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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.