By A. Jaffe (Chief Editor)

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**Additional resources for Communications in Mathematical Physics - Volume 205**

**Sample text**

Matrices JV ,V (λ) and RV ,V (λ). Let V = CN be the vector representation of A = Uq (gl(N)). Let vj = (0, . . , 0, 1j , . . , 0) be the standard basis in V . We have fi vj = δi,j vi+1 , ei vj = δi+1,j vi , where fi , ei are the Chevalley generators of Uq ((N )). Introduce a basis Eij in End (V ) by Eij vk = δj k vi . The Uq (gl(N))-module V ⊗ V has the weight decomposition, V ⊗ V = ⊕N a=1 Vaa ⊕ ⊕a**
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**Q λN ). ,ak (q λ1 , . . , q λN )} , of meromorphic functions on T , where a1 , . . , ak run through all k element subsets of {1, . . , N}, such that for any subset a1 , . . ,ak (q λ1 , . . ,ak (q λ1 , . . , q λN ) = 1. Let k be the set of all multiplicative k-forms. ,ak (q λ1 , . . ,ak (q λ1 , . . ,ak (q λ1 , . . ,ak (q λ1 , . . , q λN )} are multiplicative k-forms. ,ak (q λ1 , . . , q λN ) ≡ 1}. k . The zero element 44 P. Etingof, A. Varchenko For any a = 1, . . , N, introduce an endomorphism δa of the multiplicative group of nonzero meromorphic functions f (q λ1 , . **

We consider A as a polarized Hopf algebra with the Zgrading and polarizations defined by deg (ei ) = 1, deg (fi ) = −1, deg (Ki±1 ) = 0, A+ = Uq (n+ ), A− = Uq (n− ), A0 = Uq (h), A 0 = Uq (b+ ), A 0 = Uq (b− ). Remark. Let aij = 2 < αi , αj > / < αi , αi > be the Cartan matrix. Let di be coprime positive integers such that the matrix di aij is symmetric. Let hi ∈ h be the elements such that αi (hj ) = aij . Then one can think of the generators Ki±1 as of elements of the form q ±di hi , see p. 281 in [CP].

### Communications in Mathematical Physics - Volume 205 by A. Jaffe (Chief Editor)

by Joseph

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