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The orthogonality and qKZB-heat 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 Macdonald-Ruijsenaars 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 qKZB-heat equation. For g = sl2, this statement is the trigonometric degeneration of a conjecture from [FV2], proved in [FV2] for the 3-dimensional irreducible V. We also establish the orthogonality relation and qKZB-heat 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 qKZB-heat equation coincides with the q-Macdonald-Mehta identity, proved by Cherednik [Ch1].
Quantum Group as Semi-infinite Cohomology
, 2008
"... We obtain the quantum group SLq(2) as semi-infinite 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 semi-infinite 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 q-deformed 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 Baker-Akhiezer type, satisfying the resonance relations. We give an alternative representation-theoretic 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 anti-invariant hypergeometric qKZB solutions with values in a tensor product of finite-dimensional 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.