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Hypergeometric solutions of trigonometric KZ equations satisfy dynamical difference equations
 Adv. Math
"... {yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the ..."
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{yavmar, anv} @ email.unc.edu Abstract. The trigonometric KZ equations associated to a Lie algebra g depend on a parameter λ ∈ h where h ⊂ g is a Cartan subalgebra. A system of dynamical difference equations with respect to λ compatible with the KZ equations is introduced in [TV]. We prove that the standard hypergeometric solutions of the trigonometric KZ equations associated to slN also satisfy the dynamical difference equations.
QUANTIZATION OF CLASSICAL DYNAMICAL rMATRICES WITH Nonabelian Base
, 2003
"... . We construct some classes of dynamical rmatrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of rmatrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of o ..."
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Cited by 13 (1 self)
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. We construct some classes of dynamical rmatrices over a nonabelian base, and quantize some of them by constructing dynamical (pseudo)twists in the sense of Xu. This way, we obtain quantizations of rmatrices obtained in earlier work of the second author with Schiffmann and Varchenko. A part of our construction may be viewed as a generalization of the DoninMudrov nonabelian fusion construction. We apply these results to the construction of equivariant starproducts on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini.
DOUBLE AFFINE HECKE ALGEBRAS AND BISPECTRAL QUANTUM KNIZHNIKZAMOLODCHIKOV EQUATIONS
, 2008
"... We use the double affine Hecke algebra of type GLN to construct an explicit consistent system of qdifference equations, which we call the bispectral quantum KnizhnikZamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik’s quantum affine KZ equations associated to principal series repre ..."
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Cited by 13 (5 self)
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We use the double affine Hecke algebra of type GLN to construct an explicit consistent system of qdifference equations, which we call the bispectral quantum KnizhnikZamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik’s quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of qdifference equations acting on the central character of the principal series representations. We construct a meromorphic selfdual solution Φ of BqKZ which, upon suitable specializations of the central character, reduces to symmetric selfdual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for the Ruijsenaars’ commuting trigonometric qdifference operators. Under this correspondence Φ becomes a selfdual HarishChandra series solution Φ + of the bispectral problem. Specializing the central character as above, we recover from Φ + the symmetric selfdual Macdonald polynomials.
Yangians and Mickelsson Algebras I
"... category of modules over the Lie algebra gl m to the category of modules over the degenerate affine Hecke algebra of GLN ..."
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Cited by 12 (5 self)
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category of modules over the Lie algebra gl m to the category of modules over the degenerate affine Hecke algebra of GLN
DYNAMICAL DIFFERENTIAL EQUATIONS COMPATIBLE WITH RATIONAL QKZ EQUATIONS
, 2004
"... Abstract. For the Lie algebra glN we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the glN rational quantized KnizhnikZamolodchikov difference operators. We describe the transformations of the dynamical o ..."
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Cited by 9 (3 self)
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Abstract. For the Lie algebra glN we introduce a system of differential operators called the dynamical operators. We prove that the dynamical differential operators commute with the glN rational quantized KnizhnikZamolodchikov difference operators. We describe the transformations of the dynamical operators under the natural action of the glN Weyl group. Department of Mathematical Sciences,
Selberg Type Integrals Associated with sl3
, 2003
"... We present several formulae for the Selberg type integrals associated with the Lie algebra sl3. ..."
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Cited by 5 (0 self)
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We present several formulae for the Selberg type integrals associated with the Lie algebra sl3.
HOW TO REGULARIZE SINGULAR VECTORS AND KILL THE DYNAMICAL WEYL GROUP
, 2002
"... Abstract. Let g be a simple Lie algebra, and let Mλ be the Verma module over g with highest weight λ. For a finitedimensional gmodule U we introduce a notion of a regularizing operator, acting in U, which makes the meromorphic family of intertwining operators Φ: Mλ+µ → Mλ ⊗ U holomorphic, and conj ..."
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Abstract. Let g be a simple Lie algebra, and let Mλ be the Verma module over g with highest weight λ. For a finitedimensional gmodule U we introduce a notion of a regularizing operator, acting in U, which makes the meromorphic family of intertwining operators Φ: Mλ+µ → Mλ ⊗ U holomorphic, and conjugates the dynamical Weyl group operators Aw(λ) ∈ End(U) to constant operators. We establish fundamental properties of regularizing operators, including uniqueness, and prove the existence of a regularizing operator in the case g = sl3. 1.