Bilinear Forms, Clifford Algebras, Q-Commutation Relations, and Quantum Groups

Journal of Algebra 228, 227᎐256Ž. 2000 doi:10.1006rjabr.1999.8257, available online at http:rrwww.idealibrary.com on

Bilinear Forms, Clifford Algebras, q-Commutation Relations, and Quantum Groups

K. C. Hannabuss

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Communicated by Susan Montgomery

Received July 12, 1999

Constructions are described which associate algebras to arbitrary bilinear forms, generalising the usual Clifford and Heisenberg algebras. Quantum groups of symmetries are discussed, both as deformed enveloping algebras and as quantised function spaces. A classification of the equivalence classes of bilinear forms is also given. ᮊ 2000 Academic Press Key Words: Clifford algebra; q-commutation relations; quantum groups; non- commutative geometry.

1. INTRODUCTION

In this paper we shall show how to classify arbitrary non-degenerate bilinear forms on a finite-dimensional vector space, define associated Clifford and Grassmann algebrasŽ. where possible , and see how quantum groups arise naturally as symmetries. Although offering no new fundamen- tal insights this approach does provide simple alternative constructions of various familiar algebras and quantum groups, as well as a source of other straightforward examples. Let B be a non-degenerate bilinear form on a finite-dimensional vector space V over an algebraically closed field ކ, of characteristic different from 2.Ž. In the examples we shall always assume that ކ s ރ. When B is a symmetric form one may construct the Clifford algebra, and when B is skew symmetric one has the Heisenberg or CCRŽ canonical commutation relation. algebra, but we shall explore what happens when B is not necessarily in either of these classes.Ž Besides the obvious mathematical interest, we recall that Einstein’s unified field theory featured an asymmetric metricwx 5, Appendix II , and note that in the non-commutative Rieman- nian geometry of Brzezinski´ and Majidwx 3 asymmetric metrics also appear

227 0021-8693r00 $35.00 Copyright ᮊ 2000 by Academic Press All rights of reproduction in any form reserved. 228 K. C. HANNABUSS naturally.. Our aim is to construct a quotient of the tensor algebra by an ideal generated by elements of the form PuŽ.Ž.m ¨ y Bu, ¨ 1, where P is a linear operator on V m V, but there is a consistency requirement which we now explore. ¨ g ¨ g ¬ ¨ ¨ Each V defines linear functionals, B12: u V BŽ., u and B : ¬ ¨ u BuŽ., . The non-degeneracy of B means that B12and B are isomor- phisms from V to its dual, so that we may define a non-singular linear map s y1 ª T B12B : V V satisfying ¨ s ¨ s ¨ s ¨ BuŽ.Ž.Ž.Ž, B21u BT .Ž.Žu BT, u ..1 Ž. We shall see in Section 2 that T determines B up to equivalence.Ž This is not surprising, since when T sy1 it is known that symplectic forms can always be put into Darboux form, whilst Sylvester’s law of inertia tells us that when T s 1 a non-degenerate real symmetric form can be expressed as a sum and difference of squares, and over ރ one can write it just as a sum..Ž.Ž.Ž. Since, by repetition Bu, ¨ s BT¨, u s BTu, T¨ , we see that T preserves B, and also, replacing ¨ by Ty1 ¨, we have BuŽ , Ty1 ¨ .Ž.s B ¨, u , providing an alternative formula for transposing the arguments of B. The last expression also gives BuŽ , Ty1 ¨ .Ž.s BTu, ¨ , and similarly we obtain BTŽ y1 u, ¨ .Ž.s Bu, T¨ , showing that Ty1 is the adjoint of T, on both sides.Ž For general forms B the adjoints on the two sides are usually different.. We may also consider B as a linear functional on V m V, and then, denoting the transposition u m ¨ ¬ ¨ m u by ␴ , we have BuŽ.m ¨ s BTŽŽm 1 .␴ Žu m ¨ .., so that B s B(Tˆˆ, where T s Ž.T m 1 ␴ . In other words B annihilates imwx 1 y Tˆ , and this means that the ideal should contain the image of PŽ.1 y Tˆ . This will impose the fewest constraints and give the largest possible algebra if we require PŽ.1 y Tˆ s 0, and we shall henceforth assume this condition. We note that whenever P commutes with Tˆˆ, we haveŽ. 1 y TPs 0, so that it projects onto kerŽ. 1 y Tˆ. This subspace is non-trivial as we shall now show. UUU ¨ s ¨ We may define a dual form B on V by BBuŽ.Ž.11, B Bu, , and U regard B as an element of V m V. It does not matter whether we use U ¨ s U ¨ s B12or B in this definition, since BBuŽ.22, B BBTuŽ. 11, BT U BTuŽ.Ž., T¨ s Bu, ¨ . Then B is the unique element of V m V satisfying Ž2. U m ¨ s U ¨ s ¨ BBŽ , u . BBuŽ.Ž.22, B Bu, . Using the fact that T and its inverse are adjoint with respect to B, we obtain

U U BTBŽ2. Ž.ˆ , TuˆŽ.m ¨ s BBŽ2. Ž, u m ¨ .Ž.Ž.s Bu, ¨ s BT¨ , u U s BBŽ2. Ž., TuˆŽ.m ¨ ,2Ž. UU U from which we deduce that TBˆ s B and B g kerŽ. 1 y Tˆ . CLIFFORD ALGEBRAS AND QUANTUM GROUPS 229

The minimal polynomial mxŽ.of Tˆ therefore has a factor of x y 1 and can be written as mxŽ.s Žx y 1 .Ž.px. This suggests that we should simply take P s pTŽ.ˆˆ, to ensure that PŽ.Ž1 y T s 0. Indeed if we want P to be a function of Tˆ then, up to multiples, this is the only non-trivial possibility.. The generalised Heisenberg᎐Clifford algebra ABŽ., P is defined as the quotient of the tensor algebra mV by the ideal generated by PuŽ.m ¨ y BuŽ., ¨ 1 with u, ¨ g V, that is, ABŽ., P s m VrÄ4Pu Žm ¨ .Ž.y Bu, ¨ 1:u, ¨ g V .3Ž. We shall see laterŽ. Theorem 3.6 that when T has non-trivial Jordan blocks the kernel and image of 1 y Tˆ usually intersect in an isotropic subspace. However, when T is diagonalisable the intersection is trivial and V m V is the direct sum of the two subspaces. In that case it is sufficient to factor out by the ideal generated by Ä X y BXŽ.1: X g ker Ž 1 y Tˆ .4. It will be useful to note that P can be regarded as an orthogonal projection on V m V with respect to the natural non-degenerate bilinear form BŽ2. s B m B. Using the fact that the two-sided adjoint of T is Ty1 ; whilst ␴ is self-adjoint, the orthogonal complement of imwx 1 y Tˆ is kerwx 1 y Tˆy1 .4Ž. ŽWere B to take values in a non-commutative algebra then ␴ might not be self-adjoint, but this could easily be accommodated within the same general framework.. We can further simplify things by noting that wxwx1 y Tˆy1 s TTˆy1 ˆy 1,Ž. 5 so that kerwxwx 1 y Tˆy1 s ker 1 y Tˆ .

EXAMPLE 1.1. If B is symmetric then T s 1, and P is the projection y ␴ s 1 q ␴ m ¨ onto the kernel of 1 , which is given by P 2 Ž.Ž.1 or Pu s 1 m ¨ q ¨ m 2 Ž.u u , so that the algebra is the algebra generated by V 1 ¨ q ¨ s ¨ subject to the relations 2 Ž.Ž.u u Bu, , that is, the usual Clifford sy s 1 y ␴ algebra. Similarly, if B is antisymmetric then T 1, P 2 Ž.1 , and 1 ¨ y ¨ s ¨ the algebra is generated by V subject to the relations 2 Ž.Ž.u u Bu, , that is, the normal Heisenberg algebra for the commutation relations.

EXAMPLE 1.2. It is often useful to choose a basis e , e ,...,e for V 12 n X and identify elements of V m V with matrices via the map u m ¨ ¬ u¨ Ž.where prime denotes transpose . Then ␴ transposes the matrices, and the X elements of kerwx 1 y Tˆ are matrices X satisfying X s TX . It is easy to check that T is given in terms of the Gram matrix G s ŽŽBe, e ..by X XXrs T s ŽGG.y1 , so that the condition amounts to GXs GX .Ž We note that X detŽ.T s det ŽG .y1 detŽ.G s 1. . When n s 2 and 01 G s y ,6Ž. ž/yq 1 0 230 K. C. HANNABUSS we have

qy1 0 T sy ,7Ž. ž/0 q and unless q s "1Ž. the cases already considered we have X a multiple of

01 ,8Ž. ž/yq 0 m y m corresponding to the tensor e12e qe 21e . This gives the algebra generated by V subject to the relation

1 y s 2 Ž.ee12qe 21 e 1.Ž. 9

This is a q-CCR algebra recently considered by many authorsw 1, 2, 5᎐10, 13xwx , but especially 2 . We could have arrived at the same result in a basis-free way, by noting s s m y s y ␴ that if TXˆŽ. X then, writing T111T 1, Ž.Ž.T 1 X T 1 X. Now A s X y ␴ Ž.X is an antisymmetric tensor and, if 1 is not an s y y1 eigenvalue of T, we have X Ž.T111 TA, so that X is determined by its antisymmetric part. The dimension of kerŽ. 1 y Tˆ is thus at most that of the space of antisymmetric tensors, which for two-dimensional V is just U one-dimensional, so that kerŽ. 1 y Tˆ must be spanned by B . Since the y antisymmetric tensor A has even rank, Ž.T1 1 must have even rank, y which means that in dimension 2, T1 1 is either zero or onto, that is, if 1 is an eigenvalue of T, Then T s 1. Thus either T s 1 and we are in the symmetric case or 1 is not an eigenvalue and kerŽ. 1 y Tˆ is a one-dimen- U U sional space spanned by B . It is easy to check that the Gram matrix of B Xy1 U s m with respect to the dual basis is G , from which we find that B e1 y m e221qe e . In order to make the connection with other approaches clearer, it is useful to work with the crossed product algebra containing an element U such that for all ¨ g V, Uy1 Ü s qTyŽ1r2. ¨.Ž By treating U as a group-like element, so that, in particular ⌬Ž.U s U m U, we may extend the Hopf s algebra structure to the crossed product.. We may then define ê11eU s and ê22Ue to obtain a formula involving ordinary commutators,

s 2 y s y12y 2 ˆˆe1, e 2eU 1 e 2Ue 21 e U eqTeU 1Ž. 2 qe21 e U s y 2 s 2 Ž.ee12qe 21 e U 2U .1Ž.0

y1 2 In the physics literature e12and e are taken to be adjoints and q U is interpreted in terms of a number operator. One can also obtain a comulti- CLIFFORD ALGEBRAS AND QUANTUM GROUPS 231 plication by taking

⌬ s m q m ⌬ s m y1 q y1 m Ž.e11e U U e1, Ž.e 22e U U e2.11Ž.

Changing q to yq gives the q-Clifford algebra relation

1 q s 2 Ž.ee12qe 21 e 1.Ž. 12

EXAMPLE 1.3. When studying the symmetries of the asymmetric tensor B the usual co-commutative comultiplication on linear transformations is inappropriate, and it should be replaced by more general ⌬. To preserve B a linear transformation S should satisfy B(⌬Ž.S s B, or the infinitesimal Ž.Lie algebra version B(⌬ Ž.S s 0, which is the appropriate substitute for the usual condition that BSuŽ.Ž., ¨ q Bu, S¨ s 0 when ⌬ Ž.S s S m 1 q 1 m S. The most obvious symmetry of B is provided by T, and in that case we ⌬ s m ¨ s ¨ ⌬ s can use Ž.T T T, but if we define LjjjBe Ž, .e and set ŽL j . m y m s Ljj1 T L , for j 1, 2, . . . , n, then ⌬ m ¨ s ¨ y ¨ BŽ.Ž.LujjjŽ.BLuŽ.Ž., BTu, L s ¨ y ¨ BLuŽ.Ž.jj, BL , u s ¨ y ¨ s BeŽ.Ž.Ž.Ž.jj, uBe, Be jj, Be, u 0, Ž. 13 showing that each Lj is an infinitesimal symmetry.Ž When B is skew sy ⌬ s m symmetric and T 1 we have the usual comultiplication Ž.LjjL q m ⌬ s m y m 1 1 Ljj, and when B is symmetric, Ž.L Lj1 1 Ljis the same as the usual Clifford algebra comultiplication when one takes ac- count of the grading of the tensor product.. Let us now specialise to the case of B and T defined byŽ. 6 and Ž. 7 , where we also have

y1 s y2 y1 s 2 TL1122 T q L , TL T q L ,14Ž. ¨ s ¨ q ¨ and, for 11e 22e , wx¨ s ¨ y ¨ L12, L BeŽ.Ž. 12, eBe 2, e 1Be Ž.Ž. 21, eBe 1, e 2 sy y1 ¨ q y1 ¨ q 11e q 22e ,15Ž. which is easily seen to be identical toŽ 1 y qT2 .y1 Ž y Ty1 .¨, so that the ᒐᒉ 2 three elements T, L12, and L generate the quantum group 2Žq ..

EXAMPLE 1.4. We know that

BTŽ.Ž.Ž.Ž.y qu, ¨ s BT¨ , Ž.T y qu s B Ž1 y qT .¨ , u ,16 Ž. 232 K. C. HANNABUSS from which, taking u g kerŽ.T y q ,

H kerŽ.ŽT y q : im 1 y qT . Ž.17 so that if q is an eigenvalue thenŽ. 1 y qT is not onto, and so qy1 is also an eigenvalue. In particular, in a two-dimensional space if T has an eigenvalue yq /"1, then its other eigenvalue is yqy1, and it is equivalent to

yqy1 0 ,18Ž. ž/0 yq which we already studied in Example 1.2. We observed earlier that 1 is an eigenvalue if and only if T is the identity, which is the case q sy1 above. The only other possibility is that T has the repeated eigenvalue y1. This can either happen with diagonal T, the case q s 1 above, orŽ. in two dimensions with the indecomposable Jordan block y11 ,19Ž. ž/0 y1 and B having Gram matrix equivalent to a multiple of

02 G s .2Ž.0 ž/y21 This exceptional case turns out to be linked to the exceptional R-matrix of Demidov et al.wx 4Ž. see Section 5 . For a two-dimensional space this exhausts the possibilities for B and T, and gives an indication of why the q deformations are natural in low dimensions.

EXAMPLE 1.5. A series of higher dimensional q-CCR algebras is easily U constructed by taking V s W m W , for some real vector space W with U dual W , and setting

BxŽ.Ž, ␰ .Ž, y, ␩ .s ␩ Ž.x y qy1␰ Ž.y ,21Ž.

U for x, y g W and ␰ , ␩ g W . Then TyŽ., ␩ sy Žqyy1 , q␩., and one can U show that the algebra is generated by W and W subject to the relations

␰ y y qy␰ s 2␰ Ž.y 1,Ž. 22

U for all ␰ g W and y g W. This is not quite the same as the usual q-commutation relations which consist of a number of mutually commuting copies of the two-dimensional q-commutation relations. CLIFFORD ALGEBRAS AND QUANTUM GROUPS 233

Nonetheless it is easy to find a representation of the above relations: We take for the representation space the tensor algebra of W and let x g W act by left tensor multiplication by 2 x, and inductively define the U action of ␩ g W to annihilate degree zero tensors and satisfy ␩Ž.w m Z s qw m ␩ Ž.Ž.Z q ␩ wZ,23 Ž. for Z gmk W and w g W. This definition ensures that ␩ x y qx␩ s 2␩Ž.x 1,Ž. 24 but there is no kind of commutativity between the action of distinct x and y g W. Commutativity can be arranged by restricting the tensor product to symmetric tensors, and in two dimensionsŽ. with a suitable completion this gives the usual Fock representation of the q-commutation relationswx 2 .

EXAMPLE 1.6. A KMS state ␾ on a von Neumann algebra A provides a natural example of an asymmetric bilinear form BaŽ., b s ␾ Ž.ab for a, b g A wx14 . The KMS condition gives ␾Ž.ab s ␾ Žb⌬a⌬y1 ., where ⌬ is the modular operator, and so TaŽ.s ⌬y1a⌬. For the Heisenberg algebra or Clifford algebra, generated by elements cŽ.¨ with ¨ g V, one also gets a bilinear form BuŽ., ¨ s ␾ ŽŽ.Ž..cuc¨ on V, and often the KMS condition gives an explicit formula for T here as well. For example, a short calculation shows that the ␤-KMS state for the Clifford algebra of ޒ 2 with evolution at time t given by rotation through ␻ t gives rise to the above q-Clifford algebra with q s expŽ."␤␻ .

EXAMPLE 1.7. Finally we note that although one might construct generalised GrassmannŽ. or exterior quadratic algebras by factoring out the ideal generated by the image of P, there is an alternative more useful U approach, which uses a square root S s T 1r2. The right adjoint S defined U by BSuŽ.Ž, ¨ s Bu, S ¨ . satisfies

U BŽ.Ž.Ž.Ž.Ž.¨ , S 2 Tu s BS2 ¨ , Tu s BT¨ , Tu s B ¨ , u ,25 U so that S 2 s Ty1.IfS can be chosen to be a normal operatorŽ so that S U U U U and S commute.Ž then SS.2 s SS2 2 s 1 so that I s SSis an involu- tion. We also have U BSŽ.Ž.Ž.Ž.Ž.¨ , S u s BS2 ¨ , u s BT¨ , u s Bu, ¨ ,26 so that BSŽ¨, Iu .Ž.s BSu, ¨ . Thus the form BSuŽ., ¨ is symmetric on the Ž.q1 -eigenspace of I and antisymmetric on the Ž.y1 -eigenspace. This suggests using the quotient of the tensor algebra mV by the ideal generated by elements of the form Su m ¨ q S¨ m Iu or, more simply, by elements U u m ¨ q S¨ m S u.27Ž. 234 K. C. HANNABUSS

For the form BxŽŽ, ␰ ., Žy, ␩ ..s ␩ Žx . y qy1␰ Ž.y of Example 1.5, we have TxŽ., ␰ sy Žqxy1 , q␰ .Ž, so that we may take Sx, ␰ .s "iqŽyŽ1 r2. x, "q1r2␰ ., with any combination of signs. A brief calculation shows that U SxŽ., ␰ s "i Ž"qx1r2 , qyŽ1r2.␰ ., where the signs inside and outside the bracket match those for S. The signs outside the bracket cancel in the formula for elements of the ideal which can be written as Ž.Ž.x, ␰ m y, ␩ y Ž.qyŽ1r2. y, " q1r2␩ m Ž."q1r2 x, qyŽ1r2.␰ ,28Ž. where the signs match in the two factors of the middle term. This gives the U algebra generated by W and W subject to the relations xy s "yx, ␰␩ s "␩␰ , x␩ s q␩ x,29Ž. U for x, y g W, ␰ , ␩ g W . According to the choice of upper or lower signs, these give deformations of the symmetric or exterior algebra. In general, U for any involutions I12and I on W and W , respectively, one sees that ␰ s yŽ1 r2. 1r2 ␰ SxŽ.Ž, iq Ix12, qI. is a square root of T, but it is a normal operator only when I12and I commute. We can, however, narrow the possibilities to those previously considered, and ensure that S is normal, by insisting that the square roots be in the algebra generated by T. In fact, whenever T has a quadratic minimal polynomial of the form T 2 y bT q 1 s 0, we readily check that the operators S s "Ž.b q 2⑀ yŽ1 r2.Ž.T q ⑀ , with ⑀ 2 s 1, are square roots of T.ŽŽ In the example where b sy q q qy1 ., these reduce to the square roots already given above. When T has the non-trivial Jordan form of Example 1.4 one must take ⑀ sy1.. We then have

U y r y r S s "Ž.Ž.Ž.b q 2⑀ Ž.1 2 Ty1 q ⑀ s " b q 2⑀⑀Ž.1 2 TTy1 Ž.q ⑀ s ⑀Ty1S s ⑀Sy1 ,3Ž.0 U so that SS s ⑀. Returning to the general case, unfortunately one cannot usually construct Clifford algebras by this approach. The obvious idea would be to factor the tensor algebra by the ideal generated by elements of the form U u m ¨ q S¨ m Suy 2 BuŽ., ¨ 1, giving an algebra generated by V with relations U u¨ q Ž.Ž.S¨ S u s 2 Bu Ž, ¨ .1. Ž.31 In the case of the standard example, defined by Eqs.Ž. 6 and Ž. 7 , one U obtains the algebra generated by W and W subject to the inhomogeneous version of the relations inŽ. 29 , xy s "yx, ␰␩ s "␩␰ , x␩ y q␩ x s 2␩Ž.x 1, Ž 32 . CLIFFORD ALGEBRAS AND QUANTUM GROUPS 235

U for x, y g W, ␰ , ␩ g W . From the third relation we deduce that xy␩ y qx␩ y s 2␩Ž.yxand x␩ y y q␩ xy s 2␩ Ž.xy, and taking a linear combination xy␩ y q 2␩ xy s 2␩Ž.yxq 2 q␩ Ž.xy.33Ž. Interchanging x and y, we see that yx␩ y q 2␩ yx s 2␩Ž.xyq 2 q␩ Ž.yx.34Ž. The relation wxx, y s 0 now gives 2␩Ž.yxq 2 q␩ Ž.xys 2␩ Ž.xyq 2 q␩ Ž.yx,35 Ž . and, unless q s 1, we obtain ␩Ž.xys ␩ Ž.yx.36Ž. When dimŽ.W ) 1 we may take linearly independent x and y, and a linear functional ␩ whose kernel contains x but not y, to deduce that x s 0, which contradicts the other relations.Ž The ideal generates the whole algebra, giving a trivial quotient.. For two-dimensional V where x and y must always be linearly dependent there is no problem and the above relations define the qy1 commutation relations. This example show both the strengths and weaknesses of this approach: it gives more useful Grassmann algebras than the other approach, but does not give useful Clifford algebras in higher dimensions. The remainder of the paper is organised as follows. In the next section we classify the possible forms in terms of T.Ž One would expect this to have been done long ago, but searches and enquiries have failed to find it in the literature.. Then in Section 3 we show how to construct the projection P when it exists. We indicate the close relation to Manin’s work on quantised function algebras in Section 4, following that with a brief discussion of the relationship to the R matrix in Section 5, before showing how quantum enveloping algebras arise as symmetries of asymmetric forms in Section 6.

2. THE CLASSIFICATION OF NON-DEGENERATE BILINEAR FORMS

In this section we shall assume that V is a complex vector space, and classify the possible forms B in terms of the Jordan form of T or, equivalently, the primary decomposition of V as

s y k j V [ kerŽ.T qj ,37Ž. j 236 K. C. HANNABUSS

y k j where the minimal polynomial of T is Ł jjŽ.x q . Each summand may be further broken down into cyclic subspaces, but these must be chosen carefully in relation to B. To this end we note that each T-invariant subspace U has a well defined orthogonal complement U H since

Ä4¨ g V : BuŽ., ¨ s 0, u g U s Ä4¨ g V : BŽ.¨ , u s 0, u g U .38Ž.

ŽŽ.If Bu, ¨ s 0 for all u g U, then BŽ.Ž¨, u s BTu, ¨ .s 0, and similarly the other way around.. As usual, we call a subspace U isotropic if B ¬ U = U s 0, and non-degenerate if U H lU s 0. Our strategy will be to identify non-degenerate cyclic subspaces of the primary decomposition. Then one studies the orthogonal complement and an induction on the dimension gives a complete classification. The first step is the following generalisation of a well-known result.

PROPOSITION 2.1. Let u g kerŽ.T y ␭ k and ¨ g kerŽ.T y ␮ m. Then BuŽ., ¨ s 0 unless ␭␮ s 1. Proof. Using the unitary of T with respect to B, we easily show by induction that BTŽ k u, ¨ .Žs Bu, Tyk ¨ .Žs BT1yk ¨, u. for any positive integer k, and taking linear combinations of these identities we see that

BrTuŽ.Ž., ¨ s BuŽ.Ž., rTŽ.Ž.Ž.y1 ¨ s BTrTy1 ¨ , u ,39 for any polynomial r. On the other hand

q Ž.Ž.1 y ␭␮ k m Bu, ¨ q s BuŽ., Ž.␭Ž.Ž.T y ␮ q 1 y ␭T k m¨ k k q m q kyr mqr s ␭m rBu,1Ž.Ž.y ␭TTy ␮ ¨ ,40 Ž. Ý ž/k y r Ž. rsym which, by our earlier remark, can be rewritten as

q Ž.Ž.1 y ␭␮ k m Bu, ¨ k k q m q kyr y q mqr s ␭m rBTŽ.y ␭ Tuk r , ŽT y ␮ .¨ .41 Ž. Ý ž/k y r Ž. rsym

One or another argument of B vanishes for each value of r, so that Ž.1 y ␭␮ kqmBuŽ., ¨ s 0 and, unless ␭␮ s 1, we deduce that BuŽ., ¨ s 0.

This means that the q and qy1 summands in the primary decomposition of V are orthogonal to all others, and must span a non-degenerate CLIFFORD ALGEBRAS AND QUANTUM GROUPS 237 subspace. This provides a T-invariant orthogonal decomposition, so that it suffices just to study these non-degenerate subspaces. We shall first show how to construct non-degenerate subspaces continuing just one or two indecomposable Jordan blocks for T, and then show how to put B into a canonical form within these subspaces. We treat the case of q /"1 first, and suppose that its primary summand has minimal polynomial Ž.T y q k. There must exist u g V such that Ž.T y quky1 / 0, and then, by non-degeneracy, there also exists ¨ g V such that BTŽŽy qu .ky1 , ¨ . / 0. By the earlier comments on orthogonality we may as well take ¨ g kerŽT y qy1 .k, and without loss of generality we take BTŽŽy qu .ky1 , ¨ . s 1. /" ¨ PROPOSITION 2.2. For q 1 the T-in ariant subspace Vq generated by u and ¨ is non-degenerate with respect to B. Proof. Since T is non-singular the vectors

y ky1 u, Ž.T y qu,..., Ž.T y quk 1 , ¨ , Ž.Ž.Ty1 y q ¨ ,..., Ty1 y q ¨ Ž.42 form a basis for Vq , and so any element can be written in the form pTŽ.Žy quq rTy1 y q.¨ for some polynomials p and r of degree at most y y q y1 y ¨ k 1. If pTŽ.Žqu rT q. is orthogonal to the whole of Vq then we have s BpTŽ.Ž.y quq rTŽ.Ž.y1 y q ¨ , Ty1 y q ¨ s 043Ž. for all s. Using the earlier relations this gives

s s 0 s BpTŽ.Ž.y qu, Ž.Ty1 y q ¨ s BTŽ.Ž.Ž.Ž.y qpTy qu, ¨ .44 Choosing first s s k y 1 we see from the definition of u and ¨ that p has vanishing constant term, and then inductively we check that all the other coefficients in p vanish, so that p s 0. Similarly r s 0, showing that l Hs VqqV 0, which is the non-degeneracy condition. Preceding inductively, this shows that we can further decompose the [ H space into a B-orthogonal direct sum of spaces of the form VqqV . The following result plays an important role in the classification of B.

PROPOSITION 2.3. For any non-zero q g ކ we ha¨e the identities

y1 Ty1 y qy1 syqTy2 Ž.y q Ž.1 q qTy1 Ž.y q ,45 Ž. and in the subspace kerŽ.T y q k, for s G 1,

kysy1 s s q j y 1 q q Ty1 y qy1 syŽ.1 s jsqTy2 syj Žy q .j .46 Ž. Ž.Ý j js0 ž/ 238 K. C. HANNABUSS

Proof. The first result follows immediately from

y1 Ty1 s qy1 Ž.1 q qTy1 Ž.y q

y1 s qy1 1 y qTy1 Ž.y q Ž.1 q qTy1 Ž.y q .47 Ž.

Then, taking its sth power and expanding the inverse by the binomial theorem, we obtain the second identity. We note that, in particular,

ky1 y y Ž.Ty1 y qy1 syŽ.1 k 1 qTy2 kq2 Žy q .k 1 ,48 Ž.

ky2 y y Ž.Ty1 y qy1 syŽ.1 k 2 qTy2 kq4 Žy q .k 2 y y qyŽ.Ž1 k 1 k y 2 .qTy2 kq3 Žy q .k 1 .49 Ž. ¨ g PROPOSITION 2.4. For any u, Vq and natural numbers r and s one may express BŽŽ T y qu .r , ŽT y qy1 . s¨ .Žin terms of BŽ. T y quj , ¨ ., for j G r q s. When r q s G k, BTŽŽy qu .r , ŽT y qy1 . s¨ . s 0. Proof. Proposition 2.3 shows that

r s BTŽ.Ž.y qu, Ž.T y qy1 ¨ ys rqs ys syŽ.qBT2 ž/Ž.y q Ž.1 q qTy1 Ž.y qu, ¨ ,50 Ž. and by expanding the negative power in a geometric series we obtain the result.ŽŽ When r q s G k we use T y qu.k s 0.. y This result tells us that B is determined on Vq , by the values BTŽŽ qu. j , ¨ ., for j s 0,...,k y 1. There is, however, still some flexibility as we might have chosen different vectors u and ¨. To investigate this another technical result is useful. X X PROPOSITION 2.5. For any ¨ectors u , u g kerŽ.T y qk and ¨ , ¨ g 00X kerŽT y qy1 .k, the ¨ectors u s u q aTŽ.y qkyry1 u and ¨ s ¨ q bTŽ X a 0 b 0 y qy1 .kyry1¨ satisfy y r ¨ s y ky1 X ¨ BTŽ.Ž.quab, aBŽ. Ž. T qu, 0 qy kyry1 y2Žkyry1. y ky1 ¨X Ž.1 qbBTŽ.Ž.qu0 , q y r ¨ BTŽ.Ž.qu00, ,5Ž.1 and, for s ) r y ss¨ s y ¨ BTŽ.Ž.Ž.quab, BT Ž.qu00, .5Ž.2 CLIFFORD ALGEBRAS AND QUANTUM GROUPS 239

Proof. For any s G r ) 0 we have

y s ¨ s y kqsyry1 X ¨ BTŽ.Ž.quab, aBŽ. Ž. T qu, 0 y y q y s y y1 k r 1¨X bBž/Ž. T qu0 , Ž.T q q y s ¨ BTŽ.Ž.qu00,

q y y XXkyry1 q abBž/Ž. T y quk s r 1 , Ž.T y qy1 ¨ , Ž.53 and when s ) r this reduces to

y ss¨ s y ¨ BTŽ.Ž.Ž.quab, BT Ž.qu00, .5Ž.4 s y r ¨ For s r we get the expression for BTŽŽqu . ab, ., y y y ky1 XX¨ q y1 y y1 k r 1 y r ¨ aBŽ.Ž. T qu, 00bBž/Ž. T qTŽ.qu, q y r ¨ BTŽ.Ž.qu00, ,5Ž.5 whence the result follows using Proposition 2.3. This gives us a way to modify some of the matrix elements in the subspace Vq without affecting others. ¨ ¨ PROPOSITION 2.6. The ectors u and in Vq may be chosen so that BTŽŽy qu .r , ¨ .Žs 0 unless r s k y 1, and BŽ T y qu.ky1 , ¨ . s 1. X s s ¨X s ¨ s ¨ s y Proof. We take u u00u, and r k 2 in Proposition 2.5 to get

y ky1 ¨ s y ky1 ¨ BTŽ.Ž.quab, BTŽ. Ž.qu00, Ž.56 and

y ky2 ¨ s y y2 y ky1 ¨ BTŽ.Ž.quab, Ž.a qbBTŽ.Ž.qu, q y ky2 ¨ BTŽ.Ž.qu00, ,57Ž. so that by choosing a y qby2 syBTŽŽy qu .ky2 , ¨ . we may ensure that y ky2 ¨ s y ky1 ¨ s BTŽŽqu . ab, .Ž0, whilst retaining BTŽqu.ab, . 1. Re- peating this with lower values of r we may ensure that BTŽŽy qu .r , ¨ . s 0 unless r s k y 1. The following result enables us to calculate the matrix element BTŽŽ y qy1 .t¨, u.Ž, for any vectors u and ¨, in terms of BTŽy qu.s , ¨ . with s G r. 240 K. C. HANNABUSS

PROPOSITION 2.7. For any u, ¨ g V, and any q g ކ one has BŽ.¨ , u y qB Ž. u, ¨ s BTŽ. Žy qu ., ¨ .5Ž.8 If BŽŽ T y qu . s , ¨ .Žs 0 for all s ) r and so, in particular, when u g kerŽT y q.k and r s k y 1,. one has rr r BTŽ.Ž.y qu, ¨ sy Ž.1 qBT2 ry1 Ž.Ž.y qy1 ¨ , u .59Ž. Proof. The first identity is immediate from the defining property of T. For the second we change u to Ž.T y qur , to get rrr qBŽ.Ž.Ž. T y qu, ¨ s B ¨ , Ž.T y qus BTŽ.Ž.y1 y q ¨ , u .60Ž. Applying Proposition 2.3 this reduces to rr r qBŽ.Ž. T y qu, ¨ sy Ž.1 qB2 r Ž.Ž. Ty qy1 ¨ , u ,61Ž. from which the result follows. Combined with the previous information about decompositions this result tells us that B is determined up to equivalence on the direct sum of such non-degenerate T-invariant subspaces. We must now turn our attention to the case of the eigenvalues "1. Within the subspace kerŽ.T y q k we can, as before, find a vector ¨ such that Ž.T y q ky1¨ / 0, and there must also be a vector u g V such that BTŽŽy q .ky1¨, u. s 1. When q s "1 there is a dichotomy at this point. y PROPOSITION 2.8. If q syŽ.1 k 1 then there is a ëctor w g kerŽ.T y q k such that BŽŽ T y qw .ky1 , w. s 1, and w generates a non-degenerate T-in- äriant subspace. Otherwise there are two isotropic subspaces of kerŽ.T y q k generated by ëctors ¨ and u whose direct sum is non-degenerate. Proof. Let u and ¨ be chosen as in Proposition 2.6. Interpreting Proposition 2.7 when q 2 s 1 and u s ¨ we see that if BTŽ.y q s¨, ¨ . s 0 for all s ) r then rr r BTŽ.Ž.y q ¨ , ¨ sy Ž.Ž.1 qBTy1 Ž.y q ¨ , ¨ ,62 Ž. from which it follows that either q syŽ.1orr BTŽŽy q .r ¨, ¨ . s 0. In particular we see that if there is any vector w g kerŽ.T y q k such that BTŽŽy qw .ky1 , w.Ž/ 0 then q sy1..ky1 s ¨ s ¨ X s ¨X s s Next we use Proposition 2.5 with u00, u u, and a b, s s ¨ ) y s s y s¨ ¨ s w uabto see that, for s r, BTŽŽqw . , w.ŽŽ.BT q , . 0, r y BTŽ.Ž.y qw, w s aBŽ. Ž. T y quk 1 , ¨ y y y qyŽ.1 k r 1 aBŽ. Ž T y q .k 1¨ , u r q BTŽ.Ž.y q ¨ , ¨ .6Ž.3 CLIFFORD ALGEBRAS AND QUANTUM GROUPS 241

Using Eq.Ž. 59 we can combine the first two terms to get

rry BTŽ.Ž.Ž.y qw, w s 1 qy Ž.Ž.1 qaBTŽ.y quk 1 , ¨ r q BTŽ.Ž.y q ¨ , ¨ rr s Ž.Ž.1 qyŽ.1 qaq BT Žy q .¨ , ¨ .64 Ž.

We shall now distinguish two cases. Ž.iIfq sy Ž1 .ky1 then we can take r s k y 1, and we see that there are certainly values of a which give non-vanishing BTŽŽy qw .ky1 , w.. Ž.ii If q / Žy1 .ky1 , and the subspace generated by ¨ is not already isotropic, let r - k y 1 be the largest integer for which BTŽŽy q .r ¨, ¨ . / 0. By our earlier observation we must have q syŽ.1r , so by the earlier equation

rr BTŽ.Ž.Ž.y qw, w s 2 a q BTŽ.y q ¨ , ¨ ,6 Ž.5

sy 1 y r ¨ ¨ and we can take a 2 BTŽŽq ., . to reduce this to zero. The largest exponent s for which BTŽŽy qw . s , w. / 0 is therefore less than r and we can use an inductive argument to reduce to the case where BTŽŽy q .r ¨, ¨ . s 0 for all exponents r, in which case the subspace generated by ¨ is isotropic. One can similarly modify u to obtain another isotropic subspace generated by u which pairs with it to give a non-degenerate subspace in the same way that the q and qy1 subspaces paired before. In the case of paired isotropic subspaces we can proceed exactly as before to find a canonical form for B. In the other case of a single non-degenerate cyclic subspace, we note that BTŽŽy qw .r , Ž.T y qws . can be expressed in terms of BTŽŽy qw . j , w. with j G r q s. However, in this indecomposable case one can go further than before. Setting q s 1 and u s ¨ s Ž.T y qwŽ1r2.Ž ky3. in Eq.Ž 59 . gives BT ŽŽy qw .Ž1r2.Ž ky1. , ŽT y qw.Ž1r2.Ž ky3. .Žs 0, which means that BTŽy qw.r , Ž.T y qws . with r q s s k y 2 are already determined by BTŽŽy qw .ky1 , w.. When q sy1 the dimension k is even and with u s ¨ s Ž.T y qwŽ1r2.Ž ky1. we have

r y r y 2 BTŽ.Ž.y qwŽ.Ž.1 2 k 11, Ž.T y qwŽ.Ž.2 k 1 r r y s BTŽ.Ž.y qwŽ.1 2 k , Ž.T y qwŽ.Ž.1 2 k 1 r y y syŽ.1 Ž.Ž.1 2 k 1 BTŽ. Žy qw .k 1 , w ,66Ž. 242 K. C. HANNABUSS again showing that BTŽŽy qw .r , Ž.T y qws . with r q s s k y 2 is determined by BTŽŽy qw .ky1 , w.. Thus we cannot hope to follow the earlier strategy of choosing a new cyclic vector with vanishing BTŽŽy qw .ky2 , w.. Ž.The usual trick turns out not to change the value of this matrix element. s ¨ s X s ¨X s We can, however, still use Proposition 2.5 with u00u w s s s ¨ ) and a b, waabu to obtain for s r

y sss y BTŽ.Ž.qwaa, w BTŽ. Ž.qw, w Ž.67 and

y r s qy kyry1 y ky1 BTŽ.Ž.qwaa, w Ž.1 Ž.1 aBŽ. Ž. T qw, w r q BTŽ.Ž.y qw, w .6Ž.8

s y y ky3 s Taking r k 3 we can choose a so as to make BTŽŽqw . aa, w . 0, and then, proceed inductively, to put the Gram matrix of B in a form where BTŽŽy qw .ky1yr , w. s 0 for all even r - k y 1. Since these deter- mine the values for odd r, this gives a canonical form. We remark at this point that when T has two identical indecomposable Jordan blocks of size k with eigenvalues q syŽ.1ky1 there is an apparent ambiguity, as these might either be paired isotropic subspaces or two orthogonal non-degenerate subspaces. However, the following argument shows that these two possibilities are equivalent. In the non-degenerate ¨ ¨ case, choose generating vectors 12and for the two non-degenerate r ¨ r ¨ subspaces as above, and let S map the basis elements T 12and T to 11rry ¨ q q ¨ q ¨ q y ¨ 22T ŽŽ1 i .12 Ž1 i . .and T ŽŽ1 i .12 Ž1 i . ., respectively. r ¨ By definition S commutes with T, but we easily check that BSTŽ j, s¨ s s s y ST j. 0 for j 1, 2 and all r, s 0,...,k 1, so that the images of the two subspaces under S are isotropic, andŽ. since B is non-degenerate they are paired. This observation removes the last ambiguity in the determination of B from T. We have now dealt with all the possible indecomposable subspaces of V and have shown in each case how to construct a non-degenerate subspace on which B as well as T has a particular form. By induction on the dimension we arrive at the following result.

THEOREM 2.9. The equi¨alence class of a non-degenerate bilinear form B o¨er ކ is uniquely determined by T. We can also describe completely the possible summands in the decomposition for V using, in particular, the condition that q syŽ.1ky1 for a non-degenerate cyclic subspace.ŽŽ. For q sy1, part ii generalises the well-known fact that symplectic forms exist only in even dimensions.. CLIFFORD ALGEBRAS AND QUANTUM GROUPS 243

THEOREM 2.10. The space V is a direct sum of non-degenerate subspaces, each of which is one of the following, Ž.i The direct sum of two indecomposable isotropic subspaces of the same dimension, on which the eigenälues of T are reciprocals of each other. Ž.ii An eën-dimensional non-degenerate indecomposable subspace on which T has the eigenälue y1. Ž.iii An odd-dimensional non-degenerate indecomposable subspace on which T has the eigenälue q1.

3. ADJOINTS AND PROJECTIONS

For general non-degenerate forms there are twoŽ. usually distinct adjoints defined by

U BSuŽ.Ž, ¨ s Bu, S ¨ .Bu Ž.Ž, S¨ s B *Su, ¨ .,69 Ž. for any linear transformation S of V. U U U U PROPOSITION 3.1. The adjoints are related by ST s TS , ŽS . s S s U U UU UU Ž S. , and TŽ S. Ty1 s S s TSTy1Ž . . Proof. We have

U U BuŽ.Ž, S¨ s BS¨ , Ty1 u .Žs B ¨ , S Ty1 u .Žs BTSTy1 u, ¨ .Ž.,70

U U so that S s TS Ty1, and the next two identities are almost tautologies, UU U U and then TS s Ž ST. s ST, and similarly for the remaining identity.

Self-adjointness is independent of which adjoint is chosen, since U U BSuŽ.Ž., ¨ s Bu, S¨ can be read both as S s S and as S s S. Proposi- tion 3.1 therefore tells us that self-adjoint S must commute with T, since

U U TS s TS s ST s ST .71Ž.

This notion of self-adjointness if therefore rather restricted, but it does have one very useful applicationŽ. see Proposition 3.3 . The earlier identification of V m V with matrices can more elegantly be g m replaced by defining the transformation LX for each X V V by

¨ s Ž2. m ¨ BLuŽ.ŽX , B X, u .,7 Ž.2 and non-degeneracy means that this map gives an isomorphism between V m V and the linear transformations of V. 244 K. C. HANNABUSS

s U PROPOSITION 3.2. This correspondence satisfies LŽSmC . XXCL S and

U s y1 s LX LT␴ Ž X . LTˆŽ X . .73Ž.

With respect to a basis e12, e ,...,en for V and dual basis f12, f ,..., fn such s ␦ that BŽ. ejk, f jk, one has s m X Ý ejXjLf.74Ž.

Proof. By definition

Ž2. m m ¨ s Ž2. U m U ¨ s U U ¨ B Ž.Ž.S CX, u B ŽX, S u C .ŽBLSX u, C . s U ¨ BCLSŽ.X u, Ž.75 s U s s whence LŽSmC . XXCL S . Taking S T and C 1, and recalling that U T s Ty1, we obtain L s LTy1. We also have TX1 X ¨ s ¨ y1 BuŽ., LXXBLŽ., T u s BŽ2. Ž.X, ¨ m Ty1 u s Ž2. ␴ y1 m ¨ s y1 ¨ B Ž.Ž.X , T u BLŽ.␴ Ž X . T u, ,76Ž. U s y1 s s m ¨ s whence LX LT␴ Ž X . LTˆŽ X .. For X e f, we have BLŽ.X u, ¨ s s s BeŽ.Ž., uBf, , and LuXjBeŽ., uf. Setting e e and taking u fk s ␦ s m gives LfXk jkf, whence, in this case, X ejLf Xj, and the general result follows by summation. s s m We note, in particular, that LXj1 corresponds to X Ýe fj, and ␴ s m then Ž.X Ý fjje gives sU s L␴ Ž X . LTX T .77Ž.

s y1 For future use we note that in terms of the Gram matrix fjkkjkÝ G e ,so that m s y1 m ÝÝejjf G jkkje e .78Ž. jjk

U U This is actually the bilinear form B on V dual to B under the isomorphism B21Ž.or under B , since that isomorphism sends Ä4fk to the U U UUs s y1 basis Äekk4Ä4dual to e , whence BeŽ j, ekj.Ž.Bf, fkkG j. Using this we may also write

s m U s y1 m X Ž.1 LBXjÝ G kekLeXj.79Ž. jk CLIFFORD ALGEBRAS AND QUANTUM GROUPS 245

Combined with the following result this will show that the most general X U has the formŽ. 1 m LB with L self-adjoint. y PROPOSITION 3.3. The tensor X is in kerŽ. 1 Tˆ if and only if LX is self-adjoint, and X g imŽ. 1 y Tˆ if and only if there exists a linear transforma- s yU ¨ Ž2. s U tion L with LXXL L. We also ha eBŽ.Ž X, Y tr LLY., for all s tensors X, Y, and, in particular, BYŽ.tr ŽLY .. s U s s Proof. By the Proposition 3.2 X TXˆŽ.if and only if LXTL ˆŽ X . s y LXX; that is, L is self-adjoint. Similarly, X Y TYˆ is equivalent to s yU y1 s y1 LXYL L Y. Since BfŽ j, Te k.Ž.Be kj, f , we see that Ä4fjand ÄTe k4 form dual bases, so that we may write

s m y1 Y Ý fjYLT e j.80Ž. s m With the earlier identity X ÝejXjLf, this gives Ž2. s y1 B Ž.X, Y Ý BeŽ.jk, fBLfŽ. XjY, LT e k j, k

s y1 Ý BLŽ.Xj f, LT Ye j j

s U y1 Ý BfŽ.jXY, L LT e j j

s U y1 Ý BTLŽ.XYLT e j, f j,81Ž. j U y1 s U which is trŽTLXY L T .Žtr LLXY.. Then s s BXŽ. Ý BeŽ.jXj, Lf trŽ.L X Ž.82 j gives the final identity. Combining Proposition 3.3 and Eq.Ž. 71 we note that if X g kerŽ. 1 y Tˆ then LX commutes with T, and also, as any real polynomial in LX is also self-adjoint it too corresponds to a tensor in the kernel.

COROLLARY 3.4. For X g kerŽ. 1 y Tˆ in two dimensions and antisym- y metric B, LX is a multiple of the identity. The elements of imŽ. 1 Tˆ are the trace-free elements. Proof. In two dimensions there is only one antisymmetric form, which we denote by ⑀, with Gram matrix

01 .83Ž. ž/y10 246 K. C. HANNABUSS

¨ y ¨ ¨ Now BLŽ.Ž.XX u, BL , u is clearly antisymmetric in u and so that ¨ y ¨ s ␭⑀ ¨ s U BLŽ.Ž.Ž.XX u, BL , u u, . But if LXLX, then the left-hand ¨ y ¨ side can be rewritten as BLŽ.Ž.XX u, B , Luwhich is antisymmetric in ¨ ␤⑀ ¨ ␤ LuXXand and so of the form Ž.Lu, , and cannot vanish since B is s ␤y1␭ not symmetric. Comparing the two expressions we have LX 1, as claimed. The image is the orthogonal complement of the kernel so it consists of elements whose trace inner product with the identity vanishes.

THEOREM 3.5. For any T in two dimensions and more generally wheneër 1 and y1 are not eigenälues of T, the kernel and image of 1 y Tˆ intersect triïally. Proof. An element X is in the image if and only if it is orthogonal to the kernel, so the intersection consists of those elements which are both in s U the kernel and orthogonal to every element of the kernel. Since LXXL , we know that LX commutes with T, and it will be sufficient to concentrate our attention on a subspace where

yqy110 T s .84Ž. ž/0 yq1

There we readily check that LX has the block diagonal form A 0 L s X .85Ž. X ž/0 A Moreover, to be in the intersection of the image and kernel we now s U s g y require trŽ.Ž.LLXYtr LL XY 0 for all Y kerŽ. 1 Tˆ , or equivalently s trŽ.AB 0 for all B, which means that A and also LX vanish. It is worth noting that since LX in the intersection must be orthogonal to itself we 2 s have trŽ LXX. 0, which means that L is nilpotent. In two dimensions we already saw that LX is a multiple of the identity, and so nilpotent only if it vanishes.

EXAMPLE 3.1. In two dimensions it follows from the identity s y 11q LXXŽ.L 22trŽ.L X1 tr Ž.L X186Ž. s 11s s 1m that LPX 22trŽLX .1 BX Ž.1 so that PX 2BXŽ.Ýejf j. The algebra is therefore generated by V subject to the single relation 1 q s 2 Ž.ef11ef 22 1.Ž. 87 s y1 By our earlier observationŽ. following Proposition 3.2 f12detŽGGe.Ž 21 y s y1 y q Ge21 2 .Žand f2 det G.Ž.Ge12 1Ge 11 2 , giving the single relation 1 22y y q s 2 Ž.Ge22 1Gee 21 1 2Gee 12 2 1Ge 11 2 detŽ.G 1. Ž. 88 CLIFFORD ALGEBRAS AND QUANTUM GROUPS 247

It is easily checked that this agrees with our earlier formulae for the special cases.

EXAMPLE 3.2. Applying Eq.Ž. 88 to the exceptional two-dimensional form given by Eq.Ž. 20 , with indecomposable Jordan matrix T, Ž. 19 , gives ¨ ¨ an algebra generated by 12and subject to the relation ¨ 2 q ¨¨ y ¨¨ s 112212Ž.8.Ž. 89 ¨ ¨ This can be represented by taking 12to be multiplication by 2t, and to be Žt 2 y 2.drdt. It can also be obtained as a singular limit of the q-CCR ª algebra as q 1. More precisely, if e12and e satisfy y s ee12qe 21 e 2,Ž. 90 we set

¨ s y 1r2 y ¨ s y yŽ.1r2 q 11Ž.Ž.Ž.Ž.Ž.q 1 e e2, 2q 1 e 1e2,91 and note that

¨ 22y y 2¨ q q ¨¨ y ¨¨ s y s 12Ž.q 1 Ž.Ž.Žq 1 122114 ee2qe2 e1 .8, Ž. 92 which tends to the singular relation as q ª 1. In two dimensions we can see from our explicit formulae that the kernel and image of 1 y Tˆ intersect trivially even when T is not semi-simple, but this is exceptional, as the following result shows.

THEOREM 3.6. Except in the case of the two-dimensional block giën in the Example 3.1, wheneër T contains a non-triïal Jordan block the image and kernel of Ž.1 y Thaˆ ë a non-triïal intersection. Proof. Since kerŽ.Ž. 1 y Tˆˆs im 1 y T H , we also have

H imŽ. 1 y Tˆˆˆl ker Ž. 1 y T s ker Ž. 1 y T lker Ž. 1 y T ˆ.93Ž.

The intersection therefore consists of those X such that LX is self-adjoint s and satisfies trŽ.LLX 0 for all self-adjoint L. Let us first deal with the non-degenerate non-trivial indecomposable blocks with eigenvalue "1. If the eigenvalue is 1, the block has odd ) s y ky1 dimension k 1, and we take LX Ž.T 1 , which is self-adjoint by Eq.Ž. 59 , and also nilpotent. Any self-adjoint L commutes with T and so is y a polynomial in T 1 on this block. The product with LX therefore has trace 0. If the eigenvalue is y1, the block has even dimension k, and we q ky2 q 1 y q ky1 take Ž.T 1 2 Ž.Ž.k 2 T 1 , which is self-adjoint by Proposi- tion 2.7. Provided that k ) 2 it is nilpotent and the argument proceeds as in the previous case. 248 K. C. HANNABUSS

In the other cases we have paired indecomposable blocks kerŽ.T y q k [ kerŽT y qy1 .k, and the general element commuting with T can be written in the form

k L s Ž.T y qpTk Ž.Ž.y1 y q q Ty1 y qrTŽ.y q ,94 Ž. for some polynomials p and r. It is self-adjoint if and only if p s r.In s y k y1 y ky1 q y1 y k y ky1 particular, LX Ž.T qTŽ q. ŽT qT. Ž.q is self- adjoint, and nilpotent, so that we can argue, as before, to get a non-trivial intersection. U When V s W [ W with BxŽŽ, ␰ ., Žy, ␩ ..s ␩ Žx . y qy1␰ Ž.y and Tx Ž, ␰ .Žsy qxy1 , q␰ ., the elements commuting with T have the form Q [ R, X and are self-adjoint if and only if R s Q , the dual of Q. From this we see that for AB L s Ž.95 X ž/CD we have

1 q X 2 Ž.A D 0 L s X .96Ž. PX 0 1 A q D ž/2 Ž. This gives the algebra discussed at the end of the Introduction, generated by V subject to the relations

␰ y y qy1 y␰ s 2␰ Ž.y 1,Ž. 97

U for all ␰ g W and y g W. In terms of the tensors one calculates that

2 PxŽ.Ž.Ž.Ž, ␰ m y, ␩ s x m ␩ y q␩ m x.q Ž.␰ m y y qy1 y m ␰ .98Ž.

4. QUANTISED FUNCTION ALGEBRAS

We now consider symmetries of the form B. In Manin’s treatment quantum groups appear as symmetries of quadratic algebraswx 11 .Ž Super- spaces, which Manin allows, could also be incorporated into our framework.. Now deformed analogues of the symmetric and exterior algebras can be obtained in our setting by working with the algebras obtained by factoring out the ideal generated by elements of the form PuŽ.m ¨ . Suppose that a bialgebra A Ž.with multiplication ␮ and comultiplication ⌬ has a corepresentation on V, that is, a coaction ␦: V ª A m V satisfying ⌬ m (␦ s m ␦ (␦ Ž.Ž.1 1 . With respect to a basis Ä4ej of V the coaction CLIFFORD ALGEBRAS AND QUANTUM GROUPS 249 can be written in the form ␦ s m Ž.ejjppa e ,99Ž. g where ajp A Ž.and the summation convention has been used . We can define a map Ž.Ž.␮ m 1 ␾␦m ␦ from V m V to A m Ž.V m V , where ␾: A m V m A m V ª A m A m V m V simply interchanges the second and third tensor factors. In order to obtain symmetries of the Clifford or Grassmann algebras it is natural to require that this map intertwines P and 1 m P, that is, Ž.Ž.Ž.Ž.Ž.Ž.␮ m 1 ␾␦m ␦ P s 1 m P ␮ m 1 ␾␦m ␦ . 100 However, the relations PuŽ.m ¨ s 0 for the deformed Grassmann algebra generated by V are also preserved by the coaction of the map intertwines ␴ ( P and 1 m ␴ ( P. The following result shows an interesting link between this condition and Manin’s quantised function algebras. s ␦ s THEOREM 4.1. Let dim V 2 and the coaction be defined by Ž.ejjpa m ep. Then Ž.Ž.␮ m 1 ␾␦m ␦␴( P s Ž1 m ␴ ( P .Ž.Ž.Ž.␮ m 1 ␾␦m ␦ 101 if and only if for some ␭ g A commuting with the matrix A s Ž.aweha¨e X X jp AGA s ␭G and A Gy1A s ␭Gy1. When

01 ab G s y and A s Ž.102 ž/yq 1 0 ž/cd these are equi¨alent to Manin’s conditions

ab s qy1 ba, cd s qy1 dc, ac s qy1 ca, bd s qy1 db, Ž.103 bc s cb, ad y da s qbcy1 y qcb, with ␭ s ad y qy1 bc s da y qcb, the quantum determinant of A. Proof. We know that in a two-dimensional space, ␴ ( m ¨ s 1 ¨ y1 m PuŽ.2 Bu Ž.Ž., Gejk jke ,Ž. 104 so that ␮ m ␾␦m ␦␴( m s 1 y1 m m Ž.Ž.Ž.1 Perse 2 GG rsjkjpkqpq a a Ž.e e ,Ž. 105 whilst m ␴ ( ␮ m ␾␦m ␦ m s 1 y1 m m Ž.Ž.Ž.Ž.1 P 1 erse 2 GG jkpqrjskpq aa Ž.e e . Ž.106 250 K. C. HANNABUSS

We therefore require

y1 s y1 GGrs jkaa jp kqGG jk pqaa rj sk ,Ž. 107 which reduces to

X y1 s X y1 GArsŽ.Ž.G A pq AGA rsG pq , Ž.108 and is clearly satisfied if and only if

X X AGA s ␭G, A Gy1A s ␭Gy1 Ž.109 for some ␭ g A. With the standard form

01 G s y Ž.110 ž/yq 1 0 and writing ab A s , Ž.111 ž/cd these two conditions become

ab y qy1 ba ad y qy1 bc 01 s ␭ y Ž.112 ž/cb y qdacdy1 y qdcy1 ž/yq 1 0 and

ca y qac cb y qad 0 yq s ␭ . Ž.113 ž/da y qbc db y qbd ž/10 These give Manin’s conditions as stated and also ␭ s ad y qy1 bc s da y qcb, which is Manin’s quantum determinant DET Ž.A . We note that by X q simplifying AGA Gy1A in two different ways we get ␭ A s A␭, showing that the quantum determinant is central in the algebra generated by the matrix elements of A Žcf. Takeuchi’s general proof of the centrality of the quantum determinant, as given inwx 12, Theorem 4.6.1. . Remarks. The first of the two conditions found above is equivalent to the condition that the coaction A should preserve B up to some multiple ␭, for that condition can be written in the form Ž.Ž.␮ m B ␾␦m ␦ s B, s ␭ X s ␭ which gives aajr ks BeŽ. r, e s Be Ž.j, e k or AGA G. This can also be X interpreted as saying that GA Gy1 is the right inverse of A, and the other condition is equivalent to demanding that it be the left inverse too.Ž Since X qG2 s 1 it could also be interpreted as the condition that A preserves B CLIFFORD ALGEBRAS AND QUANTUM GROUPS 251 as well as A.. The conditions are closely related to Tunstall’s proof that Annsatisfies Manin’s relations for parameter q wx11, Section 4.2.9 . In higher dimensions more useful results are obtained from the algebras of Example 1.7. We therefore consider the condition that the operator U Ž␴ q S m S . should intertwine the corepresentation map introduced above, that is, U Ž.Ž.Ž.␮ m 1 ␾␦m ␦␴q S m S U s Ž.1 m Ž.Ž.Ž.Ž.␴ q S m S ␮ m 1 ␾␦m ␦ . 114 It follows from the earlier expressions for S that U Ž␴ q S m S .Ž.Ž.Ž.Ž.x, ␰ m y, ␩ s y, ␩ m x, ␰ yŽ.x, q␰ m Ž.y, qy1␩ . Ž. 115 We shall restrict ourselves to the case of BxŽŽ, ␰ ., Žy, ␩ ..s ␩ Žx . y qy1␰ Ž.y s [ U on the space V W W , using dual bases e12, e ,...,er of W and U s ␦ f12, f ,..., frjkjkof W so that BeŽ., f , and take the coaction in the block form ␦ ␰ s q ␰ m q q ␰ m Ž.Žx, a. x b. .jjejj Žc. x d. .f . Ž. 116 A lengthy calculation tells us that the matrix elements of a commute with each other, as do those of b, c, and d, but that there are commutation relations s s abjk rsqb rs a jk, cd jk rsqd rs c jk , s s acjk rsqc rs a jk, bd jk rsqd rs b jk , s Ž.117 bcjk rscb rs jk, y s y y1 adjk rsda rs jkqc rs b jk q bcjk rs between the blocks. These correspond to even Manin relations with a single deformation parameter q. Due to the degeneracy of the eigenvalues of T, there is an obvious classical symmetric under the group GLŽ. W of linear transformations of W, which corresponds to the case of b s c s 0 X and ad s 1.

5. THE R MATRIX

␴ ( m s rs m ␦ s By writing PeŽ.jke Pe jkrse and Ž.e jae jrr, the intertwining property for the coaction of the last section becomes rs s rs aaPrp sq jkP pqaa jp kq,Ž. 118 252 K. C. HANNABUSS which is strikingly reminiscent of the R-matrix relation. Naturally this is no coincidence. Whenever the R matrix Rˆs R␴ satisfies a quadratic equation

Ž.Rˆˆy ␣ Ž.R y ␤ s 0Ž. 119 the projection onto the ␤-eigenspace can be written as

y Ž.␤ y ␣ 1 Ž.Rˆy ␣ . Ž.120

This projection determines Rˆ completely, and its intertwining operators are exactly those of Rˆˆ. In our case R satisfies the Hecke relation

Ž.Ž.Rˆˆy qRq qy1 s 0,Ž. 121 and P is the projection. In fact one can even admit a spectral parameter ␭ such that Rˆ␭ s R␭ ␴ , with spectral parameter ␭, satisfies a quadratic equation

y y q ␭ q y1 s Ž.ŽRˆˆ␭ 1 R␭ 1 Ž.q q .0.Ž. 122

s y ŽDefining RˆˆqRq 1 gives a solution of the Hecke relation, so that is merely a special case.. One readily checks that our standard form with Gram matrix 01 y Ž.123 ž/yq 1 0 lies in the 1 y ␭Žq q qy1 . eigenspace. The projection onto this eigenspace y1 isŽ.Ž. 1 y Rˆ␭ r␭ q q q . For two-dimensional V this eigenspace should be just one-dimensional, spanned by the above form, so that with respect to m m m m the basis e11122122e , e e , e e , e e , we have

0 1 y Rˆ␭ 1 1 y1 s y 01yq 0 ,Ž. 124 ␭Ž.q q qy1 1 q qy2 yq 1 ž/ 00 which gives the usual R-matrix

10 0 0 01y ␭q ␭ 0 Rˆ␭ s . Ž.125 0 ␭ 1 y ␭qy1 0 000 0 1 CLIFFORD ALGEBRAS AND QUANTUM GROUPS 253

The exceptional R matrix

1000 qy1 y q 010 Rˆs y ,Ž. 126 q y q 1 100 0q y qy1 y111 of Demidov et al.wx 4 has the exceptional form with Gram matrix

0111i 0 02i 0 G ss22Ž.127 ž/y1 y1 ž/0 yi ž/y21 ž/0 yi as yqy1-eigenvector. This has

y1 y1 y2 11i 0 y11i 0 T ss22,Ž. 128 ž/0 y10ž/0 yi ž/y1 ž/0 yi and is clearly equivalent to our exceptional form, given by Eqs.Ž. 19 and ᒐᒉ Ž.20 . The higher-dimensional R matrices for q Ž.n similarly have our standard forms with Gram matrix

01 y Ž.129 ž/yq 110 as eigenvectors.

6. SYMMETRIES OF FORMS

The symmetries of this system can also be studied in the dual enveloping algebra approach. As noted in Example 1.3, we need a comultiplication ⌬ on some suitable subset of linear transformations S of V, and look for those satisfying B(⌬Ž.S s B or its infinitesimal version, B(⌬ ŽL .s 0. To generalise Example 1.3 to higher-dimensional situations we note that if ⌬ L s L m 1 q K m L, then the condition that it preserve B reduces to U 0 s BLuŽ.Ž, ¨ q BKu, L¨ .Ž.Žs BLu, ¨ q B LKu, ¨ .Ž., 130 U so that we require L q LK s 0. We shall demand that K is a unitary U element, so that K s Ky1 ŽŽ.and ⌬ K s K m K preserves B., and this means that there is a neat expression in terms of tensors, for if L s L we U X have LK s L y1 , and the condition reduces to KT1 ˆŽ X . q ˆ s q y1 ˆ s KX11TXŽ. KXŽ.KTX1Ž. 0.Ž. 131 254 K. C. HANNABUSS

Thus we have proved the following result: ⌬ s m q m PROPOSITION 6.1. For unitary K the element LXXL 1 K LX sy annihilates B if and only if K1 X TXˆŽ.. One immediate consequence of this is the identity m sy sy s s m ␴ ŽK KX .KT211ˆˆ Ž. X TKX Ž .TTX ˆ Ž . ŽT T .Ž.X . Ž.132

y1 s s s y1 Now KLX K LŽ KmK . X LŽTmT .␴ Ž X . TL␴ Ž X . T gives the conjuga- tion action of K on LX . To obtain the standard quantum groups we need X to be an eigenvector of K m K, and so of Ž.T m T ␴ . This is readily achieved by taking V to be the direct sum of dual [ U isotropic subspaces W W , with W having a basis e12, e ,...,er , satisfying sy␶ Tejjje ,Ž. 133 ␶ s y2Žryj.y1 U where j q . With respect to the dual basis f12, f ,..., fr of W , s ␦ which satisfies BeŽ.jk, f jk, and we have sy␶y1 Tfjjjf .Ž. 134

We now define the unitary elements K j by y␦ q␦ s jk jq1 k Kejk q ek,Ž. 135 from which it follows that

␦ y␦ s jk jq1 k Kfjk q fk.Ž. 136 We also introduce, for j s 1, 2, . . . , r y 1,

s m q y1␶y1 m Xjjje f q1 q jjf q1 ej Ž.137 and s m q y1␶y1 m Yjje q1 fjjjjq f e q1 ,Ž. 138 and note that m s y1 m q y2␶y1 m Ž.K jj1 X q e jjf q1 q jjf q1 ej Ž.139 is identical to

y sy m y y1␶y1 m s ␶y1 m q y1 m TXˆ jjTf q1 ejjjjq Te f q1 jq1 fjq1 ejjjqe f q1 , Ž.140 CLIFFORD ALGEBRAS AND QUANTUM GROUPS 255

m sy m and similarly Ž.K jj1 Y Ž.T 1 Yj, so that these define infinitesimal s m s m symmetries of B. We complement these with Xrrre e and Y rrrf f , which also provide infinitesimal symmetries of B. Finally we define the T-commutator to be wxs y y1 X, Y T XY TYT X. Ž.141 This reduces to the usual formula for symmetric and antisymmetric forms where T s "1, and also gives wxs y LXY, L T LL XYLLŽTmT .YX Ž.142 for operators of the kind we have been discussing, reflecting the inter- change symmetry of B.Ž It will sometimes be convenient to abbreviate wxwx LXYT, L to X, Y T.. We also have (⌬ wxs (⌬ ⌬ y ⌬ ⌬ ⌬ y1 s B Ž.X, Y T B X Y B T Y Ž.TX 0. Ž. 143 ŽIt would be possible to redefine things to give normal commutators using, wxwxs for example, the identity TX, Y T TX, Y .. ¨ THEOREM 6.2. The elements K jj, X , and Y j, defined abo e, satisfy y y ␦ q␦ y ␦ y␦ 1 s 2 jk 1< jyk < 1 s 2 jk 1< jyk < KXKjkj q Xk, KY jkj K q Yk, 2 y K j 1 Ž.144 X , Y s ␦ , jkT jkqy2 y 1 for j s 1,...,r, k s 1,...,r y 1, and y y ␦ q ␦ y ␦ y ␦ 1 s 2 jk 2 Ž jq1.r 1 s 2 jk 2 Ž jq1.r KXKjrj q Xr, KYK jrj q Yr, 2 y K j 1 Ž.145 X , Y s ␦ . jrT jrq y qy1 Proof. As all the operators are known explicitly on V this result can be obtained by direct calculation. Apart from some rescaling these relations ᒐᒍ give the quantum enveloping algebra qŽ.2r , as might seem appropriate as B gives a deformed symplectic structure.Ž We think of L as E , and Xjj y1 YKjjas F j.. We should note that these relations do not merely rely on the particular representation on V. Since comultiplication is a homomorphism, we have Ž.identifying the tensors and operators ⌬ X , Y s ⌬ X , ⌬ Y Ž.jjTTŽ.jŽ. k s ⌬ ⌬ y ⌬ y1 ⌬ Ž.Ž.XjjY Ž.TY j T Ž.X j s ⌬ ⌬ y ⌬ y1 ⌬ Ž.Ž.XjjY Ž.KYK jjjjŽ.X ,Ž. 146 256 K. C. HANNABUSS or, more explicitly, X , Y m 1 q K 2 m X , Y q XKm Y y KXm KYKy1 jkTTj jj jj j jj jjj q m y m KYjjX jKY jjX j.Ž. 147 The cancellation of the last two terms provides part of the motivation for the use of the T-commutator, but since Xjjand Y give inverse eigenvalues the third and fourth terms also cancel, leaving us with ⌬ X , Y s X , Y m 1 q K 2 m X , Y , 148 Ž.jjTTjk j jjTŽ. wx 2 y from which we readily see that XjjT, Y is a multiple of Ž K j 1. , with the representation determining which multiple.

ACKNOWLEDGMENT

The author thanks the anonymous referee for numerous helpful suggestions.

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