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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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Cited by 16 (8 self)
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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 r-MATRICES WITH Nonabelian Base
, 2003
"... . We construct some classes of dynamical r-matrices 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 r-matrices 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 r-matrices 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 r-matrices 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 Donin-Mudrov nonabelian fusion construction. We apply these results to the construction of equivariant star-products on Poisson homogeneous spaces, which include some homogeneous spaces introduced by De Concini.
DOUBLE AFFINE HECKE ALGEBRAS AND BISPECTRAL QUANTUM KNIZHNIK-ZAMOLODCHIKOV EQUATIONS
, 2008
"... We use the double affine Hecke algebra of type GLN to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (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 q-difference equations, which we call the bispectral quantum Knizhnik-Zamolodchikov (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 q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution Φ of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual 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 q-difference operators. Under this correspondence Φ becomes a self-dual Harish-Chandra series solution Φ + of the bispectral problem. Specializing the central character as above, we recover from Φ + the symmetric self-dual 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 Knizhnik-Zamolodchikov 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 Knizhnik-Zamolodchikov 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 finite-dimensional g-module 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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Cited by 5 (3 self)
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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 finite-dimensional g-module 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.
