TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 352, Number 2, Pages 935{968 S 0002-9947(99)02415-0 Article electronically published on August 10, 1999 A QUANTUM OCTONION ALGEBRA GEORGIA BENKART AND JOSEM.P´ EREZ-IZQUIERDO´ To the memory of Alberto Izquierdo Abstract. Using the natural irreducible 8-dimensional representation and the two spin representations of the quantum group Uq(D4)ofD4,weconstructa quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Trial- ity and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R- matrix of Uq(D4). The product in the quantum octonions is a Uq(D4)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at q = 1 new “representation theory” proofs for very well-known identities satisfied by the octonions. In the process of con- structing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra. Introduction Using the representation theory of the quantum group Uq(D4)ofD4, we construct a quantum analogue Oq 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)=(xx)(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 Oq of quantum octonions is shown to satisfy (see Prop. 4.12 below). We also prove that the quantum octonion algebra Oq 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 gl2 when considered as algebras under the commutator product [x, y]=x y y x. · − · Received by the editors November 28, 1997. 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 Formaci´on del Personal In- vestigador 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. c 1999 American Mathematical Society 935 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 936 GEORGIA BENKART AND JOSEM.P´ EREZ-IZQUIERDO´ By its construction, Oq is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of Uq(D4). Associative algebras with a Yang- Baxter operator (or r-algebras) arise in Manin’s work on noncommutative geome- try and include such important examples as Weyl and Clifford algebras, quantum groups, and certain universal enveloping algebras (see for example [B]). In the process of constructing Oq we define an algebra Pq, which is a q-analogue of the 8-dimensional para-Hurwitz algebra. The quantum para-Hurwitz algebra Pq 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 Pq exhibits related properties (see Props. 5.1, 5.3). While this paper was in preparation, we received a preprint of [Br], which con- structs a quantized octonion algebra using the representation theory of Uq(sl2). Although both Bremner’s quantized octonions and our quantum octonions reduce to the octonions at q = 1, they are different algebras. The quantized octonion alge- bra in [Br] is constructed from defining a multiplication on the sum of irreducible Uq(sl2)-modules of dimensions 1 and 7. As a result, it carries a Uq(sl2)-module structure and has a unit element. The quantum octonion algebra constructed in this paper using U (D ) has a unit element only for the special values q =1, 1. q 4 − An advantage to the Uq(D4) approach is that it allows properties such as the ones mentioned above to be derived from its representation theory. Using the fact that the product in Oq is a Uq(D4)-module homomorphism, we prove identities for Oq (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 Uq(D4) approach also affords connections with fixed points of graph automorphisms - although as we show in Sec- tion 7, the fixed point subalgebras of Uq(D4) are not the quantum groups Uq(G2) and Uq(B3), because those quantum groups do not have Hopf algebra embeddings into Uq(D4). In the final section of this paper we explore connections with quan- tum Clifford algebras. In particular, we obtain a Uq(D4)-isomorphism between the quantum Clifford algebra Cq(8) and the endomorphism algebra End(Oq Oq)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 Uq(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.LetA=(aα,β)α,β Π be the corresponding Cartan matrix.}⊂ Thus, ∈ aα,β =2(β,α)/(α, α), and there exist relatively prime positive integers, (α, α) (1.1) d = , α 2 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A QUANTUM OCTONION ALGEBRA 937 so the matrix (dαaα,β) is symmetric. These integers are given explicitly by A , D , E (r =6,7,8) d = 1 for all 1 i r, r r r αi ≤ ≤ B d = 2 for all 1 i r 1, and d =1, r αi ≤ ≤ − αr (1.2) C d = 1 for all 1 i r 1, and d =2, r αi ≤ ≤ − αr F4 dαi =2,i=1,2,and dαi =1,i=3,4, G2 dα1 =1anddα2 =3. Let K be a field of characteristic not 2 or 3. Fix q K such that q is not a root 1 ∈ of unity, and such that q 2 K.Form, n, d Z ,let ∈ ∈ >0 md md q q− (1.3) [m] = − , d qd q d − − and set m (1.4) [m]d!= [j]d j=1 Y and [0]d!=1.Let m [m] ! (1.5) = d . n [n] ![m n] ! d d − d Definition 1.6. The quantum group U = Uq(g) is the unital associative algebra 1 over K generated by elements Eα,Fα,Kα,andKα− (for all α Π) and subject to the relations, ∈ 1 1 (Q1) KαKα− =1=Kα− Kα,KαKβ=KβKα, 1 (α,β) (Q2) KαEβKα− = q Eβ, 1 (α,β) (Q3) KαFβKα− = q− Fβ, 1 K K− α− α (Q4) EαFβ FβEα = δα,β qdα q dα , − − − 1 aα,β s 1 aα,β 1 aα,β s s (Q5) − ( 1) − Eα− − EβE =0, s=0 − s α dα P1 aα,β s 1 aα,β 1 aα,β s s (Q6) − ( 1) − Fα− − FβF =0. s=0 − s α dα P Corresponding to each λ = α Π mαα in the root lattice ZΦ there is an element ∈ P mα Kλ = Kα α Π Y∈ in U (g), and K K = K for all λ, µ ZΦ. Using relations (Q2),(Q3), we have q λ µ λ+µ ∈ 1 (λ,β) 1 (λ,β) (1.7) KλEβKλ− = q Eβ and KλFβKλ− = q− Fβ for all λ ZΦandβ Π. ∈ ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 938 GEORGIA BENKART AND JOSEM.P´ EREZ-IZQUIERDO´ The algebra Uq(g) has a noncocommutative Hopf structure with comultiplication ∆, antipode S, and counit given by ∆(K )=K K , α α⊗ α ∆(E )=E 1+K E , α α⊗ α ⊗ α 1 ∆(F )=F K− +1 F , α α⊗ α ⊗ α (1.8) 1 S(Kα)=Kα− , 1 S(E )= K− E ,S(F)= F K , α − α α α − α α (Kα)=1,(Eα)=(Fα)=0. The opposite comultiplication ∆op has the property that if ∆(a)= a a , then ∆op(a)= a a . (1) ⊗ (2) (2) ⊗ (1) a a X X (We are adopting the commonly used Sweedler notation for components of the comultiplication.) The algebra U with the same multiplication, same identity, and op 1 same counit, but with comultiplication given by ∆ and with S− as its antipode, is also a Hopf algebra. The comultiplication ∆op can be regarded as the composition τ ∆, where τ : U U U U is given by τ(a b)=b a.Themapψ:U Uq(g) ◦ ⊗ → ⊗ 1 ⊗ ⊗ → with E F , F E , K K− is an algebra automorphism, and a coalgebra α 7→ α α 7→ α α 7→ α anti-automorphism, so U is isomorphic to Uq(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 idU with respect to the convolution product, so that for all a U, ∈ (1.9) S(a(1))a(2) = (a)idU = a(1)S(a(2)).