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Providence, RI: AMS, 1998 [EK] Etingof, P. : Representation-theoretic proof of Macdonald inner product and symmetry identities. Comp. Math. 102, 179–202 (1996) [ES1] Etingof, P. : Algebraic integrability of Schrödinger operators and representations of Lie algebras. hep-th 9403135, Comp. Math. 98, No. 1, 91–112 (1995) [ES2] Etingof, P. : Algebraic integrability of Macdonald operators and representations of quantum groups. q-alg 9603022, to appear in Comp. Math. [EV1] Etingof, P. : Geometry and classification of solutions of the classical dynamical Yang–Baxter equation.

Let g be a simple Lie algebra, αi , i = 1, . . , r, simple roots, g = n+ ⊕ h ⊕ n− the polar decomposition. Consider the quantum group A = Uq (g) with the Chevalley generators ei , fi , Ki±1 as defined on p. 280 in [CP]. Fix in A a counit , a comultiplication , and an antipode S as defined on p. 281 in [CP]. 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− ).

Formulas (13) and (14) define dynamical representations of A. Moreover, if A(λ) : W1 → W2 is a morphism of dynamical representations, then A∗ (λ) := A(λ+h)t defines morphisms W2∗ → W1∗ and ∗ W2 → ∗ W1 . 2. An H -bialgebroid associated to a function R : T → End (V ⊗ V ). In this Section we recall a construction from [EV2] of an H -bialgebroid AR associated to a meromorphic function R : T → End (V ⊗ V ), where V is a finite dimensional diagonalizable H module and R(λ) is invertible for generic λ.

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