On infinite-dimensional grassmannians and their quantum deformations

On Infinite-Dimensional Grassmannians and their Quantum Deformations.

R. FIORESI(*) - C. HACON(**)

ABSTRACT - An algebraic approach is developed to define and study infinite-di- mensional grassmannians. Using this approach a quantum deformation (i.e. a deformation of the coordinate ring) is obtained for both the ind-variety union of all finite-dimensional grassmanniansG_Q, and the Sato grassmannianUGMA introduced by Sato in [Sa1], [Sa2]. They are both quantized as homogeneous spaces, that is together with a coaction of a quantum infinite dimensional group. At the end, an infinite-dimensional version of the first theorem of in- variant theory is discussed for both the infinite-dimensional special linear group and its quantization.

1. Introduction.

A definition of the infinite-dimensional Sato grassmannian is first in- troduced by Sato in [Sa1], [Sa2], where he explicitly exhibits the points as infinite-dimensional matrices. Sato proves the remarkable fact that the points of the Sato grassmannianUGMA are in one to one correspon- dence with the solutions of the KP hierarchy.

A few years later Segal and Wilson [SW], using mainly analytic tech- niques, explore more deeply this correspondence.

In a later work [PS] Pressley and Segal study more extensively, (*) Indirizzo dell’A.: Dipartimento di Matematica, Università di Bologna, Piazza Porta San Donato 5, 40126 Bologna, Italy. E-mail: fioresiHdm.unibo.it

Investigation supported by the University of Bologna, funds for selected re- search topics.

(**) Indirizzo dell’A.: Department of Mathematics, UC Riverside, Riverside, CA 92521-0135, USA. E-mail: haconHmath.ucr.edu

along the same lines, an infinite-dimensional grassmannian closely relat- ed to the Sato Grassmannian. In particular they give a stratification and a Plucker embedding of it. They also produce an action of a certain infi- nite-dimensional linear group realizing it as an infinite-dimensional ho- mogeneous space. Though their definition appears quite different from Sato’s one, they essentially describe the same geometrical object, but in a slightly more general setting.

A more geometrical approach to the same subject is taken by Mulase in [Mu1], [Mu2]. He constructs the Sato grassmannian as a scheme of which he gives the functor of points. Also Plaza-Martin [PM] takes the same approach, with special attention given to the physical applica- tions.

Together with the Sato grassmannian, Sato, as well as all the above mentioned authors, introduces what we denote byG_Q, the union of all fi- nite-dimensional grassmannians.G_Q turns out to be an ind-variety [Ku]

and it is dense in various topologies insideUGMA.G_Qis an interesting ob- ject in itself. Using the points ofG_Q expressed as infinite wedge prod- ucts, in [Ka] Kac constructs an infinite-dimensional representation of an infinite-dimensional general linear group and shows the correspondence between points of the infinite-dimensional grassmannian and solutions of the KP hierarchy with algebraic methods.

In the present work we want to study the infinite-dimensional grass- manniansUGMA andG_Qusing only algebraic methods, exhibiting explic- itly their coordinate rings. This approach turns to be the most natural for our goal, that is to obtain their quantum deformations.

This paper is divided in three parts.

In the first part, §2, we consider the inverse and direct limit of the coordinate rings k[d_m,n] of the finite-dimensional grassmannian over the algebraically closed field k. Then we give an explicit pre- sentation for the inverse limit k[d×_Q] and the direct limit k[d_Q]. We also prove that k[d×_Q] and k[d_Q] can be in some sense regarded as the homogeneous coordinate rings of G_Q and UGMA. In fact the closed points of Proj (k[d×_Q]) and of Proj (k[d_Q]) turn to be in one-to-one correspondence with the points of G_Q and UGMA respectively. Both G_Q and UGMA admit an action of the infinite-dimensional special linear group SL_Q given by the union of all finite dimensional special linear groups over k. We also show that there is a corresponding coaction of the homogeneous coordinate ring of the ind-variety SL_Q on both k[d×_Q] and k[d_Q]. The results in this part are more or less known,

however since we could not locate rigourous proofs for most statements, we included them for completeness.

In the second part of the paper, §3, we repeat these same construc- tions in the quantum groups setting. We give explicit quantum deforma- tions for both the ind-varietyG_Qand the Sato grassmannianUGMA. Pro- ceeding in the same way as in §2, we take the inverse and direct limit of the quantum finite-dimensional grassmannian k_q[D_m,n] ([Fi1], [TT]).

We obtain two non commutative rings,kq×[D_Q] andkq[D_Q] deformations of k[d×_Q] andk[d_Q] respectively that we call quantum G_Q and quantum Sato grassmannian UGMA. We give an explicit presentation for both of them.G_QandUGMAare quantized as homogeneous spaces, that is there is a well defined coaction of the quantum special linear infinite-dimen- sional group k_q[SL_Q] on them.

In the last part, §4, we examine the following problem of classical in- variant theory for the infinite-dimensional case: given the natural right action of the special linear group of orderr,SL_{r, 0}(k) on the matrix alge- bra, find theSL_{r, 0}(k)-invariants. In complete analogy to what happens in the finite-dimensional case, the ring of invariants in the infinite-dimen- sional case coincides withk[d×_Q] the homogeneous coordinate ring for the ind-varietyG_Q. We then obtain the corresponding results for the quan- tum case, generalizing the results in the paper [FH].

The first author wishes to thank Prof. V.S. Varadarajan and Prof. I.

Dimitrov for many fruitful discussions and Prof. R. Achilles for helpful comments.

2. The infinite-dimensional grassmannians G_Q and UGMA. Let k be an algebraically closed field of characteristic 0.

LetG_(m,n)be the grassmannian ofmdimensional subspaces in a vec- tor space of dimension N4m1n.

An element ofG(m,n) is represented by a N3m matrix. We will as- sume (following Sato [Sa2]) that the row indices go from 2m to n21 while the column indices go from 2m to 21 .

Letk[a_i,j]_m,nbe the coordinate ring of the algebra of theN3Nma- trices, where we assume that both row and column indices go from2m to n21 .

The homogeneous coordinate ring ofG_(m,n) is isomorphic to the sub- ring of the matrix ringk[a_i,j]_m,ngenerated by the determinantsd_l0Rl_m21

of the minors obtained by taking the columns 2mR21 and the rows

l₀Rl_m21. We will denote such subring byk[d_m,n] and the above-men- tioned set of determinants by dm,n.

DEFINITION (2.1). Letm8 Fm,n8 Fn. Define the inverse family of rings:

k[d_m8,n8] K

e(m8,n8,m,n)

k[d_m,n] e(m8,n8,m,n)(dj₀Rj_{m8 21})4

4

. /

´

d_l0Rl_m21

0

if (j₀Rj_{m8 21})4(2m8R2m21 ,l₀Rl_m21) and 2mGl₀EREl_m21Gn21

otherwise with 2m8 Gj₀EREj_{m8 21}Gn8 21 .

We define:

k[d×_Q]4lim

Jk[d_m,n] , and denote the induced maps by

e(m,n):k[d×_Q]Kk[d_m,n] .

We observe that the maps e_(m8,n8,m,n) are induced by maps E_(m8,n8,m,n):k[a_i,j]_m8,n8Kk[a_i,j]_m,n defined by

E(m8,n8,m,n)(a_i,j)4ai,j, ( 2mGiGn21 , 2mGjG 21 E(m8,n8,m,n)(a_i,j)41 , ( 2m8 Gi4jG 2m21

E_(m8,n8,m,n)(a_i,j)40 otherwise . We define:

k[M_Q]4lim

Jk[a_i,j]_m,n.

REMARK (2.2). Any element bk[M_Q] is an element of the form b4 ]b(m,n)( such that b(m,n)k[ai,j]m,n and for all m8 Fm, n8 Fn, E_(m8,n8,m,n)(b_(m8,n8))4b_(m,n). Similarly any element xk[d×_Q] is of the form x4 ]x_(m,n)( such that x_(m,n)k[d_i,j]_m,n and for all m8 Fm, n8 Fn, e_(m8,n8,m,n)(x_(m8,n8))4x_(m,n).

There is a corresponding direct system of inclusions of projective varieties

G_(m,n)KG_(m8,n8) ( m8 Fm, n8 Fn. Define

G_Q4lim

KG(m,n) . Notice that G_Q4NG_(m,n).

We want to view G_Q as a projective ind-variety.

DEFINITION (2.3). Anind-variety over k is a set Xtogether with a filtration:

X₀%X₁%X₂%R such that

1) _nF0

0

2) Each X_nis a finite-dimensional variety over ksuch that the inclu- sion Xn%Xn11 is a closed immersion.

(See [Ku] for more details).

The ind-varietyXis naturally a topological space,U%Xbeingopenin Xif and only if, for eachn,UOX_nis open in X_n. The sheaf of regular functions on Xis defined byOX»4lim

J OX_n.Xis said to be alocally pro- jective ind-varietyif it admits a filtration such that eachXnis projective.

We will say thatXis a projective ind-variety if it admits a filtrationX_n and a line bundleLsuch that each restrictionLNX_nis very ample and the corresponding maps

H⁰(Xn,LNXn)KH⁰(Xn21,LNXn21)

are surjective. In other words, for eachnthere are compatible closed im- mersions X_n^%KP^Nn4P(H⁰(X_n,LNX_n)^q) with coordinate rings gener- ated by H⁰(X_n,LNX_n) and hence a closed immersion of ind-varieties X^%KP^Q4NP(H⁰(X_n,LNX_n)^q). We define

H⁰(X,L)»4lim

JH⁰(X_n,LNX_n) .

Let S(P^N)45dF0H⁰(P^N,OP^N(d) ) be the homogeneous coordinate ring ofP^NandI(X_n) be the homogeneous ideal ofX_n%P^Nn, then the ho- mogeneous coordinate ring ofX_n%P^Nn is given byS(X_n)4S(P^N) /I(X_n).

We define the homogeneous coordinate ring of the projective ind-vari- ety X%P^Q to be

S(X)»4lim

JS(X_n).

THEOREM (2.4). G_Q is a projective ind-variety, with homogeneous coordinate ring k[d×_Q].

PROOF. It is well-known that the maps G_(m,n)KG_(m8,n8) are closed immersions (when defined). Let X_n»4G_(n,n), then we have closed im- mersionsG_(m,n)^%KX_(n1m,n1m). ThereforeX4NX_n4NG_(m,n)4G_Qis an ind-variety. For each nD0 we have the Plücker embeddings Xn4 4Gn,n%

KP(RⁿC²ⁿ)4P^Nn. The homogeneous coordinate ringS(P^Nn) is generated by elementsx_IwhereI4 ]i₁,R,i_n(such that2nGi₁Ei₂E EREi_nGn21 and forn8 Fn, the closed immersionsP^Nn%KP^Nn8corre- spond to the surjective homomorphisms

S(P^Nn8)KS(P^Nn) defined by

e(m8,n8,m,n)(xi₁Ri_n8)4

. /

´

x_i1Rin

0

if (i₁Ri_n8)4(2n8R2n21 ,i₁Ri_n) and 2nGi₁EREi_nGn21

otherwise .

The homogeneous coordinate ring of the projective ind-variety P^Q4 4_n

0

D0P^Nnis just generated by lim

JH⁰(P^Nn,OP^Nn( 1 ) ). The line bundleLNXn

is just the pull-backck ofOP(RⁿC²ⁿ)( 1 ). The immersions P^NnKP^Nn8 and X_nKX_n8 are compatible. The corresponding homogeneous coordinate ring is S(X_n)4k[d_{n, 2n}]. The maps induced by the inclusions X_nKX_n8 are just the mapse_(n8,n8,n,n):k[d_{n8, 2n8}]Kk[d_{n, 2n}]. Therefore the homo- geneous coordinate ring of X is given by

limJk[dn, 2n]4lim

Jk[dm,n]4k[d×_Q] . QED .

We now turn our attention to the Sato grassmannianUGMAand its re- lation with G_Q.

DEFINITION (2.5). Let m8 Fm, n8 Fn. Define a direct family of rings:

k[dm,n] d_l0Rlm21

K

r(m,n,m8,n8)

O

k[dm8,n8] d_2m8,R2m21 ,l0Rlm21

for 2mGl₀EREl_m21Gn21 .

It is easy to see using the Plücker relations that this map is well de- fined. Moreover, the mape(m,n,m8,n8)is a left inverse for r(m,n,m8,n8) and in particular r(m,n,m8,n8) is injective. We define:

k[d_Q]4lim

Kk[d_m,n] .

Denote with r_(m,n) the induced inclusions k[d_m,n]Kk[d_Q].

DEFINITION (2.6). Define Maya diagram of virtual cardinality 0 (or shortly aMaya diagram) a strictly increasing sequencea_l4 ]ai(,iF1 , such thata_iZ anda_i4i for allic0 . Define the orderVa_lVof a Maya diagram to be the smallest numberisuch thata_j4jfor alljFi. Any se- quencel

*4l₁,R,l_mwithl₁GRGl_mGminduces a Maya diagrama_l4 4lA

* of order at mostm11 defined bya_i4l_ifor all 1GiGmanda_i4i for alliFm11 . For any Maya diagrama_l, let a_Gmdenote the ordered set a1EREam. Clearly if Va_lVGm, then a_l4aA_Gm.

Given a Maya diagrama_lof orderm11 withNa₁NF 2n11 , we wish to define corresponding elements d_a

lk[d_Q], and d×

a_lk[d×_Q]. Define d_a

l4r_(m,n)d_a

Gm

where a_l4aA_Gm.

k[d_Q] is generated as a ring by the d_a

l, since it is generated by the images of k[dm,n] under r(m,n).

We define a map

r(m,n):k[d_Q]Kk[dm,n] r(m,n)(d_a

l)4^./

´ d_a

Gm

0

for all mFVa_lV,nF Na₁N otherwise .

Then we define

d×

a_l4 ]r(m,n)da_l(k[d×_Q] .

PROPOSITION (2.7). a) There is an injection I: k[d_Q]Kk[d×_Q] which sends d_a

l to d×

a_l.

b) The image of I is dense in k[d×_Q]for the inverse limit topology on k[d×_Q] induced by the discrete topology on each k[dm,n].

PROOF. (a) It suffices to check that for m8 Fm and n8 Fn, one has

e_(m8,m,n8,n)(r_(m8,n8)d_a

l)4r(m,n)d_a

l.

(b) Define a fundamental set of neighbourhoods of 0k[d×_Q] byU04 4k[d×_Q], U_k4e_(k,²¹_k)( 0 ). For any x4 ]x_(m,n)(k[d×_Q], we must define a Cauchy sequence]y_k( 4

m

^!

n

Since e_(k,k)(x)k[d_k,k] we have e(k,k)(x)4

!

i41 m_k

bi^kdL_k^{i, 1}RdL_k^i,ji

where L_k^i,j4(l_{k, 1}^i,jRl_k,^i,_m^j ), 2kGl_k^i,_{, 1}^jEREl_k,^i,_m^j Gk21 . Set y_k4

!

i41 mk

b_i^kd×

L_k^{i, 1} ARd×

Lki,ji A.

Notice that the summation is finite and hencey_kis in the image ofI. For anyk8 Dkwe have,e(k,k)(y_k8)4e(k,k)(x) i.e.yk82xUk. Hence]yk(is a Cauchy sequence converging to x. QED.

We now give a presentation of the rings k[d_Q] and k[d×_Q].

Define k[j_a

l] to be the ring generated by the independent variables ja_l, where a_l is any Maya diagram of virtual cardinality 0.

There is a natural mapf:k[j_a

l]Kk[d_Q] such thatja_lKd_a

l. This in- duces a topology onk[ja_l] for which a fundamental set of neighborhoods is given byV_k»4f²¹I²¹U_k. Letk[j_a

l

×] be the completion ofk[j_a

l] with respect to the above topology. In particular the elements ofk[j_a

l

×] are of the form

!

i, 1Rja_l^i,ki where b_ik and the a_l^i,k are any Maya dia- grams of virtual cardinality 0. The corresponding natural map between completions f×:k[j_a

l

×]Kk[d×_Q] is defined by ja_lKd×

a_l.

DefineP(m,n)to be the ideal of Plucker relations ink[dm,n]. LetP4 4NP_(m,n)be the corresponding ideal ink[d_Q], and similarlyP×

4lim

JP_(m,n) be the corresponding ideal in k[d×_Q].

THEOREM (2.8). We have ring isomorphisms i) k[d×_Q]`k[j_a

l

×] /P× ii) k[d_Q]`k[j_a

l] /P PROOF. (i) The direct limit is an exact functor.

(ii) The inverse limit functor is left exact, and since the inverse sys- tem P(m,n) is a surjective system, the corresponding inverse system sequence

0KP×

Kk[j_a

l

×]KK[d×_Q]K0 is also exact. QED.

We want now to relate our constructions with the Sato grassmannian.

In [Sa2] Sato defines a set of points in an infinite-dimensional projective space. Already theorem (2.8) suggests to viewG_Qas the set of zeros of the idealP× in an infinite-dimensional projective space whose coordinate ring is given byk[j_a

l

×]. We want to make this euristic notion more precise and to relate the ring k[d_Q] with the Sato grassmannian.

Assume that the field k has cardinality strictly greater than ]0. Consider the directed system given by rings Rn4k[z1,R,zn] and homomorphisms of k[z₁,R,z_n8]Kk[z₁,R,z_n] (for n8 Dn) defined by z_iKz_i for all iGn, and z_iK0 for iDn. This corresponds to an affine ind-variety A^Q4NAⁿ given by the inclusion of affine planes Aⁿ4 4Spec (R_n)^%KAⁿ¹¹4Spec (R_n11). Let

R× 4lim

JRn.

LEMMA (2.9). The set of closed points of A^Q is in one to one corre- spondence with Spec_m(R×).

PROOF. Notice that eachR_ninjects in a natural way inR×and setR4 4NR_n%R×. Ifm%Ris maximal,R/m4Eis a field. By [La] we haveE4k and thereforemis generated byz_i2k_iwherek_iis the image ofz_iR/m. Ifm8is a maximal ideal ofR×, andf :R×

KR×/m8, then by the previous ob- servations, the induced maps fi: RiKR×/m8 have image contained in k. By the universal property of inverse limits then alsof :R×

Kk is deter- mined byk_i4f(z_i). It is clear that in order forfto be defined, one must have k_i40 for all but finitely many i. QED.

Recall thatk[j_a

l

×]4lim

JS(P^Nn). It follows thatP^Q4Proj (k[j_a

l

×] ) and that the closed points ofP^Qare given by sequencesk_a

lkwherea_l40 for all but finitely many Maya diagrams of virtual cardinality 0 we have a_l40 and two sequences are considered equivalent if there existslk* such thatk_a

l4lka8_l. The assertion can be verified locally on the open cov- er Spec (k[j_a

l

×]_(ja

l)) where k[j_a

l

×]_(ja

l) denotes the subring of elements of degree 0 in the localized ring k[j_a

l

×]_ja

l. The computation is now analog- ous to the one above for R×. Similarly one has that the closed points of Proj (k[j_a

l] ) correspond to all sequences (not necessarily bounded)ka_l kwherea_lruns over all Maya diagrams of virtual cardinality 0 and two sequences are considered equivalent if there existslk* such thatk_a

l4

4lk^a8l. It follows that

PROPOSITION (2.10). Assume that the cardinality of k is strictly greater than ]0.

i) The set of closed points of G_Q is in one to one correspondence with the set of closed points ofProj (k[d×_Q] ),i.e. with the sequences]k_a

l

k(satisfying all Plücker relations, where a_lMaya diagram of virtual cardinality 0 and k_a

l40 for all but finitely many Maya diagrams.

ii) The set of closed points of Proj (k[d_Q] ) is in one to one corre- spondence with the sequences ]ka_lk( satisfying all Plücker relations, where a_lMaya diagram of virtual cardinality 0. Moreover we have that those points coincide with UGMA the Sato grassmannian(as defined by Sato, [Sa1], [Sa2]).

REMARK (2.11). Proposition (2.10) shows that the ring k[d_Q] can be regarded as the «coordinate ring» for the Sato grassmannian in the sense that its maximal ideals are in one-to-one correspondence with the points ofUGMA. Theorem (2.9) allows us to interpret the Sato grassman- nian as the set of closed points in an infinite-dimensional projective space that are subjected to the relations P.

We now want to define an infinite-dimensional special linear group and show that it has an action on both G_Q and UGMA.

DEFINITION (2.12). For all m,n positive integers, define SL_(m,n)(k)`SL_N(k) asN3N matrices with determinant 1, whose row and column indicies are between 2m and n21 . The inclusions c(m,n,m8,n8): SL_m,n(k)KSL_m8,n8(k) are defined for all m8 Fm,n8 Fn,

c(m,n,m8,n8)(g)4diag (Id_m2m8,g,Id_n2n8). It is clear that we have an ac- tion ofSL_m,n(k) onG_(m,n)for allm,n. We have a corresponding projec- tive system of coordinate rings:

k[SL_m8,n8] K

f(m8,n8,m,n)

k[SLm,n]

f(m8,n8,m,n)(gij)4

. /

´

g_ij 1 0

if 2mGi,jGn21

if 2m8 Gi4jG 2m21 ,nGi4jGn8 21 otherwise .

OBSERVATION (2.13).

SL_Q(k)4^deflim

KSLm,n4NSLm,n(k) is an ind-variety with coordinate ring

k[SL_Q]4^deflim

Jk[SL_m,n] .

We want now to show thatk[SL_Q] has an Hopf algebra structure. Notice that while in the finite-dimensional case this is an immediate conse- quence of the fact that the varietySLm,n(k) is a group, in the infinite-di- mensional case we need to check the commutativity of certain dia- grams.

PROPOSITION (2.14). Let(m8,n8)D(m,n). The following are com- mutative diagrams:

i)

k[SL_m8,n8]

I

k[SL_m8,n8]7k[SL_m8,n8]

K

f(m8,n8,m,n)

K

f(m8,n8,m,n)7f(m8,n8,m,n)

k[SL_m,n]

I

k[SL_m,n]7k[SL_m,n] where D(m,n) is the comultiplication in the Hopf algebra k[SL_m,n].

ii)

k[SL_m8,n8]

I

k

K

f_(m8,n8,m,n)

K^id

k[SL_m,n]

I

k

where e(m,n) is the counit in the Hopf algebra k[SL_m,n].

iii)

k[SL_m8,n8]

I

k[SL_m8,n8]

K

f(m8,n8,m,n)

K

f(m8,n8,m,n)

k[SL_m,n]

I

k[SL_m,n] where S(m,n) is the antipode in the Hopf algebra k[SLm,n].

PROOF. Direct check.

COROLLARY (2.15). k[SL_Q] has an Hopf algebra structure given by:

a) comultiplication k[SL_Q] ]a(m,n)(

K

D_Q

O

k[SL_Q]7×k[SL_Q] ]D(m,n)(a(m,n))( b) counit

k[SL_Q] ]a_(m,n)(

K

e_Q

O

k e_(m,n)(a_(m,n)) c) antipode

k[SL_Q] ]a_(m,n)(

K

S_Q

O

k[SL_Q] ]S_(m,n)(a_(m,n))(

where 7× denotes the completed tensor product and is given by k[SL_Q]7×k[SL_Q]4lim

Jk[SL_m,n]7k[SL_m,n] (see [Ku] for more details).

PROOF. (a) is immediate from proposition (2.14) and from [Ku]. (b), (c) are immediate from proposition (2.14).

The group SL_Q(k) has an action on bothG_Qand UGMA. In order to obtain a quantization of these actions we need to describe the corre- sponding coactions of k[SL_Q] on k[d×_Q] and k[d_Q].

OBSERVATION (2.16). Since SL_m,n(k) acts on G_(m,n) we have the coaction:

k[d_m,n] d_l0Rl_m21

K

l(m,n)

O

k[SL_m,n]7k[d_m,n]

mGk0Rk

!

g_l0k0Rg_lm21km217d_k0Rk_m21

One can check the commutativity of the following diagram, form8 Fm, n8 Fn:

k[d_m,n]

I

k[d_m8,n8]

K

l(m,n)

K

l(m8,n8)

k[SL_m,n]7k[d_m,n]

I

k[SL_m8,n8]7k[d_m8,n8]

PROPOSITION (2.17). There is an coaction of k[SL_Q] on k[d×_Q] and on k[d_Q].

PROOF. Fix (m₀,n0). Letm8 DmDm0,n8 DnDn0. We have a com- mutative diagram (see observation (2.16)):

k[d_m,n]

I

k[d_m8,n8]

K

l(m,n,m0 ,n0 )

K

l(m8,n8,m0 ,n0 )

k[SL_m0,n0]7k[d_m,n]

I

k[SL_m0,n0]7k[d_m8,n8] l(m,n,m₀,n₀)(d_l0Rl_m21)4def

!

mGk0Rkm21Gn21

gA_l

0k₀RgA_l

m21k_m217d_k0Rk_m21

where:

gA_ij4

. /

´

g_ij 1 0

if 2mGi,jGn21

if 2m8 Gi4jG 2m21 , nGi4jGn8 21 otherwise

Hence there is a map:

k[d×_Q]Kk[SL_m0,n0]7k[d×_Q] .

Now taking the inverse limit ofk[SL_m0,n0] on the right side we obtain a coaction of k[SL_Q] on k[d×_Q], that is a map:

k[d×_Q]Kk[SL_Q]7×k[d×_Q]

where 7× denotes the completed tensor product (see [Ku]).

By theorem (2.7) (iii)k[d_Q] can be identified with a subring ofk[d×_Q] and one can check that the given coaction is a well defined coaction when restricted to this subring of k[d×_Q]. QED.

3. The quantum infinite-dimensional grassmannians kq[G_Q] and kq[UGMA].

We want to obtain a deformation ofG_QandUGMAas quantum homo- geneous spaces for a quantumSL_Q. In the language of quantum groups this means that we need to construct deformations of the two ringsk[d×_Q] andk[d_Q] together with a coaction of a deformation of k[SL_Q] on them.

The naturality of the construction in §2 will allow us to repeat the same arguments used for the commutative case also in the non commutative case with very small changes.

DEFINITION (3.1). Let k_q4k[q,q²¹] and let k_qaa_i,jbm,n be the free algebra overkqwithaijas non commutative generators,2mGi,jGn2 21 . Definekq[ai,j]m,n, as the associativekq-algebra with unit generated by the elements a_ij, subject to the relations:

a_ija_kj4q²¹a_kja_ij, iEk, a_ija_kl4a_kla_ij, iEk,jDl or iDk,jEl a_ija_il4q²¹a_ila_ij, jEl, a_ija_kl2a_kla_ij4(q²¹2q)a_kja_il, iEk,jEl k_q[a_i,j]_m,n is a bialgebra with counit and comultiplication:

e^q(m,n)(a_ij)4dij D^q(m,n)(a_ij)4

!

DEFINITION (3.2). We define the quantum determinant obtained by taking rows i₁Ri_p,columns j₁Rj_pas an elementD_i1^j1_Ri^R_p^jpk_qaa_i,jbm,n

given by:

Di₁j1RiR_pjp

4def

4^def

!

s: (i₁Rip)K(j1Rjp)

(2q)^2l(s)a_i1s(i1)Ra_ims(ip), 2mGi₁EREi_pGn21 2mGj1EREjpGn21 wheresruns over all the bijections andl(s) is the length of the permuta- tions.pis called therankofDi₁j₁RRi_pj_p. Its image inkq[a_i,j]_m,nis then the usual quantum determinant. We shall write Di₁j1RRi_pjp for this image also, the context making clear where the element sits. (See [PW] ch. 4 for more details). We will drop the upper indices whenever they coincide with 2pR21 .

DEFINITION (3.3). Define the quantum grassmannian ring k_q[D_m,n], as the subring ofk_q[a_i,j]_m,ngenerated by the quantum deter- minantsD_i0Rim212mGi₀EREi_m21Gn21 (see [Fi1]). We will refer to the set of such determinants with Dm,n.

An explicit presentation of the ring k_q[D_m,n] in terms of generators and relations is given by (see [TT] 3.5, [Fi1], [FH]):

(c)

q^2[m2p]lJlI4lIlJ1

!

i41 N

(q²¹2q)ⁱ

!

(L,L8)C_i^{i0Ri×k1Ri×kpRim21}

(2q)2l(s(L) )2l(s(L8) )l(L,i_k1Ri_kp)ordl_(L8,ik

1Ri_kp)ord

I4(i₀Ri_m21)EJ4(j₀Rj_m21) IOJ4 ]i_k1Ri_kp( i0EREim21, j0EREjm21

1Ga₁ER

!

(2q)^{2l(z1Rz×a1Rz×asRzr1sza1Rzas)2l(za1Rzasl1Rlm2s)} lz₁Rz×_a

1Rz×_a

sRz_m1sl_(za

1Rzasl1Rlm2s)ord40 (y)

Each of the relations in the set (y) is computed for any set of fixed in- dices: 2mGz₁EREz_m1sGn21 , 2mGl₁EREl_m2sGn21 .

All the symbols that appear have been defined in [Fi2].

Notice that the relations labeled (c) reduce forq41 to state the com- mutativity of thel_I’s while the relations labeled (y) forq41 become the Young (also called symmetry) relations.

We want now to proceed in analogy with §2 and define the following inverse and direct families.