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The orthogonality and qKZBheat equation for traces of Uq(g)intertwiners
, 2003
"... In our previous paper [EV2], to every finite dimensional representation V of the quantum group Uq(g), we attached the trace function F V (λ, µ), with values in EndV [0], obtained by taking the (weighted) trace in a Verma module of an intertwining operator. We showed that these trace functions sati ..."
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In our previous paper [EV2], to every finite dimensional representation V of the quantum group Uq(g), we attached the trace function F V (λ, µ), with values in EndV [0], obtained by taking the (weighted) trace in a Verma module of an intertwining operator. We showed that these trace functions satisfy the MacdonaldRuijsenaars and the qKZB equations, their dual versions, and the symmetry identity. In this paper we show that the trace functions satisfy the orthogonality relation and the qKZBheat equation. For g = sl2, this statement is the trigonometric degeneration of a conjecture from [FV2], proved in [FV2] for the 3dimensional irreducible V. We also establish the orthogonality relation and qKZBheat equation for trace functions obtained by taking traces in finite dimensional representations (rather than Verma modules). If g = sln and V = S kn C n, these functions are known to be Macdonald polynomials of type A. In this case, the orthogonality relation reduces to the Macdonald inner product identities, and the qKZBheat equation coincides with the qMacdonaldMehta identity, proved by Cherednik [Ch1].
Quantum Group as Semiinfinite Cohomology
, 2008
"... We obtain the quantum group SLq(2) as semiinfinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c + ¯c = 26. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra ..."
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We obtain the quantum group SLq(2) as semiinfinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c + ¯c = 26. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional qdeformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SLq(2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory. Contents 1
RESONANCE RELATIONS, HOLOMORPHIC TRACE FUNCTIONS AND HYPERGEOMETRIC SOLUTIONS TO QKZB EQUATIONS.
, 2006
"... Abstract. The resonance relations are identities between coordinates of functions ψ(λ) with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of Bak ..."
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Abstract. The resonance relations are identities between coordinates of functions ψ(λ) with values in tensor products of representations of the quantum group Uq(sl2). We show that the space of hypergeometric solutions of the associated qKZB equations is characterized as the space of functions of BakerAkhiezer type, satisfying the resonance relations. We give an alternative representationtheoretic construction of this space, using the traces of regularized intertwining operators for the quantum group Uq(sl2), and thus establish the equivalence between hypergeometric and trace function solutions of the qKZB equations. We define the quantum conformal blocks as distinguished Weyl antiinvariant hypergeometric qKZB solutions with values in a tensor product of finitedimensional Uq(sl2)modules. We prove that for generic q the dimension of the space of quantum conformal blocks equals the dimension of Uq(sl2)invariants, and when q is a root of unity is computed by the Verlinde algebra. 1.