The Quantum Torus Cq : Let q =(qij) be a r×r matrix of non-zero complex

numbers satisfying the relation : qii = 1, qij= qji , for all 1≤i,j≤ r.

Let Jq be the ideal of the non-commutative Laurent polynomial ring

S[r]=C[t1 ... tr]n.c generated by the elements, titj-qijtjti, 1≤i,j≤r.

The algebra Cq:= S[r]/Jq is called the quantum torus of rank r associated to q.

Cq is said to be cyclotomic if qij is a complex roots of unity for all i.j.

The Lie tori sll+1(Cq) : Given Ml+1(Cq) = Ml+1(C) Cq, the Lie algebra sll+1(Cq)

is defined as : sll+1(Cq) = { X=( x ij) є Ml+1(Cq) :Trace(X) є[Cq , Cq]}

with commutator relations:[xa, y b] = B(x, y)I([a, b]) + [x, y] (a ○ b)/2 + (x ○ y) [a, b]/2 [b1]

[I([a, b]), I([c, d])] = I([[a, b], [c, d]]), [b2]

[I([a, b]), x c] = x [[a, b], c ], [b3]

where x, y є sll+1(C), a, b, c, d є Cq ,

[x, y] = xy − yx, x ○ y = xy + yx − 2/(l + 1)Tr(xy)I(1),

[a, b] = ab − ba, a ○ b = ab + ba, and B(x, y) = 1/(l + 1)Tr(xy).

Let Q+ = positive integer root lattice of sll+1(C) ; Q- = - Q+ , and Q = Q+ + Q-

sll+1(Cq) has a decomposition given by:

sll+1(Cq) = ( sll+1(Cq)a ) ( sll+1(Cq)0 )

!!he IdeaTLet V be a finite-dimensional irreducible representation of sll+1(Cq) generated by a vector v. Then

there exists a positive Borel subalgebra b(sq) = ( nq+

H(sq) ) of sll+1(Cq) such that

Uq( nq+ H(sq)). v Cv .

It follows from the representation theory of multiloop Lie algebra that there exists a finitely supported

functions f : (C ×)r → P+ such that :

h tm . v = ∑ f(a)(h) eva (tm ) v, for all m G(sq),

where P+ is the positive integral weight lattice and eva : sq → C denotes the evaluation map at the point

a(C×)r . This implies that the finite-dimensional irreducible sll+1(Cq) –modules are tensor products of

sll+1(Cq) -modules which are analogous to the evaluation modules defined for the multiloop Lie algebras.

KNOWN RESULT : It has been shown in [1], [17] that a rank r cyclotomic torus Cq is

isomorphic to a tensor product: Cq Q (d1) ….. Q (ds) C[z11

... zk1

],

where Q (di) is a rank 2 quantum torus associated to the matrix q(i)=(qkl[i])

with q12[i]=ζi= (q21[i])-1, where ζi is a di

throot of unity for 1 ≤ i ≤ s.

Set supp sll+1(Cq) = {(a, m) Q×Zr : sll+1(Cq)am

≠ 0 } and H(Cq) = sll+1(Cq)0 .

:OUR OBSERVATIONS

Let ß(q) := Set of maximal commutative subalgebras of Cq and let Z(q) be the center of Cq.

For sq є ß(q), set G(sq) = { m =(m1,..,mr) є Zr : tm1.. tmr = tm є sq } and

let sq ß(q) be such that sq ∩ sq = Z(q).

• One can associate with each subalgebra sq ß(q), of a normalized cyclotomic

quantum torus Cq, an abelian group G(sq ) = Zr / G(sq) of rank d1…. ds.

• Any Borel subalgebra of sll+1(Cq) is of the form ( nq+ H(sq) ) or (nq

- H(sq) ),

where nq is the subalgebra of sll+1(Cq) generated by the elements of sll+1(Cq)a for

(a, m) supp sll+1(Cq), with a Q± and H(sq) is the subalgebra of sll+1(Cq)

generated by the elements of sll+1(Cq)0 , for m G(sq).

• Let V be an irreducible sll+1(Cq)-module with finite dimensional weight spaces.

Then there exists non-zero vector v V such that Uq(nq+)v =0 and

V = Uq(sll+1(Cq))v , where Uq(a) denotes the universal enveloping algebra of any

) qC(1l+slnalogs of Evaluation Modules for ASuppose that ca : (C

×)r → P+ is a function supported at a point a(C×)r . Let v be a non-zero vector of an

irreducible finite-dimensional sll+1(Cq) module V such that :

nq+.v = 0 and h tm.v = ca(a)(h) eva (tm ) v. for all m G(sq), for some sq ß(q).

H(Cq) is not a commutative algebra, hence if V is a non-trivial sll+1(Cq)-module, then

dim Uq(H(Cq) ) > 1, implying, h ts.v Cv for s Zr \ G(sq).

However nq+.v = 0, implies nq

+. h ts.v = 0, for all s Zr . In particular,

h ts.v is a highest weight vector of V for s Zr \ G(sq).

Hence there exists a positive Borel subalgebra b(cq) with cq ß(q) such that :

b(cq). h ts.v C. h ts.v , for all s Zr \ G(sq).

Irreducibility of the module V and the fact that the center of the algebra H(Cq) acts on all

the highest weight vectors by the same scalar, imply that there exists z G(sq) such that :

h tm. h ts.v = ca(a.zs)(h) eva.zs (tm ) h ts.v , for s Zr \ G(sq) , m G(sq),

where ca(a. zs) = ca(a), for all s Zr \ G(sq). Further owing to the bracket operation [b1] in H(Cq) ,

it is seen that the module generated by v is an irreducible sll+1(Cq)-module if and only if

ca(a) is a miniscule weight of sll+1(C) .

Let F(sq) be the set of all finitely supported functions f : (C×)r → P+ such that f(a) is a miniscule

weight for all a support of f, and let F(sq, Z(q)) be the subset of F(sq) consisting of all functions f

such that

eva(td) ≠ evb(t

d), for a,b supp f and td є Z(q),

where d є Zr

denotes a multi-index element .

Given ca є F(sq, Z(q)) and z є G(sq) , let (sq, ca, z ) denote the set of all finitely supported functions

g є F(sq, Z(q)) for which supp g = zs .a for s є G(sq).

Then the analogs of the evaluation modules for sll+1(Cq) is given by V (sq ,ca ,z) which is a module

generated by a highest weight vector v on which h sq acts by the function ca and h sq acts on

any other highest weight vector of V (sq ,ca ,z)by a function of the form zs.ca, for s є G(sq).

Given sq ß(q) , f =∑i=1

cai F(sq, Z(q)), z G(sq)

r ,

set : V (sq , f , z) = V (sq ,ca ,z).

r

a є supp f

r

:MAIN RESULTS

* Let V be a finite dimensional modules for the Lie algebra sll+1(Cq). Then V

is of the form V(sq, f,z), where f F(sq, Z(q)) and z G(sq)| f |

,

* Let sq , cq ß(q) and f1 F(sq, Z(q)), f2 F(cq, Z(q) ) with

|f j| = rj, j=1,2 and let z G(sq)r1 and h G(cq)

r2 . Then

there exists a sll+1(Cq)-module isomorphism

g : V(sq, f1, z )→ V(cq, f2,h) if and only if

subalgebra a of sll+1(Cq) .

Tanusree Khandai and Punita Batra

Harish-Chandra Research Institute, India

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±

i. sq = cq. ii. For each highest weight vector v V(sq, f1, z ), g (v) is a

highest weight vector of V(cq, f2, h) such that upto a scaling

factor f(1,v) (sq, f1, z ) is G(sq) - equivariant to f(2,g (v)) (cq. f2, h),

where f(i,w) denotes the finitely supported function by which h sq acts on the

highest weight vector w for some sq ß(q) .

±1

_

-1

th

±1

±

m

m

(a, m) є Q×Zr m є Zr

mm

±

• The multiloop Lie algebra sll+1(C) sq is a subalgebra of sll+1(Cq) for all sq ß(q).

m

a є supp f

×

where |f| = supp f.