The Alternative Clifford Algebra of a Ternary Quadratic Form
aa r X i v : . [ m a t h . R A ] J a n THE ALTERNATIVE CLIFFORD ALGEBRA OF A TERNARYQUADRATIC FORM
ADAM CHAPMAN AND UZI VISHNEA bstract . We prove that the alternative Cli ff ord algebra of a nondegenerateternary quadratic form is an octonion ring whose center is the ring of polyno-mials in one variable over the field of definition.
1. I ntroduction
The Cli ff ord algebra of a quadratic form is both an important invariant of theform and an interesting object in its own right. We propose a detailed study of thealternative Cli ff ord algebra, defined in the same manner as the classical Cli ff ordalgebra, but in the realm of alternative algebras.Let F be a field of arbitrary characteristic and let ϕ : V → F be a quadratic formon an n -dimensional vector space V over F . The classical Cli ff ord algebra C( ϕ ) isdefined as the tensor algebra of V modulo the relations v = ϕ ( v ) for every v ∈ V .Its structure is well understood: C( ϕ ) is a simple algebra of dimension 2 n over F ,whose center is F when n is even, and a quadratic ´etale extension, correspondingto the discriminant disc( ϕ ), when n is odd. The natural involution is defined bymapping v
7→ − v for all v ∈ V . The algebra thus has order dividing 2 in theBrauer group of its center. Indeed, the Cli ff ord algebra is the second cohomologicalinvariant of quadratic forms (see [2, Section 14]).An alternative analogue was recently introduced by Musgrave [5], who definedthe alternative Cli ff ord algebra C alt ( ϕ ) as the alternative tensor algebra of V mod-ulo the same relations v = ϕ ( v ). Explicitly, the alternative Cli ff ord algebra is thefree alternative algebra generated by x , . . . , x n over F , modulo the relations( u x + · · · + u n x n ) = ϕ ( u , . . . , u n )for every choice of u , . . . , u n ∈ F . (Chapter 13 of [9] is devoted to the free alterna-tive algebra.) The main motivation is to study, through an algebraic construction,the higher cohomological invariants of quadratic forms (see [2, Section 15 and 16]and [8]). When n ≤ ff ord algebra is identical to the associa-tive one. Musgrave asks if the alternative Cli ff ord algebra is finite dimensional,especially when n = Mathematics Subject Classification.
Key words and phrases.
Cli ff ord algebras, Alternative algebras, Octonion algebras, Quadraticforms.The second named author was partially supported by an Israel Science Foundation grant / In this paper we provide a complete description of the alternative Cli ff ord alge-bra C alt ( ϕ ) for nondegenerate ternary forms. Theorem 1.1.
Let ( V , ϕ ) be a nondegenrate ternary quadratic space over a field F.The alternative Cli ff ord algebra C alt ( ϕ ) is an octonion ring whose center is the ringof polynomials in one variable over F. See Theorem 4.2 and its corollaries for an explicit description of the algebra.The proof revolves around the “alternative discriminant”, studied in Section 3,which turns out to be a transcendental element in the center of the alternative al-gebra. Following the description of the alternative Cli ff ord algebra in Section 4,we observe that C alt ( ϕ ) has finitely many associative simple quotients, and conse-quently (Corollary 5.2): Corollary 1.2. C alt ( ϕ ) has a central localization which is “alternative Azumaya”in the sense that its simple quotients are all octonion algebras. In Section 6 we discuss the alternative Cli ff ord algebra in the context of coho-mological invariants. A cknowledgements We thank Ivan Shestakov and for an insightful discussion and for his carefulcomments on an earlier version of this paper, and an anonymous referee for sug-gesting significant shortcuts in the proof of the main result.2. B ackground
We briefly provide necessary background on alternative algebras.2.1.
Nonassociative rings and algebras.
Let A be a nonassociative ring. The commutator of x , y ∈ A is the element [ x , y ] = xy − yx . The associator of x , y , z ∈ A is the element ( x , y , z ) = ( xy ) z − x ( yz ). The nucleus (also the “associative center”)of A is the set N ( A ) = { a ∈ A : ( a , A , A ) = ( A , a , A ) = ( A , A , a ) = } , and the center is Z ( A ) = { a ∈ N ( A ) : [ a , A ] = } . The nucleus is an associative subring of A , and the center is a commutative subringof the nucleus.2.2. Alternative rings.
A nonassociative ring A is alternative if( a , a , b ) = ( a , b , b ) = a , b ∈ A . Linearizing, it follows that the associator alternates:( a σ (1) , a σ (2) , a σ (3) ) = sgn( σ )( a , a , a )for any a , a , a ∈ A and σ ∈ S . Any two elements of an alternative ring generatean associative subring (Artin, [9, Chapter 2, Theorem 2]). HE ALTERNATIVE CLIFFORD ALGEBRA OF A TERNARY QUADRATIC FORM 3
A simple alternative ring is either associative or an octonion algebra over itscenter, which is a field [9, Chapter 7, Section 3, Corollary 1]. A prime alternativering A is either associative or an octonion ring, provided that 3 A , L b , R b : A → A denote the multiplicationmaps L b ( c ) = bc = R c ( b ). For any b , c ∈ A , we denote b ◦ c = bc + cb . Rather thancomposition, we use the same notation for operators: R b ◦ R c = R b R c + R c R b . Remark 2.1.
Let b , c ∈ A . Then R b ◦ R c = R b ◦ c and L b ◦ L c = L b ◦ c . Corollary 2.2.
If b ◦ c = then R b , R c anticommute and L b , L c anticommute. For every a , b in an alternative ring we have that(1) [ a ◦ b , a ] = [ b , a ] , because the subring generated by a , b is associative.An alternative algebra satisfies the Moufang identities [9, Chapter 2, Lemma 7],which imply the identities ( a , [ b , a ] , c ) = [ a , ( a , b , c )] , (2) a ◦ ( a , b , c ) = ( a , b , c ) , (3)and(4) ( a , a ◦ b , c ) = ( a , b , c ) , see [9, Chapter 2, Equations (16)–(17 ′ )]. Remark 2.3.
Let S be a set of generators for an alternative algebra. An element u is central in the algebra if and only if [ u , S ] = ( u , S , S ) = u is in the nucleus if and only if( u , S , S ) = ( u , S , S S ) = ( u , S S , S S ) = ff ord algebra for the zero quadratic form (inany dimension). We do not give further details here, because the dimension of thealternative Grassmann algebra on any finite number of generators is finite, hencetoo small to be directly helpful for our cause.3. T he alternative discriminant Let ϕ : V → F be a nondegenerate ternary quadratic form, and write ϕ ( · , · ) for theunderlying symmetric bilinear form given by ϕ ( v , w ) = ϕ ( v + w ) − ϕ ( v ) − ϕ ( w ). Weconsider the generating subspace V ⊆ C alt ( V , ϕ ). Since u = ϕ ( u ) for every u ∈ V ,we have for u , v ∈ V that u ◦ v = ϕ ( u , v ) and u ◦ u = ϕ ( u ) = ϕ ( u , u ). For u , v , w ∈ V ,let δ ( u , v , w ) = u ◦ ( vw ) − ϕ ( v , w ) u = u ( vw ) − ( wv ) u . ADAM CHAPMAN AND UZI VISHNE
Proposition 3.1.
The form δ alternates.Proof. We have that δ ( u , v , v ) = ϕ ( v ) u − ϕ ( v , v ) u = δ ( u , u , w ) = u ◦ ( uw ) − ϕ ( u , w ) u = ϕ ( u ) w + uwu − ϕ ( u , w ) u = ϕ ( u ) w − ( wu ) u = . (cid:3) Consequently, the one-dimensional space F δ ( u , v , w ) is independent of the choiceof basis u , v , w of V . Notice that we can always choose a basis for which v , w areorthogonal with respect to ϕ ( · , · ), in which case δ ( u , v , w ) = u ◦ ( vw ). Proposition 3.2.
The element δ ( u , v , w ) is central in C alt ( V , ϕ ) .Proof. By Remark 2.3 one only has to verify that δ ( u , v , w ) commutes and asso-ciates with elements in V , which follows from:[ u , δ ( u , v , w )] = [ u , u ◦ ( vw )] − ϕ ( v , w )[ u , u ] = [ u , vw ] = u , v , δ ( u , v , w )) = ( u , v , u ◦ ( vw )) = ( u , v , vw ) ◦ u = ( u , v ◦ u , vw ) = . (cid:3)
4. T he alternative C lifford algebra of a ternary form Let ( V , ϕ ) be a nondegenerate 3-dimensional quadratic space. Let V be a non-degenerate 2-dimensional subspace of V . Then Q = C alt ( V , ϕ | V ) is a quaternionalgebra over F . Fix a basis { u , v } of V , and let d = ϕ ( u ) ϕ ( v ) − ϕ ( u , v ) be the de-terminant of ϕ | V with respect to this basis, which is nonzero by assumption. Since u ◦ v = ϕ ( u , v ), we have that(5) [ u , v ] = ( uv + vu ) − uvvu + vuuv ) = ϕ ( u , v ) − ϕ ( u ) ϕ ( v ) = − d . Let w ∈ V be orthogonal to V , so that V = V ⊥ Fw . It follows that u ◦ w = v ◦ w =
0. Let d = δ ( u , v , w ), which is central by Proposition 3.2.An octonion algebra is by definition the Cayley-Dickson double of a quaternionalgebra over a field. A ring R which is a faithful module over an integral do-main C ⊆ Z( R ) is an octonion ring if ( C − { } ) − R is an octonion algebra. Octonionrings can be recognized in the following manner. Remark 4.1.
Let R be an alternative ring which is a faithful module over an integraldomain C . Let Q ⊆ R be a quaternion subring, with the reduced trace tr : Q → C .If z ∈ R satisfies(6) z ◦ a = tr( a ) z for every a ∈ Q and z = γ ∈ C is nonzero, then the subalgebra generated by Q and z in R is the octonion algebra ( Q , γ ) C . HE ALTERNATIVE CLIFFORD ALGEBRA OF A TERNARY QUADRATIC FORM 5
We now prove Theorem 1.1 in the following form:
Theorem 4.2.
The alternative Cli ff ord algebra C alt ( V , ϕ ) is the octonion ring ( Q , d + d ϕ ( w )) F [ d ] , and F [ d ] is a polynomial ring over F. Remark 4.3.
The sum d + d ϕ ( w ) is well defined up to squares. Indeed, in charac-teristic not 2, the product d ϕ ( w ) is the determinant of ϕ , which is well defined upto squares. In characteristic 2, d is a square and ϕ ( w ) is well defined up to squares. Proof of Theorem 4.2.
Let z = [ w , uv ]. First we note that by Corollary 2.2,( u , v , w ) + [ w , uv ] = − u ( vw ) + w ( uv ) = − u ( w ◦ v ) = , so that z = − ( u , v , w ) vanishes in the associative Cli ff ord algebra C( V , ϕ ). Next, weclaim that C alt ( V , ϕ ) is generated over F by u , v , d and z . Indeed, z + d = w ( uv ) − ( u ◦ v ) w = w ( uv ) − ϕ ( u , v ) w , and therefore by (5)( z + d )[ u , v ] = (2 w ( uv ) − w ϕ ( u , v ))[ u , v ] = w [ u , v ] = − d w ;so w ∈ F [ d , u , v , z ]. In other words, over F [ d ], C alt ( V , ϕ ) is generated by Q = F [ u , v ]and z . We now show that (6) holds for a ∈ Q . It su ffi ces to prove this claim for a = u , v , uv , and indeed, by (3), z ◦ u = − ( u , v , w ) ◦ u = − ( u , v , w ) = z ◦ v = ( v , u , w ) ◦ v = ( v , u , w ) = z ◦ ( uv ) = [ w , uv ] ◦ ( uv ) = [ w , ( uv ) ] = [ w , ϕ ( u , v ) uv − u v ] = ϕ ( u , v ) z which is equal to tr( uv ) z because tr( uv ) = uv + vu = u ◦ v = ϕ ( u , v ).As a final preparation, we have that z ◦ w = [ w , uv ] = w ( uv ) − d = w ( uv ) − δ ( w , u , v ) = ( vu ) w = ϕ ( u , v ) w − ( uv ) w . Now let us compute z + ϕ ( u , v ) ϕ ( w ) = ([ w , uv ] + ϕ ( u , v ) w ) = (2 w ( uv ) − d ) = w ( uv )) − d ( w ( uv )) + d ) = d − uv ) w )( w ( uv )) + ϕ ( u , v ) ϕ ( w ) uv = d − ϕ ( w )( uv ) + ϕ ( u , v ) ϕ ( w ) uv = d + ϕ ( u ) ϕ ( v ) ϕ ( w );so that z = d + d ϕ ( w ). By Remark 4.1 C alt ( V , ϕ ) (cid:27) ( Q , z ) F [ d ] . (Note that by[9, Chapter 13, Theorem 12], the square of any associator in an alternative algebragenerated by three elements is central, so z ∈ F [ d ] was expected).To show that d is transcendental over F , reverse the argument. The octonion ring C ′ = ( Q , λ + d ϕ ( w )) F [ λ ] is generated by a copy of Q and an element z ′ satisfying ADAM CHAPMAN AND UZI VISHNE (6) and z ′ = λ + d ϕ ( w ). Let w ′ = − d ( z ′ + λ )[ u , v ]. Then ( t u + t v + t w ′ ) = ϕ ( t u + t v + t w ′ ) because w ′ ◦ u = w ′ ◦ v =
0, showing that the map C alt ( ϕ ) → C ′ preserving V = Fu + Fv + Fw and defined by z z ′ is well defined, and beingonto, it is an isomorphism. (cid:3) Corollary 4.4.
The subring F [ d ] is both the center and the nucleus of C alt ( ϕ ) . Corollary 4.5.
Let F be a field of characteristic not . A nondegenerate ternaryform can be presented diagonally as ϕ = h α , α , α i , and then Q = ( α , α ) F and C alt ( ϕ ) (cid:27) ( α , α , d + α α α ) F [ d ] . Corollary 4.6.
Let F be a field of characteristic . A nondegenerate ternary formcan be presented as ϕ = [ α , α ] ⊥ h α i (so that ϕ ( t , t , t ) = α t + t t + α t + α t ) and then Q = [ α α , α ) F , d = , and C alt ( ϕ ) (cid:27) [ α α , α , d + α ) F [ d ] , following the notation of [1] for Hurwitz algebras in characteristic 2.
5. S imple quotients of the alternative C lifford algebra Recall that the associative Cli ff ord algebra is either simple or the direct sumof two isomorphic quaternion algebras over F , depending on the discriminant.We now describe the simple quotients of C alt ( ϕ ), following the notation of The-orem 4.2. Proposition 5.1.
Let ¯ F be the algebraic closure of F. The simple quotients of C alt ( ϕ ) are (1) The octonion algebras ( Q , θ + d ϕ ( w )) F [ θ ] for every θ ∈ ¯ F such that θ + d ϕ ( w ) , ; (2) The (simple quotients of the) associative Cli ff ord algebra.Proof. The image of the center in a simple quotient is a field extension of F . Sup-pose d θ for θ ∈ ¯ F . If θ + d ϕ ( w ) , Q , θ + d ϕ ( w )) F [ θ ] . Otherwise, z maps to zero, making the image of Qz equalto the radical in the quotient, so by simplicity z maps to zero, and the whole alge-bra maps to the quaternion algebra Q F [ θ ] . If θ < F , this is the associative Cli ff ordalgebra. (cid:3) For example, in characteristic not 2, mapping d alt ( h α , α , α i ) → ( α , α , α ) F . (This was observed in [5,Prop. 2.17].) Corollary 5.2.
The localization of C alt ( ϕ ) at the central element ( u , v , w ) = d + d ϕ ( w ) is “alternative Azumaya” in the sense that its simple quotients are all oc-tonion algebras. On the other hand, let h z i denote the ideal of C alt ( ϕ ) generated by the associator z = ( u , v , w ). HE ALTERNATIVE CLIFFORD ALGEBRA OF A TERNARY QUADRATIC FORM 7
Remark 5.3.
The algebra C alt ( ϕ ) / h z i is associative.This is a consequence of Moufang’s theorem [7, Appendix], and coincides withthe case d + d ϕ ( w ) entral fractions of the alternative C lifford algebra For a nondegenerate ternary quadratic form ϕ , let ˆ C alt ( ϕ ) denote the algebraof central fractions of the alternative Cli ff ord algebra C alt ( ϕ ), namely ˆ C alt ( ϕ ) = F ( d ) ⊗ F [ d ] C alt ( ϕ ) where d = δ ( u , v , w ) for some basis u , v , w of V . More explicitly,assuming in this section that char F , ϕ = h α , α , α i , we have byCorollary 4.5 that ˆ C alt ( ϕ ) (cid:27) ( α , α , λ + α α α ) F ( λ ) . Assuming in this section that char F ,
2, we now put this algebra in context, bycomparing it to the cohomological invariants of quadratic forms.6.1.
Cohomological invariants.
Recall from [3, Theorem 17.3] that the coho-mological invariants over F of ternary quadratic forms are freely spanned overthe cohomology ring H ∗ ( F , µ ) by the Steifel-Whitney invariants ω i ∈ H i ( F , µ ), i = , . . . ,
3, where for ϕ = h α , α , α i the invariants are defined by ω ( ϕ ) = ω ( ϕ ) = ( α α α ), ω ( ϕ ) = ( α , α ) + ( α , α ) + ( α , α ) and ω ( ϕ ) = ( α , α , α ).For example, the associative Cli ff ord algebra C( ϕ ) is the quaternion algebra ( α i , α j )over F [ p disc( ϕ )] for any i , j . Summing up in H ( F , µ ), this invariant corre-sponds to res F [ √ − ω ( ϕ )] / F ( ω ). Computation shows that ω ∪ ω = ω + ( − ∪ ω .6.2. The associated Pfister form.
Recall that an m -fold Pfister form is a tensorproduct hh γ , . . . , γ m ii = hh γ ii⊗ · · · ⊗hh γ m ii , where hh γ ii = h , − γ i . If π is a Pfisterform, π ′ denotes the pure subform. The norm form of ˆ C alt ( h α , α , α i ) is the Pfisterform DD α , α , λ + α α α EE over F ( λ ), corresponding to the symbol µ ( h α , α , α i ) = ( α , α , λ + α α α )in H ( F ( λ ) , µ ). This element, the octonion algebra over F ( λ ) and its norm form,determine each other up to isomorphism.By symmetry we may write µ ( ϕ ) = ( α i , α j , λ + α α α ) for any i , j . Sum-ming up the three presentations, Corollary 6.1.
We have that µ ( ϕ ) = µ ( ϕ ) = ( λ + α α α ) ∪ ω ( ϕ ) . Presentation by Pfister forms.
It is of some interest to present µ ( ϕ ) in a formwhich is visibly symmetric. Let Quad k and P k denote the sets of k -dimensionalforms and k -folds Pfister forms over F up to isomorphism, respectively. Remark 6.2.
There is a one-to-one correspondenceQuad ←→ Quad × P by ϕ = h a , b , c i 7−→ ( h abc i , hh− ab , − ac ii ) = ( h δ i , ω ) and h δ i ω ′ ←−7 ( h δ i , ω ). Thus ϕ is isotropic i ff ω is hyperbolic. ADAM CHAPMAN AND UZI VISHNE
Proposition 6.3.
The norm form of ˆ C alt ( h α , α , α i ) is DD λ + α α α EE ⊗ DD − α i α j , − α i α k EE . Namely, if ϕ = h α , α , α i corresponds to ( h δ i , ω ), then the norm form of ˆ C alt ( ϕ )is DD λ + δ EE F ( λ ) ⊗ ω . Proof.
Compute, using that D , − ( λ + α α α ) E and (cid:28) − α α α , λ + α α α α α α (cid:29) areisomorphic over F ( λ ): µ ( h α , α , α i ) = h , − α , − α , α α i⊗ DD λ + α α α EE = h , − α , − α , − α i⊗ DD λ + α α α EE = D , − ( λ + α α α ) , − α , − α , − α E ⊥ D − ( λ + α α α ) E h− α , − α , − α i = * − α , − α , − α , − α α α , λ + α α α α α α + ⊥ D − ( λ + α α α ) E h− α , − α , − α i = h− α , − α , − α , − α α α i⊥ D − ( λ + α α α ) E h− α , − α , − α , − α α α i = h− i DD λ + α α α EE ⊗h α , α , α , α α α i = h− α α α i DD λ + α α α EE ⊗hh− α α , − α α ii = DD λ + α α α EE ⊗hh− α α , − α α ii . where the last step follows from − α α α being a value of DD λ + α α α EE . (cid:3) Corollary 6.4. ˆ C alt ( ϕ ) splits if and only if ϕ is isotropic. Residues.
Faddeev’s exact sequence (see [4, Section 6.4]) shows that µ ( ϕ )is determined by its residues. Since deg( λ + α α α ) is even, the only nonzeroresidues are obtained from divisors of λ + α α α : Remark 6.5.
The nonzero residues of µ ( ϕ ) are:(1) The associative Cli ff ord algebra C( ϕ ) at λ + α α α if disc( ϕ ) . α i , α j ) at λ ± √− α α α , if disc( ϕ ) ≡ Corollary 6.6.
The associative Cli ff ord algebra C( ϕ ) (up to isomorphism overF) and the localized alternative Cli ff ord algebra ˆ C alt ( ϕ ) (up to isomorphism overF ( λ ) ) determine each other. R eferences [1] A. Elduque and O. Villa. A note on the linkage of Hurwitz algebras. Manuscripta Math. ,117(1):105–110, 2005.
HE ALTERNATIVE CLIFFORD ALGEBRA OF A TERNARY QUADRATIC FORM 9 [2] R. Elman, N. Karpenko, and A. Merkurjev.
The algebraic and geometric theory of quadraticforms , volume 56 of
American Mathematical Society Colloquium Publications . American Math-ematical Society, Providence, RI, 2008.[3] S. Garibaldi, A. Merkurjev, and J.-P. Serre.
Cohomological invariants in Galois cohomology ,volume 28 of
University Lecture Series . American Mathematical Society, Providence, RI, 2003.[4] P. Gille and T. Szamuely.
Central simple algebras and Galois cohomology , volume 101 of
Cam-bridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 2006.[5] S. M. Musgrave. Representations of alternative Cli ff ord algebras of quadratic forms. Glasg.Math. J. , 57(3):579–590, 2015.[6] I. Shestakov and N. Zhukavets. The free alternative superalgebra on one odd generator.
Internat.J. Algebra Comput. , 17(5-6):1215–1247, 2007.[7] M. F. Smiley. The radical of an alternative ring.
Ann. of Math. (2) , 49:702–709, 1948.[8] V. Voevodsky. Motivic cohomology with Z / ffi cients. Publ. Math. Inst. Hautes ´Etudes Sci. ,98:59–104, 2003.[9] K. A. Zhevlakov, A. M. Slin’ko, I. P. Shestakov, and A. I. Shirshov.
Rings that are nearly as-sociative , volume 104 of
Pure and Applied Mathematics . Academic Press, Inc. [Harcourt BraceJovanovich, Publishers], New York-London, 1982. Translated from the Russian by Harry F.Smith.
E-mail address : [email protected] D epartment of C omputer S cience , T el -H ai C ollege , U pper G alilee , 12208 I srael E-mail address : [email protected] D epartment of M athematics , B ar -I lan U niversity , R amat G an , 55200 I, 55200 I