thus, the product · : V_{ζ} ⊗ V_{η} → V for q = 1 gives an 8-dimensional unital composition algebra over K whose underlying vector space is V = V_{ζ} = V_{η}. Such an algebra must be the octonions. Thus, we have

Proposition 3.7. Let V be the 8-dimensional vector space over K with basis {x_{±i} | i = 1, …, 4}. Then V with the product · defined by x · y = ȷ(x) ∗ ȷ(y), where ȷ is as in (2.12) and ∗ is as in Table 2.7, is the algebra of octonions when q = 1.

Because of this result we are motivated to make the following definition.

Quantum Octonions.

Definition 3.8. Let V be the 8-dimensional vector space over K with basis {x_{±i} | i = 1, …, 4}. Then V with the product “·” defined by

x · y = ȷ(x) ∗ ȷ(y),

where ȷ is as in (2.12) and ∗ is given by Table 2.7, is the algebra of quantum octonions. We denote this algebra by 𝕆_{q} = (V, ·).

The multiplication table for the quantum octonions is obtained from Table 2.7 by interchanging the entries in the columns and rows labelled by x_{−4} and x_{4}. As a result, we have

Table 3.9 ; \begin{matrix}· & x_{1} & x_{2} & x_{3} & x_{−4} & x_{4} & x_{−3} & x_{−2} & x_{−1} \\ x_{1} & 0 & 0 & 0 & 0 & q^{− \frac{3}{2}} x_{1} & q^{− \frac{3}{2}} x_{2} & q^{− \frac{3}{2}} x_{3} & q^{− \frac{3}{2}} x_{−4} \\ x_{2} & 0 & 0 & −q^{− \frac{1}{2}} x_{1} & −q^{− \frac{1}{2}} x_{2} & 0 & 0 & q^{− \frac{3}{2}} x_{4} & q^{− \frac{3}{2}} x_{−3} \\ x_{3} & 0 & q^{\frac{1}{2}} x_{1} & 0 & −q^{− \frac{1}{2}} x_{3} & 0 & −q^{− \frac{1}{2}} x_{4} & 0 & q^{− \frac{3}{2}} x_{−2} \\ x_{4} & 0 & q^{\frac{1}{2}} x_{2} & q^{\frac{1}{2}} x_{3} & 0 & q^{\frac{1}{2}} x_{4} & 0 & 0 & q^{− \frac{3}{2}} x_{−1} \\ x_{−4} & −q^{\frac{3}{2}} x_{1} & 0 & 0 & −q^{− \frac{1}{2}} x_{−4} & 0 & −q^{− \frac{1}{2}} x_{−3} & −q^{− \frac{1}{2}} x_{−2} & 0 \\ x_{−3} & −q^{\frac{3}{2}} x_{2} & 0 & q^{\frac{1}{2}} x_{−4} & 0 & q^{\frac{1}{2}} x_{−3} & 0 & −q^{− \frac{1}{2}} x_{−1} & 0 \\ x_{−2} & −q^{\frac{3}{2}} x_{3} & −q^{\frac{3}{2}} x_{−4} & 0 & 0 & q^{\frac{1}{2}} x_{−2} & q^{\frac{1}{2}} x_{−1} & 0 & 0 \\ x_{−1} & −q^{\frac{3}{2}} x_{4} & −q^{\frac{3}{2}} x_{−3} & −q^{\frac{3}{2}} x_{−2} & −q^{\frac{3}{2}} x_{−1} & 0 & 0 & 0 & 0 \end{matrix}

The product map: p : V_{ζ} ⊗ V_{η} → V, p(x ⊗ y) = x · y, of the quantum octonion algebra is the composition p = ∗ ∘ (j ⊗ j) of two U_{q}(D_{4})-module maps, so is itself a U_{q}(D_{4})-module homomorphism.

Let π : U_{q}(D_{4}) → End(V) be the natural representation of U_{q}(D_{4}). Then we have

\begin{aligned}π(a)(x · y) & = a · p(x ⊗ y) = p(Δ(a)(x ⊗ y)) \\ & = p( ∑_{a} ζ(a_{(1)})x ⊗ η(a_{(2)})y) \\ & = ∑_{a} ȷ(ζ(a_{(1)}x)) ∗ ȷ(η(a_{(2)}y)) \\ & = ∑_{a} (ζ(a_{(1)})x) · (η(a_{(2)})y). \end{aligned}  (3.10)

Now when q = 1 we can identify U_{q}(D_{4}) with the universal enveloping algebra U(𝔤) of a simple Lie algebra 𝔤 of type D_{4}. The comultiplication specializes in this case to the usual comultiplication on U(𝔤), which has Δ(a) = a ⊗ 1 + 1 ⊗ a for all a ∈ 𝔤. In particular, for elements a ∈ 𝔤, (3.10) becomes

π(a)(x · y) = (ζ(a)x) · y + x · (η(a)y).  (3.11)

Equation (3.11) is commonly referred to as the Principle of Local Triality (see for example, [J, p. 8] or [S, p. 88]). As a result we have,

Proposition 3.12. The quantum octonion algebra 𝕆_{q} = (V, ·) satisfies the q-Principle of Local Triality,

π(a)(x · y) = ∑_{a} (ζ(a_{(1)})x) · (η(a_{(2)})y)

for all a ∈ U_{q}(D_{4}) and x, y ∈ V, where ζ and η are the graph automorphisms ζ = (1 3) and η = (1 4) of U_{q}(D_{4}), and Δ(a) = ∑_{a} a_{(1)}⊗a_{(2)} is the comultiplication in U_{q}(D_{4}).

A quantum para-Hurwitz algebra.

At q = 1, the map ȷ : V → V sending x_{i} ↦ −x_{−i} for i ≠ ±4, x_{4} ↦ −x_{−4}, and x_{−4} ↦ −x_{4}, agrees with the standard involution x → \overline{x} on the octonions so that

x ∗ y = ȷ(x) · ȷ(y) = \overline{x} · \overline{y}.  (3.13)

Thus, the product “∗” displayed in Table 2.7 gives what is called the para-Hurwitz algebra when q = 1. (Results on para-Hurwitz algebras can be found in [EM], [EP], [OO1], [OO2].)

Definition 3.14. We say that the algebra ℙ_{q} = (V, ∗) with the multiplication ∗ given by Table 2.7 is the quantum para-Hurwitz algebra. It satisfies

x ∗ y = ȷ(x) · ȷ(y)

where 𝕆_{q} = (V, ·) is the quantum octonion algebra and ȷ is the transformation on V defined in (2.12).

Our definitions have been predicated on using the mapping · : V_{ζ} ⊗ V_{η} → V. We can ask what would have happened if we had used the projection V_{η} ⊗ V_{ζ} → V instead? To answer this question, it is helpful to observe that the mapping ȷ determines a U_{q}(D_{4})-module isomorphism ȷ : V_{ζηζ} = V_{(3 4)} → V, which is readily apparent from comparing the following picture of V_{(3 4)} with the diagram for V in (2.6).

V_{(3 4)}
x_{1}∘
↓ 1
x_{2}∘
↓ 2
x_{3}∘
4 ↙  ↘ 3
x_{4} ∘  ∘ x_{−4}
3 ↘  ↙ 4
x_{−3}∘
↓ 2
x_{−2}∘
↓ 1
x_{−1}∘

Suppose as before that p : V_{ζ} ⊗ V_{η} → V is the product p(x ⊗ y) = x · y in the quantum octonion algebra. Consider the mapping ȷ ∘ p ∘ (ȷ ⊗ ȷ) : V_{η} ⊗ V_{ζ} → V. We demonstrate that this is a U_{q}(D_{4})-module map. Bear in mind in these calculations that ȷ : V_{ζ} → V_{−} = V_{ζη} and ȷ : V_{η} → V_{+} = V_{ηζ} for ζ = (1 3) and η = (1 4). Then for a ∈ U_{q}(D_{4}) we have

ȷ ∘ p ∘ (ȷ ⊗ ȷ)(a(x ⊗ y)) = ȷ ∘ p ∘ (ȷ ⊗ ȷ)( ∑_{a} η(a_{(1)})x ⊗ ζ(a_{(2)})y)

= ȷ ∘ p( ∑_{a} j(η(a_{(1)})x) ⊗ j(ζ(a_{(2)})y))

= ȷ ∘ p( ∑_{a} ηζ(a_{(1)})j(x) ⊗ ζη(a_{(2)})j(y))

= ȷ ∘ p( ∑_{a} ζ^{2}ηζ(a_{(1)})j(x) ⊗ η^{2}ζη(a_{(2)})j(y))

Now using the fact that ηζη = ζηζ and that (ϕ ⊗ ϕ)Δ = Δϕ for all graph automorphisms ϕ together with (3.10), we see that the last expression can be rewritten as

ȷ ∘ p((ζ ⊗ η)(Δ(ζηζ(a)))(ȷ(x) ⊗ ȷ(y)))

= ȷ(ζηζ(a)p(ȷ(x) ⊗ ȷ(y))) (p is a U_{q}(D_{4}) − homomorphism)

= a(ȷ ∘ p ∘ (ȷ ⊗ ȷ))(x ⊗ y).

Thus, ȷ ∘ p ∘ (ȷ ⊗ ȷ) : V_{η} ⊗ V_{ζ} → V is a U_{q}(D_{4})-module homomorphism. Since the space Hom_{U_{q}(D_{4})}(V_{η} ⊗ V_{ζ}, V) is one-dimensional, we may assume the product p′ : V_{η} ⊗ V_{ζ} → V is given by the map we have just found, so that p′ = ȷ ∘ p ∘ (ȷ ⊗ ȷ). Alternately, ȷ∘p′ = p∘(ȷ⊗ȷ). This says that had we defined the quantum octonions using the product p′ rather than p so that x ·′ y = p′(x ⊗ y), the two algebras would be isomorphic via the map ȷ.

§4. Properties of Quantum Octonions

We use U_{q}(D_{4})-module homomorphisms to deduce various properties of the quantum octonions. At q = 1 these properties specialize to well-known identities satisfied by the octonions, which can be found for example in [ZSSS]. Since the vector spaces V, V_{ζ}, and V_{η} are the same, we will denote the identity map on them simply by “id” in what follows.

Proposition 4.1. The following diagram commutes

( |) ⊗ id
S^{2}(V_{ζ}) ⊗ V_{η}  ⟶  V_{η}
id ⊗ p ↓  ↓ id
p ∘ (ȷ ⊗ id)
V_{ζ} ⊗ V  ⟶  V_{η},

so that

∑_{k} ȷ(u_{k}) · (v_{k} · y) = ∑_{k}(u_{k}|v_{k})y  (4.2)

for all ∑_{k} u_{k} ⊗ v_{k} ∈ S^{2}(𝕆_{q}), y ∈ 𝕆_{q}. In particular, at q = 1 this reduces to the well-known identity satisfied by the octonion algebra 𝕆:

\overline{x} · (x · y) = (x|x)_{1}y.  (4.3)

Proof. Since p : V_{ζ} ⊗ V_{η} → V is a U_{q}(D_{4})-module homomorphism, we have that π(a) ∘ p = p((ζ ⊗ η)Δ(a)), which implies that π(η(a)) ∘ p = p((ζ ⊗ η)Δ(ηa)) =

p((ζη ⊗ id)Δ(a)). This says that p : V_{ζη} ⊗ V → V_{η} is also a U_{q}(D_{4})-module homomorphism. Since ȷ : V_{ζ} → V_{ζη} is a U_{q}(D_{4})-module homomorphism, it follows that (p∘(ȷ⊗id))∘(id⊗p) ∈ Hom_{U_{q}(D_{4})}(S^{2}(V_{ζ})⊗V_{η}, K). As that space of homomorphisms has dimension one, the maps in the diagram must be proportional.

To find the proportionality constant, let us consider the actions using the test element:

(x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}) ⊗ x_{1} →^{( |)⊗id} x_{1}

and

\begin{matrix}p ∘ (ȷ ⊗ id) ∘ (id ⊗ p) & ((x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}) ⊗ x_{1}) = \\ p ∘ (ȷ ⊗ id)( − q^{\frac{3}{2}} x_{1} ⊗ x_{4}) = q^{\frac{3}{2}} x_{1} · x_{4} = x_{1}. \end{matrix}

Therefore, the two maps are equal as claimed. At q = 1 we may substitute x ⊗ x ∈ S^{2}(𝕆) and y ∈ 𝕆 into the above identity to obtain the corresponding one for the octonions. □

Proposition 4.4. The following diagram commutes

id ⊗ ( |)
V_{ζ} ⊗ S^{2}(V_{η})  ⟶  V_{ζ}
p ⊗ id ↓  ↓ id
p ∘ (id ⊗ ȷ)
V ⊗ V_{η}  ⟶  V_{ζ},

so that

∑_{k} (y · u_{k}) · ȷ(v_{k}) = ∑_{k}(u_{k}|v_{k})y  (4.5)

for all ∑_{k} u_{k} ⊗ v_{k} ∈ S^{2}(𝕆_{q}), y ∈ 𝕆_{q}. In particular, at q = 1 this gives the identity

(y · x) · \overline{x} = (x|x)_{1}y  (4.6)

which is satisfied by the octonions.

Proof. Both maps are U_{q}(D_{4})-module homomorphisms and so must be multiples of each other. We compute their images on x = x_{1} ⊗ (x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}):

(id ⊗ ( |))x = ((x_{1}|x_{−1}) + (x_{−1}|x_{1}))x_{1} = x_{1},
p ∘ (id ⊗ ȷ) ∘ (p ⊗ id)x = p ∘ (id ⊗ ȷ)(q^{− \frac{3}{2}} x_{−4} ⊗ x_{1}) = −q^{− \frac{3}{2}} x_{−4} · x_{1} = x_{1}.

Hence we have the desired equality. □

Proposition 4.7. The following diagram commutes

p ⊗ id
V_{ζ} ⊗ V_{η} ⊗ V  ⟶  V ⊗ V
id ⊗ (ȷ ∘ p ∘ (id ⊗ ȷ))  ↓  ↓ ( |)
( |)
V_{ζ} ⊗ V_{ζ}  ⟶  K

so that

(x · y|z) = (x|ȷ(y · ȷ(z))) = (ȷ(x)|y · ȷ(z))  (4.8)

for all x, y, z in the quantum octonions. At q = 1 this reduces to the identity

(x · y|z)_{1} = (x|z · \overline{y})_{1}.  (4.9)

satisfied by the octonions.

Proof. Recall we have shown that ȷ∘p∘(ȷ⊗ȷ) : V_{η} ⊗V_{ζ} → V is a U_{q}(D_{4})-module homomorphism, which is equivalent to saying

a ∘ ȷ ∘ p ∘ (ȷ ⊗ ȷ) = ȷ ∘ p ∘ (ȷ ⊗ ȷ)((η ⊗ ζ)Δ(a))

for all a ∈ U_{q}(D_{4}). This implies

ζ(a) ∘ ȷ ∘ p ∘ (ȷ ⊗ ȷ) = ȷ ∘ p ∘ (ȷ ⊗ ȷ)((η ⊗ ζ)Δ(ζ(a)))
= ȷ ∘ p ∘ (ȷ ⊗ ȷ)((η ⊗ ζ)(ζ ⊗ ζ)Δ(a))
= ȷ ∘ p ∘ (ȷ ⊗ ȷ)((ηζ ⊗ id)Δ(a))

so that ȷ ∘ p ∘ (ȷ ⊗ ȷ) : V_{ηζ} ⊗ V → V_{ζ} is a U_{q}(D_{4})-module homomorphism. From this we deduce that

ȷ ∘ p ∘ (id ⊗ ȷ) = ȷ ∘ p ∘ (ȷ ⊗ ȷ) ∘ (ȷ ⊗ id) : V_{η} ⊗ V → V_{ζ}

is a U_{q}(D_{4})-module homomorphism as well. Thus, the mappings in the above diagram are scalar multiples of one another. The scalar can be discovered by computing the values of them on x_{1} ⊗ x_{−3} ⊗ x_{−2}:

(x_{1} · x_{−3}|x_{−2}) = q^{− \frac{3}{2}} (x_{2}|x_{−2}) = \frac{−q^{− \frac{3}{2}} q^{−2}}{q^{3} + q^{−3}} = \frac{−q^{− \frac{7}{2}}}{q^{3} + q^{−3}}

(x_{1}|ȷ(x_{−3} · ȷ(x_{−2}))) = (x_{1}|ȷ(q^{− \frac{1}{2}} x_{−1})) = \frac{−q^{− \frac{1}{2}} q^{−3}}{q^{3} + q^{−3}} = \frac{−q^{− \frac{7}{2}}}{q^{3} + q^{−3}} .

Hence, they agree and we have the result. The fact that ȷ is an isometry gives last equality in (4.8). □

Proposition 4.10. (x · y|z) = (ȷ(y)|ȷ(z) · K_{2ρ}^{−1}x) for all x, y, z ∈ 𝕆_{q} = (V, ·), which at q = 1 gives

(x · y|z)_{1} = (y|\overline{x} · z)_{1}  (4.11)

24 GEORGIA BENKART JOSÉ M. PÉREZ-IZQUIERDO

for the octonions.

Proof. This is an immediate consequence of the calculation,

\begin{matrix}(x · y|z)=^{1}(x|ȷ(y · ȷ(z))) & =^{2}(ȷ(y · ȷ(z))|K_{2ρ}^{−1}x) \\ = (y · ȷ(z)|ȷ(K_{2ρ}^{−1}x)) =^{3} (ȷ(y)|ȷ(z) · K_{2ρ}^{−1}x), \end{matrix}

where (1) is from Proposition 4.7, (2) uses (2.20), and (3) is Proposition 4.7. □

Note K_{2ρ}^{−1} acts on 𝕆_{q} as an algebra automorphism because it is group-like. Proposition 4.12. The following diagram commutes

id ⊗ ( |) ⊗ id

V_{η} ⊗ S^{2}(V_{ζ}) ⊗ V_{η}  ⟶  V_{η} ⊗ V_{η}
(ȷ ∘ p ∘ (ȷ ⊗ ȷ)) ⊗ p  ↓  ↓ ( |)
( |)
V ⊗ V ⟶ K.

When q = 1, this gives the identity

(x · y|x · z)_{1} = (x|x)_{1}(y|z)_{1}  (4.13)

for the octonions.

Proof. First, it is helpful to observe that

Hom_{U_{q}(D_{4})}(V_{η} ⊗ S^{2}(V_{ζ}) ⊗ V_{η}, K) ≅ Hom_{U_{q}(D_{4})}(S^{2}(V_{ζ}), V_{η} ⊗ V_{η})
= Hom_{U_{q}(D_{4})}(L(0), L(0)),

so these spaces have dimension 1. We evaluate the mappings on x = x_{−1} ⊗ (x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}) ⊗ x_{1}:

( |) ∘ (id ⊗ ( |) ⊗ id)(x) = (x_{−1}|x_{1}) = \frac{q^{3}}{q^{3} + q^{−3}}
( |) ∘ ((ȷ ∘ p ∘ (ȷ ⊗ ȷ)) ⊗ p)(x) = (ȷ(x_{−1} · x_{1})|x_{−1} · x_{1})

= −q^{3}(x_{−4}|x_{4}) = \frac{q^{3}}{q^{3} + q^{−3}} . □

Proposition 4.14. The following diagram commutes

id ⊗ ( |) ⊗ id
V_{ζ} ⊗ S^{2}(V_{η}) ⊗ V_{ζ}  ⟶  V_{ζ} ⊗ V_{ζ}
p ⊗ (ȷ ∘ p ∘ (ȷ ⊗ ȷ))  ↓  ↓ ( |)
( |)
V ⊗ V  ⟶  K

In particular, at q = 1 this gives the following identity satisfied by the octonions:

(y · x|z · x)_{1} = (y|z)_{1}(x|x)_{1}  (4.15)

Proof. The proof is virtually identical to that of Proposition 4.12. We check the maps on x = x_{−1} ⊗ (x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}) ⊗ x_{1}:

\begin{aligned}( |) ∘ (p ⊗ (ȷ ∘ p ∘ (ȷ ⊗ ȷ)))(x) & = (x_{−1} · x_{1}|ȷ(x_{−1} · x_{1})) = (ȷ(x_{−1} · x_{1})|x_{−1} · x_{1}) \\ & = −q^{3}(x_{−4}|x_{4}) = \frac{q^{3}}{q^{3} + q^{−3}} , \end{aligned}

which equals ( |) ∘ (id ⊗ ( |) ⊗ id)(x). □

r-algebras.

The notion of an r-algebra (that is, an algebra equipped with a Yang-Baxter operator) arises in Manin’s work [M] on noncommutative geometry. Noteworthy examples of r-algebras are Weyl and Clifford algebras, noncommutative tori, certain universal enveloping algebras, and quantum groups (see for example, Baez [B]).

Definition 4.16. Suppose A is an algebra over a field F with multiplication p : A⊗A → A. Assume R ∈ End(A^{⊗2}) is a Yang-Baxter operator, i.e. R_{1,2}R_{2,3}R_{1,2} = R_{2,3}R_{1,2}R_{2,3}, where R_{1,2} = R ⊗ id ∈ End(A^{⊗3}) and R_{2,3} = id ⊗ R ∈ End(A^{⊗3}). Then A is said to be an r-algebra if

R ∘ (p ⊗ id_{A}) = (id_{A} ⊗ p) ∘ R_{1,2} ∘ R_{2,3}

R ∘ (id_{A} ⊗ p) = (p ⊗ id_{A}) ∘ R_{2,3} ∘ R_{1,2}. (4.17)

When A has a unit element 1, the Yang-Baxter operator R is also required to satisfy R(1 ⊗ x) = x ⊗ 1, R(x ⊗ 1) = 1 ⊗ x for all x ∈ A.

Our quantum octonions satisfy r-algebra properties similar to those in (4.17) with respect to the R-matrix of U_{q}(D_{4}).

Proposition 4.18. (a) In Hom_{U_{q}(D_{4})}(V_{ζ} ⊗ V_{η} ⊗ V, V ⊗ V) we have

Ř_{V,V} ∘ (p ⊗ id) = (id ⊗ p) ∘ Ř_{V_{ζ},V} ∘ Ř_{V_{η},V} .

(b) In Hom_{U_{q}(D_{4})}(V ⊗ V_{ζ} ⊗ V_{η}, V ⊗ V) we have

Ř_{V,V} ∘ (id ⊗ p) = (p ⊗ id) ∘ Ř_{V,V_{η}} ∘ Ř_{V,V_{ζ}}.

Proof. Suppose that π_{1} : V ⊗ V → L(2ω_{1}), π_{2} : V ⊗ V → L(ω_{2}), and π_{3} : V ⊗ V → L(0) are the projections of V ⊗ V onto its irreducible summands. Let T_{ℓ} and T_{r} denote the transformations on the left and right in (a). We compute T_{ℓ} and T_{r} on several elements of V_{ζ} ⊗V_{η}⊗V which have been chosen so that Θ acts as the identity. We use the fact that η(ϵ_{1}) = 1/2(ϵ_{1} + ϵ_{2} + ϵ_{3} + ϵ_{4}), ζ(ϵ_{1}) = 1/2(ϵ_{1} + ϵ_{2} + ϵ_{3} − ϵ_{4}), η(ϵ_{2}) = 1/2(ϵ_{1} + ϵ_{2} − ϵ_{3} − ϵ_{4}), and ζ(ϵ_{3}) = 1/2(ϵ_{1} − ϵ_{2} + ϵ_{3} + ϵ_{4}).

We begin by evaluating the action of T_{ℓ} and T_{r} on x_{−1} ⊗ x_{1} ⊗ x_{−1}:

(Ř_{V,V} ∘ (p ⊗ id))(x_{−1} ⊗ x_{1} ⊗ x_{−1}) = −q^{\frac{3}{2}} Ř_{V,V} (x_{4} ⊗ x_{−1}) = −q^{\frac{3}{2}} (x_{−1} ⊗ x_{4})
((id ⊗ p) ∘ Ř_{V_{ζ},V} ∘ Ř_{V_{η},V} )(x_{−1} ⊗ x_{1} ⊗ x_{−1}) = q^{\frac{1}{2}} ((id ⊗ p) ∘ Ř_{V_{ζ},V} )(x_{−1} ⊗ x_{−1} ⊗ x_{1})
= (id ⊗ p)(x_{−1} ⊗ x_{−1} ⊗ x_{1})
= −q^{\frac{3}{2}} (x_{−1} ⊗ x_{4}).

Now (x_{−1}|x_{4}) = 0 implies that x_{−1} ⊗x_{4} lies in L(2ω_{1})⊕L(ω_{2}). The vector x_{−1} ⊗x_{4} is not an eigenvector of Ř_{V,V} because Ř_{V,V} (x_{4} ⊗ x_{−1}) = x_{−1} ⊗ x_{4}. Therefore, π_{1}(x_{−1}⊗x_{4}) ≠ 0 and π_{2}(x_{−1}⊗x_{4}) ≠ 0. Since dim Hom_{U_{q}(D_{4})}(V_{ζ} ⊗V_{η}⊗V, L(2ω_{1})) = 1 = dim Hom_{U_{q}(D_{4})}(V_{ζ} ⊗ V_{η} ⊗ V, L(ω_{2}), we see from the above calculation that π_{1}T_{ℓ} = π_{1}T_{r} and π_{2}T_{ℓ} = π_{2}T_{r}.

Consider the action of T_{ℓ} and T_{r} on x_{3} ⊗ x_{2} ⊗ x_{−1}:

(Ř_{V,V} ∘ (p ⊗ id))(x_{3} ⊗ x_{2} ⊗ x_{−1}) = q^{\frac{1}{2}} Ř_{V,V} (x_{1} ⊗ x_{−1}) = q^{\frac{3}{2}} (x_{−1} ⊗ x_{1})
((id ⊗ p) ∘ Ř_{V_{ζ},V} ∘ Ř_{V_{η},V} )(x_{3} ⊗ x_{2} ⊗ x_{−1}) = q^{\frac{1}{2}} ((id ⊗ p) ∘ Ř_{V_{ζ},V} )(x_{3} ⊗ x_{−1} ⊗ x_{2})
= q(id ⊗ p)(x_{−1} ⊗ x_{3} ⊗ x_{2})
= q^{\frac{3}{2}} (x_{−1} ⊗ x_{1}).

The projection π_{3} comes from the form ( |), and so it is nonzero when it acts on x_{−1} ⊗ x_{1}. Consequently, applying π_{3} to both sides of T_{ℓ}(x_{3} ⊗ x_{2} ⊗ x_{−1}) = T_{r}(x_{3} ⊗ x_{2} ⊗ x_{−1}) and using dim Hom_{U_{q}(D_{4})}(V_{ζ} ⊗ V_{η} ⊗ V, L(0)) = 1, we obtain π_{3}T_{ℓ} = π_{3}T_{r}. Therefore, T_{ℓ} = (π_{1} + π_{2} + π_{3})T_{ℓ} = (π_{1} + π_{2} + π_{3})T_{r} = T_{r}. The calculations for part (b) are almost identical and are left as an exercise for the reader. □

§5. Properties of the Quantum Para-Hurwitz Algebra

Recall that the quantum para-Hurwitz algebra ℙ_{q} = (V, ∗) is obtained from identifying the underlying vector spaces of V_{θ}, V_{θ^{2}} and V, where V is the natural representation of U_{q}(D_{4}) and θ is the graph automorphism θ = (1 4 3), and using the multiplication p^{∗} : V_{θ} ⊗ V_{θ^{2}} → V, x ⊗ y ↦ x ∗ y = ȷ(x) · ȷ(y). The resulting product is displayed in Table 2.7. As we saw in the last chapter, the quantum octonion algebra enjoys very nice composition properties relative to the bilinear ( |). It is known that the para-Hurwitz algebra ℙ (that is the q = 1 version of ℙ_{q}), which is obtained from the octonions using the product x ∗ y = \overline{x} · \overline{y}, has an associative bilinear form admitting composition. Here we establish similar results for the bilinear form on ℙ_{q} = (V, ∗)

Proposition 5.1. For the quantum para-Hurwitz algebra ℙ_{q} = (V, ∗),

(x ∗ y|z) = (x|y ∗ z)  and  (x ∗ y|z) = (y|z ∗ K_{2ρ}^{−1}x)

for all x, y, z ∈ ℙ_{q}. At q = 1 these relations reduce to the well-known properties of the para-Hurwitz algebra (x ∗ y|z)_{1} = (x|y ∗ z)_{1} and (x ∗ y|z)_{1} = (y|z ∗ x)_{1}.

Proof. The proof amounts to using the fact that ȷ is an isometry together with the fact that ȷ commutes with K_{2ρ}^{−1}. So we have

(x∗y|z) = (ȷ(x)·ȷ(y)|z) = \{ \begin{matrix}(x|ȷ(y) · ȷ(z)) = (x|y ∗ z) \\ (y|ȷ(z) · K_{2ρ}^{−1}ȷ(x)) = (y|ȷ(z) · ȷ(K_{2ρ}^{−1}x)) = (y|z ∗ K_{2ρ}^{−1}x). \end{matrix} □

Proposition 5.2. Ignoring the U_{q}(D_{4})-module structures we have

id ⊗ ( |) ⊗ id
V ⊗ S^{2}(V) ⊗ V  ⟶  V ⊗ V
(ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ)) ⊗ p^{∗}  ↓  ↓ ( |)
( |)
V ⊗ V  ⟶  K
id ⊗ ( |) ⊗ id
V ⊗ S^{2}(V) ⊗ V  ⟶  V ⊗ V
p^{∗} ⊗ (ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ))  ↓  ↓ ( |)
( |)
V ⊗ V  ⟶  K

Proof. These diagrams commute when p is used in place of p^{∗}. Now when p^{∗} = p ∘ (ȷ ⊗ ȷ) is taken, we have

( |) ∘ (ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ)) ⊗ p^{∗} = ( |) ∘ ((ȷ ∘ p) ⊗ (p ∘ (ȷ ⊗ ȷ)))

=^{1} ( |) ∘ p ⊗ (j ∘ p ∘ (ȷ ⊗ ȷ))

= ( |) ∘ id ⊗ ( |) ⊗ id

where (1) follows from the fact that ȷ is an isometry. The other diagram can be shown to commute in a similar way. □

Proposition 5.3. The following diagram commutes and shows at q = 1 that the relation (x · y) · x = (x|x)_{1}y = x · (y · x) holds for the para-Hurwitz algebra.

V_{θ^{2}} ⊗V \underleftarrow{id⊗(ȷ∘p^{∗}∘(ȷ⊗ȷ))} S^{2}(V_{θ^{2}})⊗V_{θ} =S^{2}(V)⊗V =S^{2}(V_{θ})⊗V_{θ^{2}}  →^{id⊗p^{∗}} V_{θ} ⊗V
↓p^{∗}  ( |)⊗id↓  ȷ∘p^{∗} ∘(ȷ⊗ȷ) ↓
V_{θ}  =  V  =  V_{θ^{2}}

Proof. Again it suffices to check the maps on the element x = (x_{1} ⊗x_{−1} +x_{−1} ⊗ x_{1}) ⊗ x_{1}, and for it we have

(ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ)) ∘ (id ⊗ p^{∗})(x) = ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ)(x_{1} ⊗ (−q^{\frac{3}{2}} )x_{4})
= −ȷ(x_{1}) = x_{1} = ( |) ⊗ id(x), while

p^{∗} ∘ (id ⊗ (ȷ ∘ p^{∗} ∘ (ȷ ⊗ ȷ)))(x) = p^{∗}(x_{1} ⊗ q^{\frac{3}{2}} x_{−4}) = x_{1} = ( |) ⊗ id(x). □

§6. Idempotents and Derivations of Quantum Octonions

It is apparent from Table 3.9 that e_{1} = q^{− \frac{1}{2}} x_{4} and e_{2} = −q^{\frac{1}{2}} x_{−4} are orthogonal idempotents in the quantum octonion algebra 𝕆_{q} = (V, ·). Their sum e = e_{1} + e_{2} acts as a left and right identity element on the span of e_{1}, e_{2}, x_{j}, x_{−j}, j = 2, 3. Moreover, the relations e · x_{1} = q^{2}x_{1}, x_{1} · e = q^{−2}x_{1}, e · x_{−1} = q^{−2}x_{−1}, and x_{−1} · e = q^{2}x_{−1} hold in 𝕆_{q}. Hence, at q = 1 or −1, the element e is an identity element. Relative to the basis e_{1}, e_{2}, x_{±j}, j = 1, 2, 3, the left and right multiplication operators L_{e_{1}}, R_{e_{1}}, L_{e_{2}}, R_{e_{2}} can be simultaneously diagonalized. The operators L_{e_{1}}, R_{e_{1}} determine a Peirce decomposition of 𝕆_{q},

𝕆_{q} = ⊕_{(γ,δ)}(𝕆_{q})_{γ,δ} where (𝕆_{q})_{γ,δ} = {v ∈ 𝕆_{q} | e_{1} · v = γv, and v · e_{1} = δv}. (6.1)

Now (𝕆_{q})_{0,q^{−2}} = Kx_{1}, (𝕆_{q})_{1,0} = Kx_{2} + Kx_{3}, (𝕆_{q})_{1,1} = Ke_{1}, (𝕆_{q})_{0,0} = Ke_{2}, (𝕆_{q})_{0,1} = Kx_{−2} + Kx_{−3}, and (𝕆_{q})_{q^{−2},0} = Kx_{−1}. The idempotent e_{2} gives a similar Peirce decomposition.

Suppose d is a derivation of 𝕆_{q} such that d(e_{1}) = 0 = d(e_{2}). It is easy to see that d must map each Peirce space (𝕆_{q})_{γ,δ} into itself. Thus, we may suppose that

\begin{matrix}d(x_{1}) = b_{1}x_{1}  d(x_{−1}) = b_{−1}x_{−1} \\ d(x_{2}) = c_{2,2}x_{2} + c_{3,2}x_{3}  d(x_{3}) = c_{2,3}x_{2} + c_{3,3}x_{3} \\ d(x_{−2}) = d_{2,2}x_{−2} + d_{3,2}x_{−3}  d(x_{−3}) = d_{2,3}x_{−2} + d_{3,3}x_{−3}. \end{matrix}  (6.2)

Applying d to the relation x_{1}·x_{−1} = q^{− \frac{3}{2}} x_{−4} = −q^{−2}e_{2} (or to the relation x_{−1}·x_{1} = −q^{\frac{3}{2}} x_{4} = −q^{2}e_{1}) shows that b_{−1} = −b_{1}. Now from x_{k} · x_{k} = 0, we deduce that c_{j,k} = 0 for j ≠ k, and similarly from x_{−k} · x_{−k} = 0 it follows that d_{j,k} = 0. There are relations x_{1} · x_{−j} = ξx_{k}, x_{−j} · x_{1} = ξx_{k}, x_{−1} · x_{j} = ξx_{−k}, and x_{j} · x_{−1} = ξx_{−k} (where j ≠ k and ξ in each one indicates some appropriate scalar that can be found in Table 3.9), and applying d to those equations gives b_{1} + d_{j,j} = c_{k,k} and b_{−1} + c_{j,j} = d_{k,k}. The second is equivalent to the first since b_{−1} = −b_{1}. From x_{±j} · x_{±k} = ξx_{±1} we see that c_{2,2} + c_{3,3} = b_{1} and d_{2,2} + d_{3,3} = −b_{1}. Finally, applying d to x_{j} · x_{−j} = ξe_{1} or x_{−j} · x_{j} = ξe_{2}, we obtain d_{j,j} = −c_{j,j} for j = 2, 3. Every nonzero product between basis elements in the quantum octonions is one of these types, or it is one involving the idempotents e_{1}, e_{2}, so we have determined all the possible relations. Consequently, we can conclude that for any derivation d such that d(e_{1}) = 0 = d(e_{2}), there are scalars b = b_{1} and c = c_{2,2} so that the following hold:

\begin{matrix}d(e_{1}) = 0 = d(e_{2}), \\ d(x_{1}) = bx_{1}  d(x_{−1}) = −bx_{−1} \\ d(x_{2}) = cx_{2}  d(x_{−2}) = −cx_{−2} \\ d(x_{3}) = (b − c)x_{3}  d(x_{−3}) = −(b − c)x_{−3}. \end{matrix}  (6.3)

The same calculations show that any transformation defined by (6.3) where b, c are arbitrary scalars is a derivation of 𝕆_{q}.

Now suppose that d is an arbitrary derivation of 𝕆_{q}, and let d(e_{1}) = a_{0}e_{1} + a′_{0}e_{2} + ∑_{j=−3,j≠0}^{3} a_{j}x_{j} and d(e_{2}) = b_{0}e_{1} + b′_{0}e_{2} + ∑_{j=−3,j≠0}^{3} b_{j}x_{j}. Computing d(e_{1}) · e_{1} + e_{1} · d(e_{1}) = d(e_{1}), we obtain

2a_{0}e_{1} + q^{−2}a_{1}x_{1} + a_{2}x_{2} + a_{3}x_{3} + q^{−2}a_{−1}x_{−1} + a_{−2}x_{−2} + a_{−3}x_{−3} = d(e_{1}),

which shows that a_{0} = a′_{0} = a_{−1} = a_{1} = 0. A similar calculation with the relation d(e_{2}^{2}) = d(e_{2}) gives b_{0} = b′_{0} = b_{−1} = b_{1} = 0. From d(e_{1})·e_{2} +e_{1} ·d(e_{2}) = 0 it follows that b_{2} = −a_{2} and b_{3} = −a_{3}, and from d(e_{2}) · e_{1} + e_{2} · d(e_{1}) = 0, the relations b_{−2} = −a_{−2}, b_{−3} = −a_{−3} can be determined.

Now e_{1} · d(x_{1}) = −d(e_{1}) · x_{1} shows that the coefficients of x_{2} and x_{3} in d(x_{1}) are q^{\frac{3}{2}} a_{−3} and q^{\frac{3}{2}} a_{−2} respectively; while d(x_{1}) · e_{2} = −x_{1} · d(e_{2}) gives that those coefficients are −q^{− \frac{3}{2}} b_{−3} = q^{− \frac{3}{2}} a_{−3} and −q^{− \frac{3}{2}} b_{−2} = q^{− \frac{3}{2}} a_{−2}. Thus, when q is not a cube root of unity, we obtain that a_{−2} = 0 = a_{−3}. Likewise the relations a_{2} = 0 = a_{3} can be found from analyzing d(x_{−1})·e_{1} = −x_{−1}d(e_{1}) and e_{2} ·d(x_{−1}) = −d(e_{2}) · x_{−1}. Consequently, when q is not a cube root of unity, every derivation of 𝕆_{q} must annihilate e_{1} and e_{2}, and so must be as in (6.3) for suitable scalars b, c. To summarize we have

Theorem 6.4. The derivation algebra of the quantum octonion algebra 𝕆_{q} = (V, ·) is two-dimensional. Every derivation d of 𝕆_{q} is given by (6.3) for some scalars b, c ∈ K.

Simplicity of the quantum octonions.

The idempotents also provide a proof of the following

Theorem 6.5. The quantum octonion algebra 𝕆_{q} = (V, ·) is simple.

Proof. Let I be a nonzero ideal of 𝕆_{q}. Since e_{1}I ⊆ I and Ie_{1} ⊆ I, it follows that

I = ⊕_{γ,δ}I_{γ,δ},

where I_{γ,δ} = {x ∈ I | e_{1} · x = γx, and x · e_{1} = δx}. Assume first x_{i} ∈ I for some i = {±1, ±2, ±3, ±4}. If we multiply x_{i} by x_{−i} when i = ±1, ±2, or ±3, we see that e_{1} or e_{2} belongs to I. Therefore, either e_{1}𝕆_{q} + 𝕆_{q}e_{1} ⊆ I or e_{2}𝕆_{q} + 𝕆_{q}e_{2} ⊆ I, and as a consequence, I = 𝕆_{q}. It remains to consider the possibility that some nonzero linear combination ax_{2} + bx_{3} or ax_{−2} + bx_{−3} is in I. Multiplying by one of the elements x_{2}, x_{3}, x_{−2}, or x_{−3}, we obtain that x_{1} or x_{−1} is in I, and that implies I = 𝕆_{q} as before. Consequently, 𝕆_{q} is simple. □

§7. Quantum Quaternions

The subalgebra ℍ_{q} of 𝕆_{q} generated by x = x_{2} and y = x_{−2} is 4-dimensional and has x, y, e_{1}, e_{2} as a basis, where e_{1} = q^{− \frac{1}{2}} x_{4} and e_{2} = −q^{\frac{1}{2}} x_{−4} are the orthogonal idempotents of Section 6. The sum e = e_{1} + e_{2} is the identity element of ℍ_{q}, and multiplication in this algebra is given by the following:

Table 7.1 ; \begin{matrix}· & e_{1} & e_{2} & x & y \\ e_{1} & e_{1} & 0 & x & 0 \\ e_{2} & 0 & e_{2} & 0 & y \\ x & 0 & x & 0 & q^{−1}e_{1} \\ y & y & 0 & qe_{2} & 0 \end{matrix}

It is easy to see that at q = 1 the algebra above is isomorphic to the algebra M_{2}(K) of 2 × 2 matrices over K via the mapping e_{1} ↦ E_{1,1}, e_{2} ↦ E_{2,2}, x ↦ E_{1,2}, y ↦ E_{2,1}, which sends basis elements to standard matrix units. Since the split quaternion algebra is isomorphic to M_{2}(K), this algebra reduces to the quaternions at q = 1. Another way to see this is to observe that the bilinear form on 𝕆_{q}, when restricted to ℍ_{q}, satisfies (3.6) at q = 1. Thus, at q = 1 we have a 4-dimensional unital algebra with a nondegenerate bilinear form admitting composition, and so it must be the quaternions.

The algebra ℍ_{q} is neither associative nor flexible if q ≠ ±1 since

(x · y) · x = q^{−1}x ≠ qx = x · (y · x).

However, it is Lie admissible. Indeed, if h = q^{−1}e_{1} − qe_{2}, then

[h, x] = (q + q^{−1})x, [h, y] = −(q + q^{−1})y, and [x, y] = h.

Therefore, the elements x, \frac{2}{q + q^{−1}} y, \frac{2}{q + q^{−1}} h determine a standard basis for sl_{2}.

Since e = e_{1} + e_{2} is the identity, we see that ℍ_{q} is the Lie algebra gl_{2} under the commutator product [a, b] = ab − ba.

The subalgebra generated by x_{3} and x_{−3} is isomorphic to the algebra ℍ_{q}. However, that is not true if we consider the subalgebra ℍ′_{q} generated by x_{1} and x_{−1}. Let u = x_{−1} and v = x_{1}. Then multiplication in ℍ′_{q} is given by

Table 7.2 ; \begin{matrix}· & e_{1} & e_{2} & u & v \\ e_{1} & e_{1} & 0 & q^{−2}u & 0 \\ e_{2} & 0 & e_{2} & 0 & q^{2}v \\ u & 0 & q^{2}u & 0 & q^{2}e_{1} \\ v & q^{−2}v & 0 & q^{−2}e_{2} & 0 \end{matrix}

There is no identity element in ℍ′_{q}. But at q = 1, the element e_{1} + e_{2} becomes an identity element, and the algebra is isomorphic to M_{2}(K). The bilinear form at q = 1 has the composition property, so the algebra ℍ′_{q} too might rightfully be termed a quantum quaternion algebra. The elements h′ = q^{2}e_{1} − q^{−2}e_{2}, u, v determine a standard basis for sl_{2}:

[h′, u] = 2u,  [h′, v] = −2v,  [u, v] = h′.

Since [q^{2}e_{1} + q^{−2}e_{2}, h′] = [q^{2}e_{1} + q^{−2}e_{2}, u] = [q^{2}e_{1} + q^{−2}e_{2}, v] = 0, the algebra ℍ′_{q} under the commutator product is isomorphic to gl_{2}. It is neither associative nor flexible because

((e_{1} + v) · e_{1}) · (e_{1} + v) = e_{1} + q^{−4}v ≠ e_{1} + q^{−2}v = (e_{1} + v) · (e_{1} · (e_{1} + v)).

§8. Connections with U_{q}(B_{3}) and U_{q}(G_{2})

In this section we assume that the field K has characteristic zero. We consider the subalgebras of U_{q}(D_{4}) that are described as the fixed points of graph automorphisms by

\begin{aligned}U_{ζ,η} & = \{a ∈ U_{q}(D_{4}) | ζ(a) = a = η(a)\} \\ U_{δ} & = \{a ∈ U_{q}(D_{4}) | δ(a) = a\}, \end{aligned}  (8.1)

where δ = (3 4) = ζηζ. Then U_{δ} ⊇ U_{ζ,η}. In the non-quantum case, the enveloping algebra U(G_{2}) ⊆ U_{ζ,η} ⊆ U(D_{4}) and U(B_{3}) ⊆ U_{δ} ⊆ U(D_{4}). We consider the structure of V, V_{ζ}, V_{η} as modules for the subalgebras given in (8.1).

Now for any a ∈ U_{δ}, we have that

ȷ(ax) = ȷ(δ(a)x) = aȷ(x)

for all x ∈ V so that any a ∈ U_{δ} commutes with ȷ. In particular, the eigenspaces of ȷ are U_{δ}-invariant. But the eigenspaces of ȷ are 1-dimensional (spanned by x_{4} −x_{−4}), and 7-dimensional (spanned by {x_{i} | i = ±1, ±2, ±3} ∪ {x_{4} + x_{−4}}). Consider the 7-dimensional space under the action of U_{ζ,η}. Since F_{α_{2}}, F_{α_{1}} +F_{α_{3}} +F_{α_{4}}, E_{α_{2}}, E_{α_{1}} + E_{α_{3}} + E_{α_{4}} ∈ U_{ζ,η}, it is easy to check that the 7-dimensional space is an irreducible U_{ζ,η}-module, hence an irreducible U_{δ}-module as well. The action of U_{ζ,η} on V_{ζ} and V_{η} is the same as its action on V. As a result we have

Proposition 8.2. The modules V, V_{ζ} ≅ V_{−}, and V_{η} ≅ V_{+} are isomorphic as modules for the subalgebra U_{ζ,η} in (8.1). They decompose into the sum of irreducible U_{ζ,η}-modules of dimensions 1 and 7, which are the eigenspaces of ȷ.

For q = 1, Proposition 8.2 amounts to saying that V, V_{ζ}, V_{η} are isomorphic U(G_{2})-modules which decompose into irreducible modules of dimension 1 and 7.

The elements F_{α_{1}}, F_{α_{2}}, F_{α_{3}} + F_{α_{4}}, E_{α_{1}}, E_{α_{2}}, E_{α_{3}} + E_{α_{4}} belong to U_{δ}, so if we regard V_{ζ} or V_{η} as a module for U_{δ}, then we obtain a diagram

x_{4}

↗^{F_{α_{1}}}

x_{1} →^{F_{α_{3}} +F_{α_{4}}} x_{2} →^{F_{α_{2}}} x_{3} …

↘^{F_{α_{3}} +F_{α_{4}}}

x_{−4}

which is gotten by superimposing the two diagrams of V_{ζ} and V_{η}. Using that, it is easy to argue that both these modules remain irreducible for U_{δ}. Moreover, for any a ∈ U_{δ}, we have

\begin{aligned}ȷ(ζ(a)x) & = ȷ((ζδ)(a)x) = ȷ((ηζ)(a)x) \\ & = ȷ(θ^{2}(a)x) = η(a)ȷ(x) \end{aligned}

where θ = (1 4 3). Consequently, V_{ζ} ≅ V_{η} as U_{δ}-modules.

At q = 1 these computations show that V_{ζ} and V_{η} (the two spin modules of U(D_{4})) remain irreducible for U(B_{3}), and in fact they are the spin representation of that algebra. To summarize we have

Proposition 8.3. When regarded as modules for the subalgebra U_{δ} of U_{q}(D_{4}), V_{ζ} ≅ V_{−}, and V_{η} ≅ V_{+} are irreducible and isomorphic.

In light of these propositions it is natural to expect that these fixed point subalgebras of the graph automorphisms might be the quantum groups U_{q}(G_{2}) and U_{q}(B_{3}). However, that is not true. In fact the next “nonembedding” result shows that there are no Hopf algebra homomorphisms of them into U_{q}(D_{4}) except for the trivial ones given by the counit.

Proposition 8.4. When q is not a root of unity, then any Hopf algebra homomorphisms

U_{q}(G_{2}) → U_{q}(B_{3}), U_{q}(B_{3}) → U_{q}(D_{4}), U_{q}(G_{2}) → U_{q}(D_{4})

are trivial (a ↦ ϵ(a)1 for all a).

The method we use to establish this result is due originally to Hayashi [H]. We begin with a little background before presenting the proof.

Suppose U_{q}(𝔤) is a quantum group and (X, π), π : U_{q}(𝔤) → End(X), is a representation of U_{q}(𝔤). This determines another representation (X, π′), π′ = π ∘ S^{2}, of U_{q}(𝔤) where S is the antipode. But it is well-known [Jn, p. 56] that S^{2} has the following expression:

S^{2}(a) = K_{2ρ}^{−1}aK_{2ρ} for all a ∈ U_{q}(𝔤)

where ρ = 1/2 ∑_{γ>0} γ = ∑_{α∈Π} c_{α}α (the half-sum of the positive roots of 𝔤) and K_{2ρ} = ∏_{α∈Π} K_{α}^{c_{α}} (where Π is the set of simple roots of 𝔤). The two representations (X, π) and (X, π′) are in fact isomorphic by the following map

ϕ : (X, π) → (X, π′), x ↦ π(K_{2ρ}^{−1})x

because ϕ(π(a)x) = π(K_{2ρ}^{−1})π(a)x = π(K_{2ρ}^{−1}a)x = π(S^{2}(a)K_{2ρ}^{−1})x = π(S^{2}(a))π(K_{2ρ}^{−1})x = π′(a)ϕ(x) for all a ∈ U_{q}(𝔤).

Examples 8.5. Let 𝔤 = D_{4} and let X = V, V_{−}, V_{+} (the natural or spin representations of U_{q}(D_{4})). Here ρ = 3α_{1} + 5α_{2} + 3α_{3} + 3α_{4} so K_{2ρ}^{−1} = K_{α_{1}}^{−6}K_{α_{2}}^{−10}K_{α_{3}}^{−6}K_{α_{4}}^{−6}. Relative to the basis {x_{1}, x_{2}, x_{3}, x_{4}, x_{−4}, x_{−3}, x_{−2}, x_{−1}} we have as before

ϕ_{D_{4}} = ϕ_{D_{4}}^{−} = ϕ_{D_{4}}^{+} = diag{q^{−6}, q^{−4}, q^{−2}, 1, 1, q^{2}, q^{4}, q^{6}}.  (8.6)

Next let 𝔤 = G_{2} and let X be the 7-dimensional irreducible U_{q}(G_{2})-representation. Here ρ = 5α_{1} + 3α_{2} so K_{2ρ}^{−1} = K_{α_{1}}^{−10}K_{α_{2}}^{−6}. With respect to the following basis (see [Jn], Chap. 5),

{x_{2α_{1}+α_{2}}, x_{α_{1}+α_{2}}, x_{α_{1}}, x_{0}, x_{−α_{1}}, x_{−(α_{1}+α_{2})}, x_{−(2α_{1}+α_{2})}},  we have  (8.7)

ϕ_{G_{2}} = diag{q^{−10}, q^{−8}, q^{−2}, 1, q^{2}, q^{8}, q^{10}}.  (8.8)

Finally, assume X is the 7-dimensional natural representation or the 8-dimensional spin representation of U_{q}(B_{3}). In this case 2ρ = 5α_{1} + 8α_{2} + 9α_{3} so that K_{2ρ}^{−1} = K_{α_{1}}^{−5}K_{α_{2}}^{−8}K_{α_{3}}^{−9}. If we assume as in [Jn] that a short root satisfies (α, α) = 2, then setting (ϵ_{i}, ϵ_{i}) = 2 for i = 1, 2, 3, we have that the matrix of the bilinear form relative to the basis α_{1} = ϵ_{1} − ϵ_{2}, α_{2} = ϵ_{2} − ϵ_{3}, α_{3} = ϵ_{3} of simple roots is

( \begin{matrix}4 & −2 & 0 \\ −2 & 4 & −2 \\ 0 & −2 & 2 \end{matrix} ) .

From this we can compute that (−2ρ, ϵ_{i}) = −10, −6, −2 for i = 1, 2, 3, respectively. Since we know that for any weight vector x_{μ} that K_{α}x_{μ} = q^{(μ,α)}x_{μ}, we obtain that relative to the basis

{x_{ϵ_{1}}, x_{ϵ_{2}}, x_{ϵ_{3}}, x_{0}, x_{−ϵ_{3}}, x_{−ϵ_{2}}, x_{−ϵ_{1}}},  (8.9)

for the natural representation,

ϕ_{B_{3}} = diag{q^{−10}, q^{−6}, q^{−2}, 1, q^{2}, q^{6}, q^{10}}.  (8.10)

The spin representation of U_{q}(B_{3}) has a basis

{x_{ξ_{1}}, x_{ξ_{2}}, x_{ξ_{3}}, x_{ξ_{4}}, x_{−ξ_{4}}, x_{−ξ_{3}}, x_{−ξ_{2}}, x_{−ξ_{1}}},

where ξ_{1} = (1/2)(ϵ_{1} + ϵ_{2} + ϵ_{3}), ξ_{2} = (1/2)(ϵ_{1} + ϵ_{2} − ϵ_{3}), ξ_{3} = (1/2)(ϵ_{1} − ϵ_{2} + ϵ_{3}), and ξ_{4} = (1/2)(−ϵ_{1} + ϵ_{2} + ϵ_{3}). The corresponding matrix of ϕ_{B_{3}}^{+} is given by

ϕ_{B_{3}}^{+} = diag{q^{−9}, q^{−7}, q^{−3}, q^{−1}, q, q^{3}, q^{7}, q^{9}}.

Now suppose that f : U_{q}(𝔤_{1}) → U_{q}(𝔤_{2}) is a Hopf algebra homomorphism between two quantum groups. Consider representations (X, π) and (X, π′) for U_{q}(𝔤_{2}) as above. Then (X, π ∘ f), (X, π′ ∘ f) are representations for U_{q}(𝔤_{1}). Since π′ ∘ f = π ∘ S^{2} ∘ f = π ∘ f ∘ S^{2}, it follows that π′ ∘ f = (π ∘ f)′. Moreover, the map

ϕ_{𝔤_{2}} : (X, π) → (X, π′)

satisfies

ϕ_{𝔤_{2}}((π ∘ f)(a)x) = π′(f(a))ϕ_{𝔤_{2}}(x) = (π ∘ f)′(a)ϕ_{𝔤_{2}}(x)

for all a ∈ U_{q}(𝔤_{1}) and x ∈ X so it is an isomorphism as U_{q}(𝔤_{1})-modules: ϕ_{𝔤_{2}} : (X, π ∘ f) → (X, (π ∘ f)′).

Proof of Proposition 8.4. Suppose f : U_{q}(G_{2}) → U_{q}(B_{3}) is a Hopf algebra homomorphism. Consider the natural representation Y = (X, π) of U_{q}(B_{3}), and the corresponding representation

ϕ_{B_{3}} : (X, π ∘ f) → (X, (π ∘ f)′)  (8.11)

of U_{q}(G_{2})-modules. Either these are the 7-dimensional irreducible U_{q}(G_{2})-module or they are the sum of 7 trivial 1-dimensional modules. In the first case ϕ_{B_{3}} is a multiple of ϕ_{G_{2}}, so the eigenvalues of ϕ_{B_{3}} are {λq^{±10}, λq^{±8}, λq^{±2}, λ}. This is impossible since q is not a root of unity, so it must be that π ∘ f is trivial.

Analogously, consider the spin representation Y^{+} = (X^{+}, π^{+}) of U_{q}(B_{3}). Then as a U_{q}(G_{2})-module, such a representation should be the sum of representations of dimension 1 or 7. Reasoning as in (8.11), we see that if a 7-dimensional U_{q}(G_{2})- module occurs, then ϕ_{B_{3}}^{+} restricted to the 7-dimensional submodule should be proportional to ϕ_{G_{2}}. This would mean that

{λq^{±10}, λq^{±8}, λq^{±2}, λ} ⊆ diag{q^{±9}, q^{±7}, q^{±3}, q^{±1}}

for some scalar λ, which is impossible. Thus π^{+} ∘ f is trivial.

We have seen that the natural and spin representations for U_{q}(B_{3}) are trivial modules for f(U_{q}(G_{2})). Thus, f(a) − ϵ(f(a))1 is in the annihilator of those representations. Since any finite-dimensional irreducible U_{q}(B_{3})-representation of type I can be obtained from the tensor product of those, and since f is a Hopf homomorphism, it follows that f(U_{q}(G_{2})) acts trivially on any finite-dimensional irreducible U_{q}(B_{3})-representation of type I. However, ∩ ann(V) = (0) for all such representations (see [Jn, Prop. 5.11]). Consequently, f(a) = ϵ(f(a))1 for all a ∈ U_{q}(G_{2}).

The other two cases in Proposition 8.4 are similar. For a Hopf homomorphism f : U_{q}(G_{2}) → U_{q}(D_{4}), it boils down to the fact that any finite-dimensional irreducible U_{q}(D_{4})-representation occurs in tensor products of copies of the natural or spin representations and

{λq^{±10}, λq^{±8}, λq^{±2}, λ} ⊈ {q^{±6}, q^{±4}, q^{±2}, 1}.

In the final case, f : U_{q}(B_{3}) → U_{q}(D_{4}), the argument amounts to the impossibility of

{λq^{±10}, λq^{±6}, λq^{±2}, λ} ⊆ {q^{±6}, q^{±4}, q^{±2}, 1}, and

{λq^{±9}, λq^{±7}, λq^{±3}, λq^{±1}} ⊆ {q^{±6}, q^{±4}, q^{±2}, 1}. □

§9. Quantum Octonions and Quantum Clifford Algebras

In this section we investigate the representation of the quantum Clifford algebra C_{q}(8) on 𝕆_{q} ⊕ 𝕆_{q}. Our main reference for the quantum Clifford algebra is [DF], but we need to make several adjustments to make the results of [DF] conform to our work here. It is convenient to begin with the general setting of the quantum group of D_{n} and then to specialize to D_{4}.

A QUANTUM OCTONION ALGEBRA 37

Explicit form of Ř_{V,V} for U_{q}(D_{n}).

The natural representation V of the quantum group U_{q}(D_{n}) has a basis

{x_{1}, x_{2}, …, x_{2n}} = {x_{ϵ_{1}}, …, x_{ϵ_{n}}, x_{−ϵ_{n}}, …, x_{−ϵ_{1}}},  (9.1)

where E_{α}, F_{α}, K_{α} act as in (2.2). A straightforward computation shows that relative to the basis {x_{1}, x_{2}, …, x_{2n}}, the matrix of Ř = Ř_{V,V} is given by

\begin{aligned}Ř & = q^{−1} ∑_{i=1}^{2n} E_{i,i} ⊗ E_{i,i} + ∑_{i≠j,j′} E_{j,i} ⊗ E_{i,j} + (q^{−1} − q) ∑_{j>i} E_{j,j} ⊗ E_{i,i} \\ & + q ∑_{i=1}^{2n} E_{i′,i} ⊗ E_{i,i′} + (q − q^{−1}) ∑_{i<j}(−q)^{ρ_{j′} −ρ_{i′}} E_{i′,j} ⊗ E_{i,j′}, \end{aligned}  (9.2)

where i′ = 2n + 1 − i and ρ = (ρ_{1}, …, ρ_{2n}) = (n − 1, n − 2, …, 1, 0, 0, −1, …, 2 − n, 1 − n). The projection ( , ) : V ⊗ V → K is given by

(x_{i}, x_{j}) = δ_{i,j′}(−1)^{n−1}(−q)^{ρ_{i′}} .

If we modify the basis slightly by defining

\begin{matrix}e_{i} = & (−1)^{n−1+ρ_{i′}} x_{i} \\ e_{i′} = & x_{i′} \end{matrix}  (9.3)

for i = 1, …, n, then we obtain the usual expression for Ř:

\begin{aligned}Ř & = q^{−1} ∑_{i=1}^{2n} E_{i,i} ⊗ E_{i,i} + ∑_{i≠j,j′} E_{j,i} ⊗ E_{i,j} + (q^{−1} − q) ∑_{j>i} E_{j,j} ⊗ E_{i,i} \\ & + q ∑_{i=1}^{2n} E_{i′,i} ⊗ E_{i,i′} + (q − q^{−1}) ∑_{i<j} q^{(ρ_{j′} −ρ_{i′})}E_{i′,j} ⊗ E_{i,j′}, \end{aligned}  (9.4)

and the bilinear form is given by

(e_{i}, e_{j}) = δ_{i,j′}q^{ρ_{i′}} .

The quantum Clifford algebra.

In [DF], Ding and Frenkel define the quantum Clifford algebra C_{q}(2n) as the quotient of the tensor algebra T (V) of V by the ideal generated by {v ⊗ w + qŘ′_{2,1}(v ⊗ w) − (w, v)1 | v, w ∈ V } where Ř′ is Ř with q and q^{−1} interchanged, and Ř′_{2,1} = σ ∘ Ř′ ∘ σ where σ(v ⊗ w) = (w ⊗ v). This quantum Clifford algebra can be thought of as a unital associative algebra generated by ψ_{1}, …, ψ_{n}, ψ_{1}^{∗}, …, ψ_{n}^{∗} with relations,

\begin{matrix}ψ_{i}ψ_{j} = −q^{−1}ψ_{j}ψ_{i}  ψ_{i}^{∗}ψ_{j}^{∗} = −qψ_{j}^{∗}ψ_{i}^{∗} (i > j) \\ ψ_{i}ψ_{i} = 0 = ψ_{i}^{∗}ψ_{i}^{∗}  ψ_{i}ψ_{j}^{∗} = −q^{−1}ψ_{j}^{∗}ψ_{i} (i ≠ j) \\ ψ_{i}ψ_{i}^{∗} + ψ_{i}^{∗}ψ_{i} = (q^{−2} − 1) ∑_{i<j} ψ_{j}ψ_{j}^{∗} + 1. \end{matrix}  (9.5)

If we make the change of notation Y_{i} = ψ_{n−i}^{∗}, Y_{i}^{∗} = ψ_{n−i}, i = 1, …, n, then the relations in (9.5) become:

\begin{matrix}Y_{i}Y_{j} = −q^{−1}Y_{j}Y_{i}  Y_{i}^{∗}Y_{j}^{∗} = −qY_{j}^{∗}Y_{i}^{∗} (i > j) \\ Y_{i}Y_{i} = 0 = Y_{i}^{∗}Y_{i}^{∗}  Y_{i}Y_{j}^{∗} = −qY_{j}^{∗}Y_{i} (i ≠ j) \\ Y_{i}Y_{i}^{∗} + Y_{i}^{∗}Y_{i} = (q^{−2} − 1) ∑_{i>j} Y_{j}Y_{j}^{∗} + 1. \end{matrix}  (9.6)

In order to construct a representation of C_{q}(8) on 𝕆_{q} ⊕ 𝕆_{q} we will need the following proposition:

Proposition 9.7. Let I be the ideal of the tensor algebra T (V) generated by {v ⊗ w + q^{−1}Ř_{2,1}(v ⊗ w) − (w, v)1 | v, w ∈ V } where Ř_{2,1} = σ ∘ Ř ∘ σ. Then C_{q}(2n) ≅ T (V)/I.

Proof. Consider the following basis of V:

{y_{i} = q^{ρ_{i}}e_{i}, y_{i}^{∗} = e_{i′}, | i = 1, …, n}  (9.8)

which satisfies

(y_{i}, y_{j}^{∗}) = δ_{i,j}  (y_{j}^{∗}, y_{i}) = δ_{i,j}q^{2ρ_{i}}.  (9.9)

Let g : V ⊗ V → I be defined by

g(v ⊗ w) = v ⊗ w + q^{−1}Ř_{2,1}(v ⊗ w) − (w, v)1.  (9.10)

From the explicit form of Ř on the basis {e_{i}}, one can check that the following hold:

\begin{matrix}g(y_{i} ⊗ y_{j}) = y_{i} ⊗ y_{j} + q^{−1}y_{j} ⊗ y_{i} (i > j) \\ g(y_{j}^{∗} ⊗ y_{i}^{∗}) = y_{j}^{∗} ⊗ y_{i}^{∗} + q^{−1}y_{i}^{∗} ⊗ y_{j}^{∗} (i > j) \\ g(y_{i} ⊗ y_{i}) = (1 + q^{−2})y_{i} ⊗ y_{i}  g(y_{i}^{∗} ⊗ y_{i}^{∗}) = (1 + q^{−2})y_{i}^{∗} ⊗ y_{i}^{∗} \\ g(y_{j}^{∗} ⊗ y_{i}) = y_{j}^{∗} ⊗ y_{i} + q^{−1}y_{i} ⊗ y_{j}^{∗} (i ≠ j) \\ g(y_{i}^{∗} ⊗ y_{i}) = y_{i}^{∗} ⊗ y_{i} + y_{i} ⊗ y_{i}^{∗} + (1 − q^{−2}) ∑_{i>j} y_{j} ⊗ y_{j}^{∗} − 1, \end{matrix}  (9.11)

These give the relations in (9.6) in T (V)/I. However, we also have the relations:

\begin{matrix}g(y_{j} ⊗ y_{i}) = q^{−1}y_{i} ⊗ y_{j} + q^{−2}y_{j} ⊗ y_{i} (i > j) \\ g(y_{i}^{∗} ⊗ y_{j}^{∗}) = q^{−1}y_{j}^{∗} ⊗ y_{i}^{∗} + q^{−2}y_{i}^{∗} ⊗ y_{j}^{∗} (i > j) \\ g(y_{i} ⊗ y_{j}^{∗}) = q^{−1}y_{j}^{∗} ⊗ y_{i} + q^{−2}y_{i} ⊗ y_{j}^{∗} (i ≠ j) \\ g(y_{i} ⊗ y_{i}^{∗}) = y_{i}^{∗} ⊗ y_{i} + q^{−2}y_{i} ⊗ y_{i}^{∗} \\ + (1 − q^{−2})q^{2ρ_{i}}( ∑_{j=1}^{n} y_{j} ⊗ y_{j}^{∗} + ∑_{j>i} q^{−2ρ_{j}}y_{j}^{∗} ⊗ y_{j}) − q^{2ρ_{i}}. \end{matrix} (9.12)

The first three identities don’t say anything new. So to conclude the proposition, it suffices to verify that the relation g(y_{i} ⊗ y_{i}^{∗}) = 0 holds in C_{q}(2n), that is:

Lemma 9.13. In C_{q}(2n) we have

q^{−2}Y_{i}Y_{i}^{∗} + Y_{i}^{∗}Y_{i} + (1 − q^{−2})q^{2ρ_{i}}( ∑_{j=1}^{n} Y_{j}Y_{j}^{∗} + ∑_{j>i} q^{−2ρ_{j}}Y_{j}^{∗}Y_{j}) = q^{2ρ_{i}}.

Proof. As a shorthand notation we set a_{i} = Y_{i}Y_{i}^{∗}, a′_{i} = Y_{i}^{∗}Y_{i}, and b_{i} = a_{i}+a′_{i} for i = 1, …, n. From the relations in C_{q}(2n) we know that b_{i} = (q^{−2} − 1) ∑_{j<i} a_{j} + 1 so that b_{i+1} = (q^{−2} − 1)a_{i} + b_{i} or

a_{i+1} + a′_{i+1} = q^{−2}a_{i} + a′_{i}.  (9.14)

Letting c_{i} = q^{−2}a_{i} + a′_{i} we have

c_{i} − q^{2}c_{i+1} = q^{−2}a_{i} + a′_{i} − a_{i+1} − q^{2}a′_{i+1} =^{1} (1 − q^{2})a′_{i+1}

where (1) follows from (9.14). Therefore,

\begin{aligned}c_{i} & = (1 − q^{2})a′_{i+1} + q^{2}(1 − q^{2})a′_{i+2} + ⋯ + q^{2(n−i−1)}(1 − q^{2})a′_{n} + q^{2(n−i)}c_{n} \\ & = (q^{−2} − 1)q^{2ρ_{i}}( ∑_{j>i} q^{−2ρ_{j}}a′_{j}) + q^{2ρ_{i}}c_{n}. \end{aligned} (9.15)

On the other hand,

\begin{aligned}c_{n} & = q^{−2}a_{n} + a′_{n} = (q^{−2} − 1)a_{n} + a_{n} + a′_{n} = (q^{−2} − 1)a_{n} + b_{n} \\ & = (q^{−2} − 1)a_{n} + (q^{−2} − 1)( ∑_{j<n} a_{j}) + 1 = (q^{−2} − 1)( ∑_{j≤n} a_{j}) + 1. \end{aligned} (9.16)

The left-hand side of the identity in the statement of the lemma is

c_{i} + (1 − q^{−2})q^{−2ρ_{i}}( ∑_{j=1}^{n} a_{j}) + (1 − q^{−2})q^{2ρ_{i}}( ∑_{j>i} q^{−2ρ_{j}}a′_{j})

=^{1} c_{i} + q^{2ρ_{i}}(1 − c_{n}) + (1 − q^{−2})q^{2ρ_{i}}( ∑_{j>i} q^{−2ρ_{j}}a′_{j})

=^{2} q^{2ρ_{i}} (the right side),

where (1) is (9.16) and (2) is (9.15). □ The U_{q}(D_{4}) case.

Let us return now to our quantum octonions. Consider the two “intertwining operators” Φ^{0} and Φ^{1} which are the U_{q}(D_{4})-module homomorphisms:

\begin{matrix}Φ^{0} : V_{η} ⊗ V →^{ȷ⊗id} V_{η∘ζ} ⊗ V →^{ȷ∘p∘(ȷ⊗ȷ)} V_{ζ} \\ Φ^{1} : V_{ζ} ⊗ V →^{ȷ⊗id} V_{ζ∘η} ⊗ V →^{p} V_{η}, \end{matrix}

We scale these mappings by setting Ψ^{0} = q^{− \frac{3}{2}} Φ^{0} and Ψ^{1} = q^{− \frac{3}{2}} Φ^{1}. Let 𝔖_{q} = (id+q^{−1}Ř), where Ř = Ř_{V,V}, which we know maps V ⊗ V onto S^{2}(V). Lemma 9.17. (Compare [DF, Prop. 3.1.2])

Ψ^{1} ∘ (Ψ^{0} ⊗ id) ∘ (id ⊗ 𝔖_{q}) = id ⊗ ( , )

Ψ^{0} ∘ (Ψ^{1} ⊗ id) ∘ (id ⊗ 𝔖_{q}) = id ⊗ ( , ).

Proof. It is enough to check the values on x_{1} ⊗ x_{1} ⊗ x_{−1}:

Ψ^{1} ∘ (Ψ^{0} ⊗ id) ∘ (id ⊗ 𝔖_{q})(x_{1} ⊗ x_{1} ⊗ x_{−1}) = Ψ^{1} ∘ (Ψ^{0} ⊗ id)(x_{1} ⊗ (x_{1} ⊗ x_{−1} + x_{−1} ⊗ x_{1}))
= q^{− \frac{3}{2}} Ψ^{1}(q^{− \frac{3}{2}} x_{4} ⊗ x_{1}) = q^{−3}x_{1}.

But (x_{1}, x_{−1}) = (q^{3} + q^{−3})(x_{1}|x_{−1}) = q^{−3}, so both maps agree on x_{1} ⊗ x_{1} ⊗ x_{−1}. The computation for the other map is left as an exercise. □

Now for any v ∈ V define

\begin{matrix}Ψ(v) : & V_{ζ} ⊕ V_{η} → V_{ζ} ⊕ V_{η} \\ Ψ(v)(x + y) = Ψ^{0}(y ⊗ v) + Ψ^{1}(x ⊗ v). \end{matrix}  (9.18)

The relations in the previous lemma say that

Ψ ∘ (Ψ ⊗ id) ∘ (id ⊗ 𝔖_{q}) = id ⊗ ( , ),

so for any basis elements e_{i}, e_{j} ∈ V and x ∈ V_{ζ} ⊕ V_{η}, we have

(e_{j}, e_{i})x = Ψ ∘ (Ψ ⊗ id) ∘ (id ⊗ 𝔖_{q})(x ⊗ e_{j} ⊗ e_{i})

= Ψ ∘ (Ψ ⊗ id)(x ⊗ (e_{j} ⊗ e_{i} + q^{−1}Ř(e_{j} ⊗ e_{i})))

= Ψ(e_{i})Ψ(e_{j})(x) + q^{−1} ∑_{k,ℓ}(Ř ∘ σ)_{i,j}^{k,ℓ}Ψ(e_{ℓ})Ψ(e_{k})(x),

where (Ř ∘ σ)_{i,j}^{k,ℓ} is the coordinate matrix of Ř ∘ σ in the basis {e_{i} ⊗ e_{j}}. So Ψ(e_{i}) and Ψ(e_{j}) satisfy

(e_{j}, e_{i})id = Ψ(e_{i})Ψ(e_{j}) + q^{−1} ∑_{k,ℓ}(Ř ∘ σ)_{i,j}^{k,ℓ}Ψ(e_{ℓ})Ψ(e_{k}).

Notice now that

\begin{aligned}e_{i} ⊗ e_{j} + q^{−1}Ř_{2,1}(e_{i} ⊗ e_{j}) & = e_{i} ⊗ e_{j} + q^{−1}σ ∘ Ř ∘ σ(e_{i} ⊗ e_{j}) \\ & = e_{i} ⊗ e_{j} + q^{−1} ∑_{k,ℓ}(Ř ∘ σ)_{i,j}^{k,ℓ}e_{ℓ} ⊗ e_{k}, \end{aligned}

so the operators Ψ(v) verify the relations of the quantum Clifford algebra. This shows that the map

\begin{matrix}Ψ : V & → End(V_{ζ} ⊕ V_{η}) \\ v & ↦ Ψ(v) : x + y → q^{−3}(ȷ(y · ȷ(v)) + ȷ(x) · v) \end{matrix}  (9.19)

extends to a representation Ψ : C_{q}(8) → End(V_{ζ} ⊕ V_{η}) = End(𝕆_{q} ⊕ 𝕆_{q}).

In [DF], Ding and Frenkel show that C_{q}(2n) is isomorphic to the classical Clifford algebra, so C_{q}(2n) is a central simple associative algebra of dimension 2^{8}. The same is true of End(𝕆_{q} ⊕ 𝕆_{q}). As a result we have

Proposition 9.20. The map Ψ : C_{q}(8) → End(V_{ζ} ⊕V_{η}) = End(𝕆_{q} ⊕𝕆_{q}) in (9.19) is an algebra isomorphism.

It is worth noting that at q = 1 the map in Proposition 9.20 reduces to

\begin{matrix}Ψ : & 𝕆 → End(𝕆 ⊕ 𝕆) \\ v ↦ Ψ(v) : x + y ↦ v · \overline{y} + \overline{x} · v \end{matrix}  (9.21)

so

Ψ(v)Ψ(v) : x + y ↦ Ψ(v)(v · \overline{y} + \overline{x} · v) = v · (\overline{v} · x) + (y · \overline{v}) · v = (v|v)_{1}(x + y), (see Props. 2.21, 4.3). This shows that Ψ extends to the standard Clifford algebra - that is, to the quotient T (𝕆)/ < v ⊗ v − (v|v)_{1}1 > of the tensor algebra, in this case as in [KPS].

References

[B] J.C. Baez, R-commutative geometry and quantization of Poisson algebras, Adv. in Math. 95 (1992), 61–91.

[Br] M. Bremner, Quantum octonions, preprint (1997).

[CP] V. Chari and A.N. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge, 1994.

[DF] J.T. Ding and I.B. Frenkel, Spinor and oscillator representations of quantum groups; Lie Theory and Geometry, Progress in Math. Birkhäuser, Boston, 1994, pp. 127–165.

[D] V.G. Drinfel’d, On almost cocommutative algebras, Leningrad Math. Jour. 1 (1990), 321– 342.

[EM] A. Elduque and H.C. Myung, Flexible composition algebras and Okubo algebras, Comm. Algebra 19 (1991), 1197–1227.

[EP] A. Elduque and J.M. Pérez, Composition algebras with associative bilinear form, Comm. Algebra 24 (1996), 1091–1116.

[H] T. Hayashi, q-analogues of Clifford and Weyl algebras - spinor and oscillator representations of quantum enveloping algebras, Comm. Math. Phys. 127 (1990), 129–144.

[J] N. Jacobson, Exceptional Lie Algebras, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, 1971.

[Jn] J.C. Jantzen, Lectures on Quantum Groups, vol. 6, Graduate Studies in Math. Amer. Math. Soc., Providence, 1996.

[KPS] M.-A. Knus, R. Parimala, and R. Sridharan, On compositions and triality, J. Reine Angew. Math. 457 (1994), 45–70.

[M] Yu. I. Manin, Quantum Groups and Noncommutative Geometry, Université de Montréal, Centre de Recherches Mathématiques, Montréal, 1988.

[OO1] S. Okubo and J.M. Osborn, Algebras with nondegenerate associative symmmetric bilinear form permitting composition I, Comm. Algebra 9 (1981), 1233–1261.

[OO2] S. Okubo and J.M. Osborn, Algebras with nondegenerate associative symmmetric bilinear form permitting composition II, Comm. Algebra 9 (1981), 2015–2073.

[R] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I,II, LOMI Preprint E-4-87, E-17-87 (1987-88).

[S] R.D. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, 1966.

[ZSSS] K.A. Zhevlakov, A. M. Slin’ko, I.P. Shestakov, and A.I. Shirshov, Rings That Are Nearly Associative, Pure and Applied Mathematics vol. 104, Academic Press, Inc. Harcourt Brace Jovanovich, Publ., New York-London, 1982.

Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA

   E-mail address: benkart@math.wisc.edu, perez@math.wisc.edu