The Quantum Torus C q : Let q =(q ij ) be a r×r matrix of non-zero complex numbers satisfying the relation : q ii = 1, q ij= q ji , for all 1≤i,j≤ r. Let J q be the ideal of the non-commutative Laurent polynomial ring S [r] =C[t 1 ... t r ] n.c generated by the elements, t i t j -q ij t j t i, 1≤i,j≤r. The algebra C q := S [r] /J q is called the quantum torus of rank r associated to q. C q is said to be cyclotomic if q ij is a complex roots of unity for all i.j. The Lie tori sl l+1 (C q ) : Given M l+1 (C q ) = M l+1 (C) C q, the Lie algebra sl l+1 (C q ) is defined as : sl l+1 (C q ) = { X=( x ij ) є M l+1 (C q ) :Trace(X) є[C q , C q ]} 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 sl l+1 (C), a, b, c, d є C q , [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 sl l+1 (C) ; Q - = - Q + , and Q = Q + + Q - sl l+1 (C q ) has a decomposition given by: sl l+1 (C q ) = ( sl l+1 (C q ) a ) ( sl l+ 1 (C q ) 0 ) !! he Idea T Let V be a finite-dimensional irreducible representation of sl l+1 (C q ) generated by a vector v. Then there exists a positive Borel subalgebra b(s q ) = ( n q + H(s q ) ) of sl l+1 (C q ) such that U q ( n q + H(s q )). 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 t m . v = ∑ f(a)(h) ev a (t m ) v, for all m G(s q ), where P + is the positive integral weight lattice and ev a : s q → C denotes the evaluation map at the point a (C × ) r . This implies that the finite-dimensional irreducible sl l+1 (C q ) –modules are tensor products of sl l+1 (C q ) -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 C q is isomorphic to a tensor product: C q Q (d 1 ) ….. Q (d s ) C[z 1 1 ... z k 1 ], where Q (d i ) is a rank 2 quantum torus associated to the matrix q(i)=(q kl [i]) with q 12 [i]=ζ i = (q 21 [i]) -1 , where ζ i is a d i th root of unity for 1 ≤ i ≤ s. Set supp sl l+1 (C q ) = {(a, m) Q×Z r : sl l+1 (C q ) a m ≠ 0 } and H(C q ) = sl l+1 (C q ) 0 . : OUR OBSERVATIONS Let ß(q) := Set of maximal commutative subalgebras of C q and let Z(q) be the center of C q. For s q є ß(q), set G(s q ) = { m =(m 1 ,..,m r ) є Z r : t m 1 .. t m r = t m є s q } and let s q ß(q) be such that s q ∩ s q = Z(q). • One can associate with each subalgebra s q ß(q), of a normalized cyclotomic quantum torus C q, an abelian group G(s q ) = Z r / G(s q ) of rank d 1 …. d s . • Any Borel subalgebra of sl l+1 (C q ) is of the form ( n q + H(s q ) ) or (n q - H(s q ) ), where n q is the subalgebra of sl l+1 (C q ) generated by the elements of sl l+1 (C q ) a for (a, m) supp sl l+1 (C q ), with a Q ± and H(s q ) is the subalgebra of sl l+1 (C q ) generated by the elements of sl l+1 (C q ) 0 , for m G(s q ). • Let V be an irreducible sl l+1 (C q )-module with finite dimensional weight spaces. Then there exists non-zero vector v V such that U q (n q + )v =0 and V = U q (sl l+1 (C q ))v , where U q (a) denotes the universal enveloping algebra of any ) q C ( 1 l+ sl nalogs of Evaluation Modules for A Suppose that c a : (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 sl l+1 (C q ) module V such that : n q + .v = 0 and h t m .v = c a (a)(h) ev a (t m ) v. for all m G(s q ), for some s q ß(q). H(C q ) is not a commutative algebra, hence if V is a non-trivial sl l+1 (C q )-module, then dim U q (H(C q ) ) > 1, implying, h t s .v Cv for s Z r \ G(s q ). However n q + .v = 0, implies n q +. h t s .v = 0, for all s Z r . In particular, h t s .v is a highest weight vector of V for s Z r \ G(s q ). Hence there exists a positive Borel subalgebra b(c q ) with c q ß(q) such that : b(c q ). h t s .v C. h t s .v , for all s Z r \ G(s q ). Irreducibility of the module V and the fact that the center of the algebra H(C q ) acts on all the highest weight vectors by the same scalar, imply that there exists z G(s q ) such that : h t m . h t s .v = c a (a.z s )(h) ev a.z s (t m ) h t s .v , for s Z r \ G(s q ) , m G(s q ), where c a (a. z s ) = c a (a), for all s Z r \ G(s q ). Further owing to the bracket operation [b1] in H(C q ) , it is seen that the module generated by v is an irreducible sl l+1 (C q )-module if and only if c a (a) is a miniscule weight of sl l+1 (C) . Let F(s q ) 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(s q , Z(q)) be the subset of F(s q ) consisting of all functions f such that ev a (t d ) ≠ ev b (t d ), for a,b supp f and t d є Z(q), where d є Z r denotes a multi-index element . Given c a є F(s q , Z(q)) and z є G(s q ) , let (s q , c a , z ) denote the set of all finitely supported functions g є F(s q , Z(q)) for which supp g = z s . a for s є G(s q ). Then the analogs of the evaluation modules for sl l+1 (C q ) is given by V (s q ,c a ,z) which is a module generated by a highest weight vector v on which h s q acts by the function c a and h s q acts on any other highest weight vector of V (s q ,c a ,z)by a function of the form z s .c a, for s є G(s q ). Given s q ß(q) , f =∑ i=1 c a i F( s q , Z ( q ) ) , z G(s q ) r , set : V (s q , f , z) = V (s q ,c a ,z). r a є supp f r : MAIN RESULTS * Let V be a finite dimensional modules for the Lie algebra sl l+1 (C q ). Then V is of the form V(s q, f , z), where f F(s q , Z(q)) and z G(s q ) | f | , * Let s q , c q ß(q) and f 1 F(s q , Z(q)), f 2 F(c q , Z(q) ) with |f j | = r j, j=1,2 and let z G(s q ) r 1 and h G(c q ) r 2 . 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For each highest weight vector v V(s q, f 1, z ), g (v) is a highest weight vector of V(c q, f 2, h) such that upto a scaling factor f (1,v) (s q, f 1, z ) is G(s q ) - equivariant to f (2,g (v)) (c q. f 2, h), where f (i,w) denotes the finitely supported function by which h s q acts on the highest weight vector w for some s q ß(q) . ±1 _ -1 th ±1 ± m m (a, m) є Q×Z r m є Z r m m ± • The multiloop Lie algebra sl l+1 (C) s q is a subalgebra of sl l+1 (C q ) for all s q ß(q). m a є supp f × where |f| = supp f.