Quantum Deformation of the Grassmannian Manifold

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1 Journal of Algebra 214, Article D jabr , available online at on Quantum Deformation of the Grassmannian Manifold R. Fioresi Department of Mathematics, Uniersity of California, Los Angeles, California rfioresi@math.ucla.edu Communicated by Susan Montgomery Received September 30, 1997 n this paper we work out a deformation of GŽ r, n., the grassmannian of r-subspaces in a vector space of dimension n over a field k of characteristic 0. Gr, Ž n. is deformed as an homogeneous space for SL Ž k. n, the special linear group n of k ; this means that kgr, Ž n., the coordinate ring of Gr, Ž n., gets deformed together with with the coaction of ksl, the coordinate ring of SL Ž k. n n, on it. Our deformation comes together with a coaction of the corresponding deformation of SL Ž k. n. At the end we give an explicit presentation of the deformed grassman- nian, in terms of generators and relations Academic Press NTRODUCTON The concept of uantum homogeneous space has been first introduced by Manin 1 in the context of the uantization of linear algebraic groups over a field k, algebraically closed and of characteristic 0. A uantum homogeneous space is a deformation of the coordinate ring of an algebraic variety on which there is a transitive action of a linear algebraic group G, together with a coaction of the uantum group associated to G. An interesting example of a homogeneous space is the grassmannian Gr, Ž n. of r-subspaces of a vector space V of dimension n. The special linear group SL Ž k. acts on Gr, Ž n.; kgr, Ž n., the coordinate ring of Gr, Ž n. n,is given by the subring of ksl, the coordinate ring of SL Ž k. n n, generated by the determinants of certain minors. A first attempt to provide a deformation of Gr, Ž n. as a uantum homogeneous space was made by Taft and Towber 2. n their paper they exhibit a deformation of the ring kgr, Ž n. together with a coaction of k M on it, where k M is a deformation of the matrix bialgebra. The n n $30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved. 418

2 DEFORMATON OF THE GRASSMANNAN MANFOLD 419 generators of the deformed ring k GŽ r, n. are in one to one correspon- 1 r l dence with the uantum determinants Di i Ý S ai Ž1. r 1 1 a k SL k, a deformation of SL k over k k, i Ž r. n n, 1 r i1 ir n and satisfy some of the relations satisfied by the Di 1 r i s. However, not all the commutation relations among the Di 1 r i are provided there explicitly. n 3 Lakshmibai and Reshetikhin take a very different approach to a generalization of the same problem. They give uantizations for all the flag manifolds of the classical groups using a uantization of the universal enveloping algebra associated to the Lie algebra of the group. Their approach is very abstract; their uantization depends on the duality between the uantized coordinate ring and the uantized universal enveloping algebra of a group G, and they do not exhibit explicitly the deformations of the Plucker relations. A similar approach has been taken also by Soibelman 4, but his treatment is even more abstract and the non-commutative ring of the uantum grassmannian is not exhibited in any explicit way. n this paper we define k GŽ r, n., the deformation of Gr, Ž n., in a very natural way as the subalgebra of k SL Ž k. n generated by the uantum 1 r determinants: D. k GŽ r, n. i i acuires naturally a uantum homoge- neous space structure: the coaction of k SL Ž k. n on it is given by the comultiplication in k SL Ž k. restricted to the subalgebra k GŽ r, n. n. The difficulty is then transferred to the determination of the commutation rules between the uantum determinants and of the uantum Plucker relations. The relations satisfied by the uantum determinants are of two types: Ž. 1 commutation relations: for 1 these state the commutativity of the coordinate ring of the grassmannian; Ž. 2 uantum Plucker relations: for 1 these reduce to the classical Plucker relations. The uestion arises whether these relations give a presentation of the deformed grassmannian. We prove that this is true if we change the base ring from k to A the local subring of k at 1, i.e., the ring of all elements x fg, f, g k, gž At the end we briefly discuss an example of physical interest: k GŽ 2, n. Žsee 5 for more details.. 1. DEFNTON OF THE QUANTUM GRASSMANNAN RNG k GŽ r, n. 1 DEFNTON 1.1. Let k k,, where k is an algebraically closed field of characteristic 0. Let k ² a : be the free algebra over k with a as ij ij

3 420 R. FORES non-commutative generators. Define k M n, the uantum matrix bialgebra, as the associative k -algebra with unit generated by n 2 elements a ij, i, j 1...n4 subject to the relations a a 1 a a, i k, a a a a, i k, j l or i k, j l ij kj kj ij ij kl kl ij aijail 1 aila ij, jl, aijaklaklaijž 1. akja il, i k, jl We shall refer to the two-sided ideal in k ² a : ij generated by these relations as M, the ideal of the Manin relations. We can summarize the Manin relations as where aijakl AŽi, j, k, l. BŽi, j, k, l. aklaij CŽ i, j, k, l.ž 1. akja il, 1 if i k, j l AŽ i, j, k, l. 1 if i k, j l 0 otherwise 1 if j l, i k BŽ i, j, k, l. 1 if j l, i k 0 otherwise 1 if i k, j l CŽ i, j, k, l. 1 if i k, j l 0 otherwise. All the congruences in k ² a : ij are modulo M unless otherwise specified. Let s define on k M the comultiplication and the counit : n Ý Ž a. a a, Ž a. ij ik kj ij ij k k M n is a bialgebra with the given and. Let s now introduce the notion of uantum determinant. DEFNTON Ž We define the uantum determinant obtained by takj 1 j ing rows i i, columns j j as an element D m k ² a : 1 m 1 m i i ij given 1 m by Ý j j lž. 1 m i i def 1 m i1 Ži 1. i m Ži m. : Ž i1 i m. Ž j1 jm. D a a, 1 i i n 1 m 1 j j n, 1 m

4 DEFORMATON OF THE GRASSMANNAN MANFOLD 421 where runs over all the bijections and lž. is the length of the j 1 j permutation. m is called the rank of D m. ts image in k M i i n is then 1 m the usual uantum determinant. We shall write D j 1 j m i1 i m for this image also, the context making it clear where the element sits. For all the properties of these uantum determinants see 6. This enables us to define the Hopf algebras 1 n k SL k M D 1 n def n 1 n ji 1 ˆi n Ž ij. def 1 ˆj n S a D ² : 1 n k GL k M T D T 1, a T Ta n def n 1 n ij ij ji ˆ 1 1 i n 1 n Ž ij. def 1 ˆj n 1 n S a D D, where S denotes the antipode. t is customary to identify the indeterminate T with D 1 1 n1 n. k GL n can be viewed as the non-commutative localization of the ring 1 n k M by the central element D. As in the commutative case, k M n 1 n n embeds into k GL. n DEFNTON Ž Define k GŽ r, n., the k -uantum grassmannian, as 1 r the subalgebra of k SLn generated by D i i, 1 i1 ir n. Observe that for 1 this is the ring kgr, Ž n.. PROPOSTON Ž k GŽ r, n. is a k SL n -comodule and the coaction is gien by. Proof. We first need to check that Žk GŽ r, n.. k SL n Ž 1 k G r, n. t suffices to check that D r. k SL k GŽ r, n. i1 ir n ž Ý / 1 r lž. i i i1 Ž1. i r Ž r. S r n n lž. D a a Ý Ý Ý i j i j j Ž1. j Ž r. Ž. a a a a S j 1 j 1 r 1 r n n lž. Ý Ý i j i j Ý 1 r j1 Ž1. jr Ž r. j11 jr1 Sr a a a a n n l 1 r Ý Ý i1j1 irjr J j11 jr1 a a D, where is the permutation that takes j j into the ordered set J.

5 422 R. FORES satisfies the coaction properties by definition; in fact the coassociativity of a coaction is the same as the coassociativity axiom for. The same is true for the counit axiom. Our aim is to give a presentation of the ring k GŽ r, n. in terms of generators and relations. The relations among the generators Di 1 r i will be of two types: the commutation relations and the uantum Plucker relations. To simplify the notation we will denote Di 1 r i with Di i whenever there is no danger of confusion. We will start working out the commutation relations among the determinants as elements of k M. From Ž 1.2. n it is clear that such relations will hold in k SL and k GL. n n DEFNTON Ž Let s define a commutation relation for D and D J, J, D, D k ² a : to be an expression R k ² a :, J ij DDJ M ij R f Ž. D D g Ž. D D F Ž. D D, DD J def J J J J KL K L K, KL where f Ž., g Ž. k are invertible elements, f Ž 1. g Ž 1. J J J J 1, F Ž. k, F Ž 1. KL J 0, and is by respect to the lexicographic order. Capital letters will indicate multi-indices, i.e., Ž i i.. The multiindices, J, K, L are all supposed to be ordered, where an index X is ordered if X Ž x x., x x. We want to determine f Ž., g Ž., and F Ž. J J KL in such a way that RDD M for all, J s. Notice that even the existence itself of the J commutation relations for all the D that define k GŽ r, n. is not at all obvious. t will turn out to be somewhat involved to determine the commutation rules. We shall do this in Sections 2 and 3. The crucial case is when J. Because of the involved nature of the commutation rules and their proofs, we have treated as an example in Section 2 the commutation rule between two 3 3 uantum determinants. The main result in Section 2 is theorem Ž We will consider separately the two cases: J and J. Remark Ž Notice that, with the form of the commutation relation defined in Ž 1.5., given a generic polynomial p in the generators D s, p Ý a D D there exists a polynomial p such that a k 1 M 1 1 M M Ý J 1 J M J 1 J M bj J k where J J 1 M Ý p p b D D, 1 M and is respect the lexicographic order.

6 DEFORMATON OF THE GRASSMANNAN MANFOLD 423 Obseration Ž n view of the nature of the relations involved it is sufficient to give a presentation for k GŽ r,2r.. n fact, if n 2r, given two determinants D, D in k GŽ r, n. i i j j their commutation relations will depend only on the rows i1 i r, j1 j r, hence we can view them as determinants in k GŽ r, 2r.. The ideal of relations in k GŽ r, n. among the generators is obtained in the following way. t consists of all the relations among the D s in k GŽ r,2r.k SL k ² a : i i 2 r ij M where i, i i 4 l 4, j 1, r 1 2r for all the sets of distinct l l 4 1 n 4. n case n 2r, since, as we will see, k GŽ r, n. 1 2r k GŽ n r, n. and n 2n 2r, we are back to the case already dis- cussed. 2. COMMUTATON RELATONS: THE CASE J DEFNTON Ž Let a a k ² a : ij. Define the commutation relation of a a applied to a a as an expression R Ž a a. eual to ij kl aijakl R Ž a a. a a AŽi, j, k, l. BŽi, j, k, l. a a aijakl where the functions A, B, C are as in 1.1. CŽ i, j, k, l.ž 1. a a, Notice that in general R Ž a a.. LEMMA 2.2. aijakl M 1 Ra a aijakl akjail if i k kj il Ra až aijakl. ij kl R a a 1 a a if i k. akjail ij kl kj il Proof. Let i k. By definition R a a a a AŽi, j, k, l. a a C i, j, k, l 1 a a aijakl ij kl ij kl kl ij kj il R a a a a AŽ k, j, i, l. a a C k, j, i, l 1 a a. akjail ij kl ij kl kl ij kj il f we subtract the second euation from the first we get R a a R a a AŽ k, j, i, l. AŽi, j, k, l. a a aijakl ij kl akjail ij kl kl ij kj il 1 CŽ i, j, k, l. CŽ k, j, i, l. a a.

7 424 R. FORES Using the definitions of A and C we see that 1 AŽ k, j, i, l. AŽi, j, k, l. if j l ½ 0 if j l 1 if j l CŽ i, j, k, l. CŽ k, j, i, l. ½ 0 if j l and this proves the result. For i k the argument is similar. PROPOSTON Ž Let Ž i i., Ži i. the complement of in 1 2r4 with, 1 i1 ir 2r, 1 i1 ir 2r and Ž,. Ž 1 2r.. Then there exists always an elementary transposition Ž m, m 1. such that Ž., Ž. Ž.. ŽNote. Gien the multiindices Ž i i., J Ž j j., by Ž, J. we mean the r s-uple Ž 1 s i1 i, j j.. r 1 s. Proof. Let p be the first integer such that i p i p1 1. This p must exist since Ž,. Ž 1 2r.. Then Ž i, i 1. p p is the reuired ele- mentary transposition. DEFNTON Ž Given and as in Proposition Ž 2.3. let s call the transposition used in the proof of Proposition Ž 2.3. a standard transposition. From the construction it is clear that such a is uniue. Define intž., the intertwining number of, to be the number of standard transpositions necessary to go from to Ž 1 r.. We define as the standard tower for a seuence Ž 1 r. where Ž. N 0 i i1 i1 and all i are standard transpositions. By definition, given, we have a uniue standard tower and its length N intž.. Notice that, from the definition of standard transposition, all of the multi-indices N 0 turn out to be ordered. DEFNTON Ž Let T k ² a : ij be the free module over k gener- ated by the monomial tensors a a, PQ p p 1 1 2r 2 r where P Ž p p., Q Ž., and p 1 2r 4, 1 r 4 1 2r 1 2r j j, j 1, 2r and the p j s are all distinct. The permutation group S acts on T in the following way: 2 r ž Ý / Ý b b. PQ PQ PQ Ž P.Q bpqk bpqk

8 DEFORMATON OF THE GRASSMANNAN MANFOLD 425 Remark Ž Notice that RDD T. Notice also that if t M T in J general Ž. t T. M DEFNTON Ž Given T a monomial tensor, Fa a is defined ij kl as follows. f aij is adjacent to akl then Fa a is the element of T ij kl obtained from by replacing a a with ij kl AŽi, j, k, l. aklaij CŽ i, j, k, l.ž 1. akjail otherwise, F 0. F is known once we know Ž i, j, k, l. a a a a. Call the ij kl ij kl monomial tensor in F containing a a the principal part, Ž F. a a kl ij a a pp, ij kl ij kl and the remaining term, if different from 0, the tail. We will refer to the generic Fa a as an operation of type F on the tensor. f t ÝbPQPQ ij kl T, Fa a t is obtained by performing the operation Fa a on only one of the ij kl ij kl monomial tensors in t and of course we need to specify which one. For example, if we have a11a22 a33 a12 a21a33 and we are performing the operation F on the second monomial tensor appearing in we have a12 a21 F a a a a a a. a a Let t Ý n i0 bpq PQ, PQ monomial tensors, bpq k and assume that i i i i i i i i some Fa a operates on P Q. We define the principal part of Fa a t as ij kl 0 0 ij kl the principal part of F and the tail as F t Ž F. a a P Q a a a a P Q pp. ij kl 0 0 ij kl ij kl 0 0 We will also say that FFa F ij a kl a i j a k l is the composition of the two operations F a, F ij a kl a i j a k l, meaning that the tensor Ft is obtained performing first the operation Fa a and then the operation Fa a on the result. i j k l ij kl Let F F F operate on t. Given S we say that F a a a a 2 r,a i1 j1 k1l1 isjs ksls composition of operations of type F operating on Ž. t,is associated to F if F Fa a Fa a and each Fa a operates on the Ži 1. j1 Ž k 1.l1 Ži s. js Ž k s.ls Ži u. ju Ž k u.lu monomial tensor Ž. PQ corresponding to the monomial tensor PQ in t on which F operates. ai j a u u kulu Notice that T is stable under the operations F a a s, i.e., if t T, ij kl F t T. aijakl LEMMA Ž Let T, monomial tensor, Ž m, m 1. S 2 r, Fa a as defined aboe. Let Fa a be the operation associated to F ij kl Ži. j Ž k.l aijakl

9 426 R. FORES operating on. Then Ž. aijakl i F Ž ii. Ž Fa a. if i, k4 m, m 14 ij kl F 1 F a a Ž. if Ž i, k. Ž m 1, m. Ž a a. a Ži. ja Ž k.l ij kl 1 F Ž. if Ž i, k. Ž m, m 1.. ij kl Proof. Part Ž. i comes from the definition of operation of type F; Ž ii. comes from Lemma Ž Remark 2.9. From 2.8 it is clear that if F is a composition of operations of type F and t T is not necessarily a monomial tensor, then Ft t. Remark Ž f Fa and F ij a kl a i j a k l act on different monomial tensors in t, then F F t F F t. a a a a a a ij kl i j k l i j k l aijakl LEMMA r D D D D. r1 2 r 1 r 1 r r1 2 r Proof. This is by induction on r. The case r 1 is clear. For general r, using the expansion along the column r, we can write Žsee. 6 r 2 r 1 r 1 r irj2 r 1 r1 1 1 r r1 2 r Ý Ý 1 ˆi r ir r1 ˆj 2 r jr i1 jr1 D D D a D a r 2 r ij3r 1 r1 1 Ý Ý 1 ˆi r r1 ˆj 2 r ir jr i1 jr1 D D a a r 2 r 1 Ž r1. ij3r 1 r1 1 Ý Ý r1 ˆj 2 r 1 ˆi r jr ir i1 jr1 D D a a. The last congruence is by the induction hypothesis. Now D 1 r1rˆ 1 r1rˆ 1 ˆi rj ˆajr ajr D1 ˆi rj ˆ ž / 1 1 Ž r1. p 1 pˆ r Ý jp 1 ˆi rj ˆ pr, pi a D.

10 DEFORMATON OF THE GRASSMANNAN MANFOLD 427 f we substitute in, D 1 1 r rdr1 1 r 2 r r Dr1 1 r 2 r D1 1 r r 1 Ž 1. r r 2 r ij3r 1 r1 Ý Ý Ž. Dr1 ˆj 2 r i1 jr1 ž / Ý Ž Ž r1. p 1 pˆ r. jp 1 ˆi rj ˆ ir pr, pi a D a. Let us look at the last term only, 2r Ž r1. 1 j2 rž r1. p 1 pˆ r Ž. Ý Ý Ž. Dr1 ˆj 2 rajp j11 pr, pi r ir 1 pˆ r Ý Ž. D1 ˆi ra ir. ž / i1 r ir 1 pˆ r 1 r i1 1 ˆi r ir pr 1 r But this is 0 since Ý D a D 0. LEMMA Ž Fix on the set a 4 ij i1, 2 r, j1, r some total ordering. Let t Ý b a a k ² a :, J J i j i j ij a homogeneous tensor of degree s. f 1 1 s s t 0 and a a, J, then t 0. i1j1 isjs Proof. Consider the k-homomorphism : k ² a : k² a :, Ž a. ij ij ij a, Ž. 1. Then we have that Ž. Ž a a a a, i, j, k, l. ij M ij kl kl ij. Assume that t 0. Then divide each bj by the highest power of 1 dividing the gcd of the b s. Let t be the tensor so obtained. Then Žt. J will give a non-trivial relation among the a ij, which is a contradiction. DEFNTON AND REMARK Ž Let PQ T a monomial tensor, P and Q as in Definition Ž Given a multi-index K and its complement K in 1 2r 4, so that Ž K, K. Ž k k. 1 2r, we can define an order relation K on the elements of P: pik pj if and only if pi k s, pj k t, s t. We say that the monomial tensor PQ T is in order Ž K, K. if p p i j, i.e., ŽK, K. i K j P. Similarly we say that t Ýb T, b k is in order Ž K, K. PQ PQ PQ if every PQ is. The given order relation allows us to define a total order on the a ij s; namely aijk akl if i K k or if i k, j l. Define lk, P as the minimum number of transpositions i of adjacent p j s necessary to go from P to Ž K, K..

11 428 R. FORES LEMMA Ž Let P Ž p p. S, P ŽK, K. 1 2r 2 r, and 1 l K, P the transpositions defined in Ž Fix i, j, i j. Then we hae the following: Ž. 1 f p p then Ž p, p., k 1, l i K j k i j K, P. Ž. 2 f p p then Ž p, p., for a uniue k 1, l. Proof. i K j k i j K, P This is obvious. LEMMA Let T be a monomial tensor. Then PQ F b b, a a PQ PQ PQ PQ PQ ij kl where bp Q 0 or P2 P Proof. This is obvious by Definition Ž 1.1. of Manin relations and by Definition Ž 2.7. of F. aijakl LEMMA Ž Let PQ T, K, 1 l as aboe, with 1 K, P Ž p, p., p p. Define F F. For Ž p, p. i i1 i K i1 PQ a a PQ m k l, 1 p i i pi1i1 p p define k K l Then: F F F F F F m m1 1 PQ ap a PQ kk pll m1 1 pp F F F F. m 1 1 PQ m1 1 PQ pp Ž.Ž i F F. is in order ŽK, K. PQ pp. lk, P 1 Ž ii. F F Ž F F. PQ PQ pp Ýi, b k bp Q P Q where lk, P 1 lk, P 1 PQ i i i i i i lk, P l K, P. i Proof. Part Ž. i is a conseuence of the definition of the s Ž Part Ž ii. is coming from Ž K LEMMA Let PQ T, K, 1 l as aboe. There exists F, a K, P PQ K composition of operations of type F, such that F is in order Ž K, K. PQ. PQ Moreoer F K can be chosen such that gien two indices p i, pj P, i j, the PQ two corresponding elements in each monomial tensor of F K PQ hae been PQ interchanged once by some F in F K if pik p j, and hae not been PQ interchanged if pik p j. Proof. By induction on l K, P. For lk, P 0 it is obvious. Let lk, P 0. By Lemma Ž there exists F F such that l 1 PQ K, P F F F F F F l l P1 1 PQ l l 1 1 PQ K, P K, K, P K, P pp F F F F, l 1 1 PQ l 1 1 PQ K, P K, P pp

12 DEFORMATON OF THE GRASSMANNAN MANFOLD 429 where Ž F F. is in order ŽK, K. PQ pp and F F PQ lk, P 1 lk, P1 1 Ž F F. PQ pp Ýi, b k bp Q P Q where lk, P l K, P. By induclk, P1 1 PQ i i i i i i i tion hypothesis for every i there exists F K operating on PQ such that PQ i i i i K Ž. K K F PQ is in order K, K. Define F Ł i F F F. We can PQ i i i i PQ PQ i i lk, P 1 take the product without specifying the order because the F K PQ operations i i operate on different monomial tensors Žsee Remark Ž Certainly K F is in order Ž K, K. PQ. Now fix p i, pj P, i j. f pi K pj then PQ Ž p, p., k by Lemma Ž k i j and by induction no operation in K Ł F interchanges a and a.f p p then by Ž i p p i K j there is a PQ i i i i j j uniue k such that Ž p, p.. This means that k i j F F F F l 1 1 PQ l 1 1 PQ K, P K, P pp Ý Ý c d, PuQu PuQu PQ PQ u, cp Qk, dp Qk u u where in each P Q for all u, ap, ap have been interchanged once by u u i i j j F, while in each P Q they have not been interchanged. So we have that k Ž u u. u u u u in P Q, Pu p1 p 2r, pi p t, pj p s, s t, i.e., ps K p t, hence u u by the induction hypothesis no operation in Ł u F K P Q will interchange u u Ž. a u u, a u u p p.n P Q, P p1 p2r pi p s, pj p t, s t, i.e., ps t t s s K pt hence by induction hypothesis some operations Ł F K P Q will inter- change a, a once in each term. pt t pss K LEMMA Let and F as aboe. Let Ž p, p. Ž PQ i j m, PQ. Ž K. m 1 S 2 r, i j, m K, m 1 K, and let F Ž P.Q be associated to K F and Žsee Definition Ž PQ. Then F Ž K. F K 1 EŽ p, p., ž PQ / Ž P.Q Ž P.Q PQ i j 0 where 0 PQ and ½ 0 if pik p j, i.e., pi m, pj m 1 EŽ p i, pj. 1 if pik p j, i.e., pi m 1, pj m. Proof. This comes from 2.17 and 2.8. THEOREM Ž Ž Main Result.. Let Ž i i., J Ž j j. the complement of in 1 2r4 with,1i1 ir 2r, 1 j j 2r. Let N intž., then N r 1 i D D D D Ý Ý L L, i0 L, L C i

13 430 R. FORES where the set Ci is defined in the following way. Consider the standard tower Žsee Ž 2.4..: Ž 1 r., Ž m, m 1.Define. N 0 N C 1, i Ž Ž C s C s 1 s 1, C s 1 i Ž i1 i1 sž i.., 0 i s N, Ž i, s. Ž 0,0. C C N i i, where with the notation for C s i we mean C s X, X Ž X, X., Ž Y, Y. Ž Y, Y i Ž 1 1 p p C s 1 s 1 X, X X, X, C s 1 Y, Y Y, Y. Ž. Ž. i1 i1 1 1 p p i 1 1 s is the subset of C s of indices Ž L, L. Ž l l. i i 1 2r such that k 1 l u, s k l, u t, k, k 1. We agree to put C and s t s i i eual to the empty set if one of the indices i or s is negatie orifis. Notice that the Ž. Ž. X, X s and Y, Y s in C s j j l l i need not be distinct. Proof. This is by induction on N. For N 0 this is Lemma Ž Let N 0. Ž n!. 2 Ý N 1 N1 PQ l l PQ l l l1 D D b, where b k and T are monomial tensors. Let PQ PQ l l l l Ž n!. 2 N 1 D D Ł l1 F F N1 N1 N1 PlQl the composition of operations of type F such that F N 1 D D D D is N1 N1 N1 N1 in order Ž,.Žsee Lemmas Ž 2.16., Ž N1 N1. Notice that since each F N 1 PQ operates on different monomial tensors it does not matter the order l l in which we take the composition Žsee Remark Ž Since m 1 N1, m, for every P Ž p p. that appears in N1 l 1 2r there exist two indices pi and pj such that pi m 1, pj m, i j. Hence pi p j. N 1 By Lemma Ž 2.18., ž / r D D F r D D F N 1 r D D D D D D N1 N1 N1 N1 Ž 1. t N 1,

14 DEFORMATON OF THE GRASSMANNAN MANFOLD 431 where t r D D and F is associated to F N 1 D D D D and. N 1 N1 N1 N1 N1 Hence by the induction hypothesis N1 1 i Ý Ý N 1 L L i0 N1 t D D. L, L C i To finish the proof of the theorem and obtain from the reuired congruence, we need to show that ž D D N1 N1/ N1 N1 F N 1 r D D N1 1 i Ý Ý L L i0 Ž L, L. Ž C N1 i. D D N 1 i Ý Ý L L i1 Ž L, L. N1 i1 D D. Let t 0 be the tensor N1 M 1 i Ý Ý 0 L L Ý PQ l l PQ l l i0 Ž L, L. C N1 i l1 t D D b for suitable b k, T monomial tensors and for some integer PQ PQ l l l l M. Let By induction we have that But by Lemma 2.12, Let s write t M N 1 F F N1 t Ł. 0 PQ l1 l l F N 1 r D D t F N 1 t. D D N1 N1 0 t 0 0 N1 N1 F N 1 r D D F N 1 t. D D N1 N1 t0 0 N1 N1 as t t t, where N1 1 i Ý Ý 1 L L i0 Ž N1 L, L. C N1 i i t D D N1 1 i Ý Ý 2 L L i0 Ž L, L. N1 i t D D.

15 432 R. FORES N 1 Ž. By definition of, L, L l l N 1 i 1 2r i we have that there exist indices s, t, s t, such that ls m 1, lt m, hence ls l t. N 1 Hence by Lemma Ž we have Ž t. Hence 0 t t F Ž N 1. t F Ž N 1. Ž t. 1 2 Žt 1. 1 Žt 2. 2 Ž t 1. Ž t 2. F N 1 t F N 1t 1 2 N1 1 1 i Ž. Ý Ž. Ý D D L L i0 Ž L, L. N1 i N1 1 1 i Ý Ý N 1 t0 0 L L i0 Ž L, L. N1 i F t D D. N1 1 1 i N 1 Ý Ý t0 0 0 L L i0 Ž L, L. N1 i F t t D D By this is ž / N1 1 i Ý Ý L L i0 Ž L, L. C N1 i D D N 1 i Ý Ý L L i1 Ž L, L. N1 i1 D D. ž D D N1 N1/ N1 N1 F N 1 r D D N1 1 i Ý Ý L L i0 Ž L, L. Ž C N1 i. D D N 1 i Ý Ý L L i1 Ž L, L. N1 i1 D D. DEFNTON Ž For a given multi-index K Ž k k. let K ord Ž ord ord. 4 ord ord k k be the multi-index such that k k k k 4, ord ord k k. Let be the permutation such that Ž K. K. K K ord

16 DEFORMATON OF THE GRASSMANNAN MANFOLD 433 PROPOSTON Let the notation be as in Then R r D D DD N i Ž 1 l L l L. Ý Ý Lord Lord i1 L, L C i D D is a commutation relation for DD. Proof. By Theorem Ž and by Definition Ž 2.20., to show that R we just need to show C ŽŽ,.. M 0. This is by induction on N. For N 0 it is clear. For n 0, Ž. C C N 1 Ž,. Ž,.. 0 N 0 N N1 N1 To show that R is a commutation relation for D, D, according to Definition Ž 1.5. we need to show: Ž. i The coefficients of D D and DD are invertible and for 1 they are eual to 1, 1, respectively, while the coefficient of D D L L is eual to 0 for 1. Ž ii. L L, L. To show Ž. i it is sufficient to show Ž,. ord ord ord Ž. Ž. Ž. Ci ord where Ci ord contains all K ord, K ord with K, K C i. This is by induction on N. For N 0 it is clear. Let N 0 and assume Ž, C N 1., N1 N1 i ord Ž. C C N 1 N 1, C N 1. i i1 i1 N i Ž. Ž N 1., C because in C N 1 appear only indices K, intž K. i1 ord i1 ord Ž. Ž int., C N 1. by induction. N i ord Let s prove Ž ii. by induction on N. f N 0 it is obvious. Let N 0. By induction for every L, L C N 1 ord ord i1 we have Lord N1 N, Ž N 1. L L. Let L C then L C N 1 ord ord N i N i. Again by induction, Ž Ž L... Since Ž m, m 1. N ord N1 i for some m and m N1, m 1 we have immediately L Ž. N1 ord N N1. Moreover since no Ž 12. Ž see how the standard transpositions are defined. i we have L Ž 1, l l. L. ord 2 r EXAMPLE Ž The commutation relation for D145 D 236. The standard tower for Ž 145. is given by 4 Ž Ž Ž Ž Ž , 45, 23,

17 434 R. FORES According to the definition of C in Ž we have that i Ž. C 0 Ž 123., Ž Ž. C 1 Ž 124., Ž Ž. C 1 Ž 123., Ž Ž. C 2 Ž 125., Ž Ž. C 2 Ž 124., Ž 356., Ž 123., Ž Ž. C 2 Ž 123., Ž Ž. C 3 Ž 135., Ž Ž. C 3 Ž 125., Ž 346., Ž 134., Ž 256., Ž 132., Ž Ž. C 3 Ž 124., Ž 356., Ž 123., Ž 546., Ž 132., Ž Ž. C 3 Ž 123. Ž Ž. C 4 Ž 145., Ž Ž. C 4 Ž 135., Ž 246., Ž 125., Ž 436., Ž 143., Ž 256., Ž 142., Ž Ž C 4 Ž 125., Ž 346., Ž 134., Ž 256., Ž 132., Ž 546., 2 Ž Ž 123., Ž 456.., Ž Ž 124., Ž 536.., Ž Ž 142., Ž Ž. C 4 Ž 123., Ž 546., Ž 132., Ž 456., Ž 124., Ž Ž. C 4 Ž 123., Ž According to Theorem 2.19 the commutation relation is D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D.

18 DEFORMATON OF THE GRASSMANNAN MANFOLD 435 We can write it using the notation of Proposition Ž 2.21., i.e., the fact that D Ž. l Ž L. D. L L ord 3 D D D D D D D D 1 2 D134 D256 D124 D D D D D D D 1 1 D123 D456 D124 D356 D124 D D D D D D D D D. Notice that in constructing the commutation relation for D 145, D 236, we get also those for D 124, D 356; D 125, D 346; D 135, D 246. According to Theorem Ž they are 3 D356 D124 D124 D356 Ž 1. D123 D D D D D Ž. D D D D D D D D D D 1 Ž. D125 D346 D134 D256 D132 D D D D D D D D D. n the notation of Proposition Ž after some simplifications this becomes 3 D356 D124 D124 D356 Ž 1. D123 D D D D D Ž. D D D D D D D D Ž. D125 D346 D134 D256 D123 D456 1 Ž. D124 D 356. All these relations can be verified by direct computation.

19 436 R. FORES 3. COMMUTATON RELATONS: THE CASE J Now we turn to the problem of determining the commutation rule for DD, J in case J. LEMMA Ž Let S be the antipode for k SL Žsee Definition Ž n. Then: 1 ˆi ˆ j j Ž j j. Ž i i. 1 i r n i i 1 ˆj ˆj n S D D. Proof. This is by induction on r. For r 1 this is the definition of S. Let r 1. For all the determinant expansion refer to 6, Chap. 4, r j k1 ĵ1 j2 j 1 j r r i i Ý i ˆ ž 1 i k ir ikj1/ k1 S D S D a r k1 ĵ1 j2 jr Ý Ž i j. i ˆi i k1 S a S D ž / k 1 1 k r r k1 Ž j ˆ 2 jr. Ž i1 i k i r. j1i k Ý k1 1 ˆ i n 1 ˆ i i ˆ k 1 k i r n D1 ˆj n D1 jˆj ˆj n r r Ž j j. Ž i i. k1 1 ˆi k n Ý 1 j1 n k1 Ž. Ž. D ˆ Ý Ž Ž. 1 ˆi ˆi ˆ 1 k i r n. li 1 jˆj ˆj ˆ k 1 2 r l n l1,..., j ˆj ˆj n4 a D, 1 2 r where k i Ž k 1,. l l s, j l j. Now k 1 k k s s1 l ik l s S D j j Ž j1 jr. Ž i1 i r. i i Ž. Ý l1 j ˆj ˆj n4 1 2 r Ý r i j 1 ˆi k n k 1 Ž. D 1 ˆ j n a ž 1 li k/ k1 1 ˆi ˆi ˆi n 1 j j j l n j1ls 1 k r Ž. D ˆ ˆ ˆ Ž. 1 2 r Ž j1 jr. Ž i1 i r. Ý l1 j ˆj ˆj n4 1 2 r

20 DEFORMATON OF THE GRASSMANNAN MANFOLD 437 mj 1 mˆ n 1 lj Ý D1 ˆj na ž 1 1 lm/ m1 ˆi ˆ 1 i r n4 1 ˆi ˆi ˆi n 1 j j j l n j1ls 1 k r Ž. D ˆ ˆ ˆ 1 2 r because r i j 1 ˆi k n mj 1 mˆ n k 1 1 Ý Ž. D1 ˆj nali Ý Ž. D1 ˆj na 1 k 1 lm k1 m1 ˆi ˆi n4 n m j 1 1 mˆ n Ý 1 ˆj n lm lj m1 1 D a. 1 1 Since Ý Ž Ž m l.. 1 ˆi ˆi ˆ 1 k i r n lm 1 jˆj ˆj ˆ 1 2 r l n l1 j ˆj ˆj n4 1 2 r a D 0, p p1 where m m p, i m i, we have S D j j Ž j1 jr. Ž i1 i r. i i Ž. j 1 ˆi ˆi ˆ 1 ls 1 k i r n Ý ljž. D1 jˆj ˆj ˆ r l n l1 j ˆj ˆj n4 1 2 r Ž j 1 ˆi ˆ 1 j r. Ž i 1 i r. 1 i r n 1 j j n Ž. D ˆ ˆ. COROLLARY Ž Let S be the antipode for k GL. Then n Ž j j. Ž i i. 1 ˆi ˆ 1 i r n 1 j1 jr 1 n i i 1 ˆj ˆj n 1 n S D D D. Proof. The proof is the same as the one of 3.1.

21 438 R. FORES DEFNTON Ž Define on k SL the antilinear involution : n a SŽ a.. ij This map endows k SL with a -Hopf algebra structure Žsee. n 8. Observe that 1 ˆj ˆ j j Ž i i. Ž j j. 1 jr n i i 1 ˆi ˆi n D D. DUALTY THEOREM Ž There is an anti-isomorphism that identifies k GŽ r, n. and k GŽ n r, n.. Proof. Consider the antilinear involution restricted to k GŽ r, n. k SL n and call A its image. By the previous observation one sees that r1 n A is generated by the determinants D1 ˆi ˆi n. Define on A the morph- ism, ž / D r1 n D 1 nr. 1 ˆi ˆi n 1 ˆi ˆi n Because of the nature of the Manin relations is well defined and it is an isomorphism. ts image is k GŽ n r, n.. is the anti-isomorphism that identifies k GŽ r, n. and k GŽ n r, n.. COROLLARY Ž There is an anti-isomorphism sending generators into generators that identifies the subring of k GL n generated by the determinants D 1 r with the subring generated by D 1 nr D 1 r1. 1 ˆi ˆi n 1 ˆi ˆi n 1 r Proof. Define the antilinear map : k GL k GL, Ž a. n n ij Sa, where S is the antipode for k GL ji n. Define also the family of isomorphisms j: A j Bj sending generators to generators, where A j1 n j and B are the subrings of k GL j n generated respectively by D1 ˆ i ˆ 1 nj 1 i j n and D,1i i n. The reuired anti-isomorphism is 1 ˆi ˆ 1 i j n 1 j 1 nr given by. THEOREM Ž Let Ž i i., J Ž j j., J, J r s r, J r s. Then N s 1 DJD DD J i1 ji Ý i lž K. lž L. Ý Lord Kord Ž Ž.. J K, LC K C i D D, where with the symbol X Ž x x. 1 s we denote the complement of the multi-index X Ž x x., in J and with the symbol CŽ X. the complement of X in J J. Notice that CŽ X. X and that CŽ X. X s because J J 2 s. N is the length of the standard tower

22 DEFORMATON OF THE GRASSMANNAN MANFOLD 439 for J J J and C Ž J i is the corresponding set defined in Theorem Proof. Consider k M J, the matrix bialgebra generated by a ij, i J, j 1, r s. t is clear that the commutations relations for D, DJ in k M are the same as the ones in k M n J. 1 1 By 3.5, D J, D commutes as DD, DD J, where D, DJ are ele- ments of k M k M k GL J J J J, the matrix bialgebra generated by a, i J J, j 1, 2 s ij and D is the determinant of maximal rank in k M. Since J, J J J by Theorem Ž we have N s 1 i D DJ Ý Ž. Ý DK D C Ž K., i0 Ž Ž.. J K, C K C i where N is the length of the standard tower for J in J. By Proposi- tion 2.21 N s 1 DDJ DD J i1 Ý Since D is central element we can write s DD 1 DD 1 DD 1 DD 1 J J By 3.5 we have the result. i Ý Ž lž. lž.. K C Ž K ord. ord Ž Ž.. J K, C K C i N Ý D D. Ý 1 i lž K. lž C Ž K.. Ž. i1 J ŽK, CŽ K.. C i D D 1 D D 1. K C Ž K. ord EXAMPLE Ž The commutation relation for D, D. Ž , J Ž 234., J 1, 2, 3, 4, 54, J 3. 4 J J 1, 2, 4, 54, s 2, Ž 24., J Ž 15.. The standard tower for J Ž 15. is given by J J Ž 15. J Ž 14. J Ž 12., Ž 24., Ž J 0 0 ord Ž. C Ž 12., Ž 45. Ž. Ž. J J C Ž 14., Ž 25., C Ž 12., Ž 45. Ž. Ž. J J C Ž 15., Ž 42., C Ž 14., Ž 25., Ž 12., Ž 54., C Ž 12., Ž 45. J 2 2 Ž.

23 440 R. FORES 2 1 D D D D Ž. D D D D D D, that is, according to Proposition 2.21, Hence D24 D15 D15 D24 D14 D25 D12 D D D D234 D135 D135 D234 D134 D235 D123 D D D. 4. QUANTUM PLUCKER RELATONS We want to give an analogue of the classical Plucker relations. n DEFNTON 4.1. Consider the n-dimensional free module V R over the ring R k M. Let e 4 n i 1 i n be the standard basis for V and e 4 the standard dual basis for V. Define i 1 i n s V s VŽ ei ej ej e i, ei e i., 1 i j n s V s V e e 1 e e, e e Ž i j j i i i., 1 i j n. We will indicate the products in s V and s V with the same symbol. As in the commutative case we have that 4 r V span e e, 1 i i n R i1 ir 4 r V span e e, 1 j j n. R j1 jr Notice that e e 4 is the dual basis of e e 4: Že j j i i j 1 e.ž e e. j i i Ž j j., Ži i.. r Let 1 n with 0 a 0 11 a1r 1. r., an1 anr where the a ij satisfy the Manin relations. n this case D 1 r e e D 1 r e e e 12 r 13 r 1 3 r D 1 r e e. nr n nr n

24 DEFORMATON OF THE GRASSMANNAN MANFOLD 441 Obseration 4.2. Let be defined as above. Consider the two maps, Ž.: V r1 V Ž.: V nr1 V. is obtained from through the identification nr V r V, after fixing the n-form e1 e n. This pairing is non-degenerate, 1 n form a basis for V and form a basis for V since the matrix a 4 1 n ij is invertible. Using such a basis we have that the linear homomorphisms and are given by the matrices Ž 0N, r A N, nr., Ž BM, r 0 M, nr., N dim r1 V M dim nr1 V, where 0n, m denotes a null matrix with n rows and m columns and A, B are suitable matrices over R of the specified orders. THEOREM Ž 4.3. ŽQuantum Plucker Relations for k GŽ r, n... Let,, as aboe. Then: Ž i. kerž. annihilates keržž... Ž ii. mž t. annihilates mž t Ž... Ž t.ž t r1 nr1 iii 0 for all V and V. Proof. This is immediate from the previous proposition using the given matrices for and Ž.. We will refer to the relations Ž iii. as uantum Plucker relations in analogy with the commutative case Žsee. 8. Under the specialization 1, the uantum Plucker relations go over to the classical Plucker relations. 5. PRESENTATON OF THE RNG A GŽ r, n. DEFNTON Ž Let s consider the local ring A fž. gž. gž k. Define A ² a : ij as the free tensor algebra over A with generators a ij s. We can define, as we did in Section 1, the matrix bialgebra A M as A ² a : and the two Hopf algebras n ij M 1 n A SL A M D 1 n def n 1 n ² : 1 n A GL A M T D T 1, a T Ta. n def n 1 n ij ij

25 442 R. FORES Define A GŽ r, n., the A-uantum grassmannian, as the subalgebra of 1 r A SLn generated by D i i,1i1 ir n. Observe that for 1 this is the ring kgr, Ž n.. Observe also that A GŽ r, n. is a homogeneous space. There is in fact a coaction of A SL n given by the comultiplication Žsee Proposition Ž We want to give a presentation of the ring A GŽ r, n. in terms of generators and relations. Remark Ž Notice that the rings A GŽ r, n. and A SL n are both torsion-free over A. LEMMA Ž Let s consider the homomorphism : A ² : A GŽ r, n., Ž. D. Then kerž. is an homogeneous ideal. Proof. Let s consider: A ² : A M n D. ker is an homogeneous ideal since M is. n fact look at r Ýb1 s D D kerž.. This means that r A ² a : ij, r Ý f 1 s 1... s D...D M. Since M is an homogeneous ideal each homogeneouss 1 s component of r is in M hence each homogeneous component of r is in KerŽ.. Notice that homogeneity of some polynomials in the D s in A ² a : ij, with respect to D or a ij, is a completely euivalent notion. Let s consider the map p: A M A SL n n. Obviously p. We want to show ker kerž.. This amounts to saying that Ž 1 r. Ž. 1 r D 1 A ² : 0, Ž 1 r. 1 r where D1 r 1 is the two-sided ideal in A Mn generated by D1 r 1. f we look at these as in A ² a :, ij 1 r D 1 A ² :. 1 r M Ž 1 r Let x Ý b D D D 1. Ž A ² :. 1 r. Divide 1 s 1 s 1 s each b by the highest power of Ž 1. dividing the gcd of these 1 s coefficient. f we set 1 we get a contradiction. THEOREM 5.4. A GŽ r,2r. A ² :, GŽ r,2r. where the are indeterminates, i i,1 i i n, and

26 DEFORMATON OF THE GRASSMANNAN MANFOLD 443 GŽ r,2r. is the ideal generated by the relations N i Ž r 1 l L l L. Ý Ý J J Lord L i1 L, L C i J, J N s 1 i lž L. lž K. J J Ý Ý Lord K i1 Ž Ž.. K K, LC K C i J, J t Ž.Ž t 0, r1 V, nr1 V where all the symbols appearing in the euations hae been defined in Ž 2.19., Ž 2.20., Ž 3.5., Ž ŽNote. and in this context hae to be intended as the functions and where we replace the determinants D s with the indeterminates s.. Proof. Consider the commutative diagram A ² : A GŽ r,2r. ord ord k² : k GŽ r,2r. where the vertical maps are the specializations at 1 and Ž. D, 1Ž. D 1 with D Žresp. D 1. denoting the uantum Ž resp. classical. determinants. The horizontal maps are surjective. Let J be the kernel of and GŽ r,2r. the two-sided ideal generated by the commutation relations Ž C. and the Plucker relations Ž P.. We know that J is homogeneous while GŽ r,2r. is trivially homogeneous. GŽ r,2r. J and we wish to show that GŽ r,2r. J. Fix a degree m and let the suffix m denote the homogeneous component of degree m Ž in the s and D s.. Let M be the number of monomials in of degree m. Then considering the basis of such monomials we have We have Ž m Ž ² : M ² : M A A, k k. J. GŽ r,2r. We now make some observations. m m m

27 444 R. FORES Ž.Ž A Ž² :. m is torsion-free as a module over A because it is free; Ž A GŽ r,2r.. m is also torsion-free as an A-module because it is a submodule of A SL. t follows from the second of these that Ž n as ŽA Ž² :. J ŽA GŽ r,2r... that if f A and ŽA Ž² :. m m m m are such that f J m, then J m. Ž. The specialization 1 maps Ž. onto kerž. GŽ r,2r. m 1 m while it maps J into kerž. so that both J and Ž. map onto kerž.. Let m 1 m m GŽ r,2r. m 1 m be a basis for kerž. and let be a set of monomials such that form a basis of Ž k² :.. Clearly Select elements 1 b 1, b 1,...,b B a, a,...,a 1 2 A a, a,...,a, b 1, b 1,...,b A 1 2 B m A B M. b, b,...,b 1 2 B 1 in GŽ r,2r. m such that bj specializes to b j. We regard the aj as monomials in Ž A ² :. m also. The argument now depends on three key facts. Ž. 1 a 1, a 2,...,aA are linearly independent oer A mod J m. Otherwise there are f, not all zero, in A such that j fa 1 1 faaa J m. Ž 1. n view of Ž. we can divide Ž. 1 by the largest power of 1 that divides all the f and so assume that for some j, we have f Ž. j j 1 0. Then, in k² :, f1ž 1. a1 faž 1. aa 0, where some f Ž. j 1 0. This contradicts the linear independence of mono- mials in k² :. Ž. 2 a, a,...,a, b, b,...,b form a basis for Ž A ² :. 1 2 A 1 2 B m oer A. Each of the a j, bp is a linear combination of the monomials. The coeffi- cients thus form a M M matrix Ž. ij 1i, jm

28 DEFORMATON OF THE GRASSMANNAN MANFOLD 445 of elements of A. To prove Ž. ij 2 we must only show that the determi- nant of this matrix is a unit in A. Since the specializations form a basis for k² :, we have a, a,...,a, b 1, b 2,...,b A 1 2 B det Ž But this means that det A. Ž. 3 J Ž. A -span of b, b,...,b. Clearly m GŽ r,2r. m 1 2 B Ý J A b. m GŽ r,2r. m j 1jB Ž. Let J. By 2 we have, for suitable, A, m i j Ý Ý a b. 1iA i i j j 1jB But, by Ž. 1, the aj are linearly independent mod J m. Hence all the i must be 0. So ÝJjb j. Remark Ž A presentation of A GŽ r, n. can be obtained from Theorem Ž 5.1. using the method described in Observation Ž 1.7., which is still valid in this context. Remark Ž The problem of giving a presentation for the ring k GŽ r, n. of the k -uantum grassmannian is still open. f one can show that det above is a power of, the argument of the proof of Theorem Ž 5.4. will show that the theorem is valid over k. n some particular cases one can prove the result directly by computation. EXAMPLE 5.7. k G 2, n. k GŽ 2, n. k ² :, i1i2 GŽ2, n. where is the two-sided ideal generated by the relations Ž GŽ2, n. refers to lexicographic ordering. 1, Ž ii. Ž j j. j1j2 i1i2 i1i2 j1j

29 446 R. FORES with i, i, j, j not all distinct. f i, i, j, j are all distinct, j ji i i i j j, i1 i2 j1 j , i j i j j1j2 i1i2 i1i2 j1j2 i1j1 i2j j1j2 i1i2 i1i2 j1j2 i1j2 j1i2 i1j1 j2i2 1 2 Ž. i j j i, i1 j1 j2i , i i j j. Ž p. i1i2 j1j2 i1j1 i2j2 i1j2 i1j The set of relations labeled Ž p. are the uantum Plucker relations. This example is interesting on its own from a physical point of view. Classically the grassmannian manifold GŽ 2, 4. contains an open affine dense subspace called the big cell that can be identified with the complex Minkowski space Žsee. 9. Hence the deformation of GŽ 2, 4. leads naturally to a deformation of the complex Minkowski space and the map defined in Ž 3.3. allows us to proceed to define a deformation of the real Minkowski space. Both the real and the complex uantum Minkowski spaces turn out to be homogeneous spaces for suitable uantum groups Žfor more details see. 5. ACKNOWLEDGMENTS thank my teacher, Professor V. S. Varadarajan, for all the discussions and suggestions that have led me to these results. also thank Professor Taft for a very helpful discussion. REFERENCES 1. Yuri. Manin, Quantum Groups and Non Commutative Geometry, Vol. 49, Centre de Recherches Mathematiues Montreal, E. Taft and J. Towber, Quantum deformation of flag schemes and Grassmann schemes.. A -deformation of the shape-algebra for GLŽ n., J. Algebra 142 Ž 1991., V. Lakshmibai and N. Reshetikhin, Quantum flag and Schubert schemes, in Contemp. Math., Vol. 134, pp , Amer. Math. Soc., Providence, R, Ya. S. Soibelman, On the uantum flag manifold, Funktsional. Anal. i Prilozhen. 26, No. 3 Ž 1992., 9092 n Russian ; translation, Functional Anal. Appl. 26, No. 3 Ž 1992., R. Fioresi, Quantization of flag manifolds and conformal space time, Re. Math. Phys. 9, No. 4 Ž B. Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89, No. 439, 1991.

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