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Duality for Knizhnik-Zamolodchikov and dynamical equations
- ACTA APPL. MATH
, 2001
"... We consider the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl k, gl n) duality. We show that the KZ and dynamical equations naturally exchange under the duality. ..."
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Cited by 31 (9 self)
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We consider the Knizhnik-Zamolodchikov (KZ) and dynamical equations, both differential and difference, in the context of the (gl k, gl n) duality. We show that the KZ and dynamical equations naturally exchange under the duality.
COMBINATORICS OF RATIONAL FUNCTIONS AND POINCARÉ-BIRCHOFF-WITT EXPANSIONS OF THE CANONICAL U(n–)-VALUED DIFFERENTIAL FORM
, 2004
"... Abstract. We study the canonical U(n–)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projection ..."
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Cited by 13 (4 self)
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Abstract. We study the canonical U(n–)-valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras Ar, Br, Cr, Dr in a conveniently chosen Poincaré-Birchoff-Witt basis. 1.
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,
HYPERGEOMETRIC SOLUTIONS OF THE QUANTUM DIFFERENTIAL EQUATION OF THE COTANGENT BUNDLE of a partial flag variety
, 2013
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QUANTUM COHOMOLOGY OF THE COTANGENT BUNDLE OF A Flag Variety As A Yangian Bethe Algebra
, 2013
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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.
IDENTITIES BETWEEN q-HYPERGEOMETRIC AND HYPERGEOMETRIC INTEGRALS OF DIFFERENT DIMENSIONS
, 2003
"... Given complex numbers m1,l1 and nonnegative integers m2,l2, such that m1 +m2 = l1 +l2, for any a,b = 0,...,min(m2,l2) we define an ˆl2-dimensional Barnes type q-hypergeometric integral Ia,b(z,µ;m1,m2,l1,l2) and an ˆl2-dimensional hypergeometric integral Ja,b(z,µ;m1,m2,l1,l2). The integrals depend ..."
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Cited by 3 (2 self)
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Given complex numbers m1,l1 and nonnegative integers m2,l2, such that m1 +m2 = l1 +l2, for any a,b = 0,...,min(m2,l2) we define an ˆl2-dimensional Barnes type q-hypergeometric integral Ia,b(z,µ;m1,m2,l1,l2) and an ˆl2-dimensional hypergeometric integral Ja,b(z,µ;m1,m2,l1,l2). The integrals depend on complex parameters z and µ. We show that Ia,b(z,µ;m1,m2,l1,l2) equals Ja,b(e µ,z;l1,l2,m1,m2) up to an explicit factor, thus establishing an equality of ˆl2-dimensional q-hypergeometric and ˆm2-dimensional hypergeometric integrals. The identity is based on the (gl k,gl n) duality for the qKZ and dynamical difference equations.
Quasi-polynomials and the Bethe Ansatz
, 2008
"... We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obt ..."
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Cited by 1 (0 self)
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We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions). To a solution of the Bethe Ansatz equation, one assigns a common eigenvector (called the Bethe vector) of the trigonometric Gaudin operators. The dynamical Weyl group projectively acts on the common eigenvectors of the trigonometric Gaudin operators. We conjecture that this action preserves the set of Bethe vectors and coincides with the action induced by the action on points of populations. We prove the conjecture for sl2.
Varchenko A., Spaces of quasi-polynomials and the Bethe ansatz
"... Abstract. We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all sol ..."
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Cited by 1 (1 self)
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Abstract. We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions). To a solution of the Bethe Ansatz equation, one assigns a common eigenvector (called the Bethe vector) of the trigonometric Gaudin operators. The dynamical Weyl group projectively acts on the common eigenvectors of the trigonometric Gaudin operators. We conjecture that this action preserves the set of the Bethe vectors and coincides with the action induced by the action on points of populations. We prove the conjecture for sl2. 1.
