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28
Quantum Dynamical coboundary Equation for finite dimensional simple Lie algebras
, 2006
"... For a finite dimensional simple Lie algebra g, the standard universal solution R(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R–matrix and from the solution ..."
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For a finite dimensional simple Lie algebra g, the standard universal solution R(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical Yang–Baxter Equation quantizes the standard trigonometric solution of the Classical Dynamical Yang–Baxter Equation. It can be built from the standard R–matrix and from the solution F(x) ∈ Uq(g) ⊗2 of the Quantum Dynamical co(x)R F12(x). F(x) can be computed explicitely as an infinite product through the use of an auxiliary linear equation, the ABRR equation. Inspired by explicit results in the fundamental representation, it has been conjectured that, in the case where g = sl(n + 1) (n ≥ 1) only, there could exist an element M(x) ∈ Uq(sl(n + 1)) such that the dynamical gauge transform R J of R(x) by M(x), Cycle Equation as R(x) = F −1
Trigonometric osp(12) Gaudin model
 J. Math.Phys
"... The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(12) classical rmatrix. The eigenvectors of the trigonometric osp(12) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the crea ..."
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The problems connected with Gaudin models are reviewed by analyzing model related to the trigonometric osp(12) classical rmatrix. The eigenvectors of the trigonometric osp(12) Gaudin Hamiltonians are found using explicitly constructed creation operators. The commutation relations between the creation operators and the generators of the trigonometric loop superalgebra are calculated. The coordinate representation of the Bethe states is presented. The relation between the Bethe vectors and solutions to the KnizhnikZamolodchikov equation yields the norm of the eigenvectors. The generalized KnizhnikZamolodchikov system is discussed both in the rational and in the trigonometric case.
REPRESENTATIONS OF AFFINE QUANTUM FUNCTION ALGEBRAS
, 2002
"... 1.1. The goal of this paper is to investigate the algebraic structure of certain quantized algebras of functions associated to affine KacMoody Lie algebras and to describe their irreducible representations. Let C be an affine Cartan matrix and g = g(C) be the associated affine Lie algebra. The main ..."
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1.1. The goal of this paper is to investigate the algebraic structure of certain quantized algebras of functions associated to affine KacMoody Lie algebras and to describe their irreducible representations. Let C be an affine Cartan matrix and g = g(C) be the associated affine Lie algebra. The main object of our interest Cq[G] is a ⋆subalgebra of the dual space Homk(U, k) generated
DIFFERENTIAL EQUATIONS COMPATIBLE WITH BOUNDARY RATIONAL QKZ EQUATION
, 908
"... Dedicated to Professor Tetsuji Miwa on his sixtieth birthday Abstract. We give differential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (C ∨ n, Cn) which in the case of t ..."
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Dedicated to Professor Tetsuji Miwa on his sixtieth birthday Abstract. We give differential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (C ∨ n, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case. 1.
CHEVALLEY RESTRICTION THEOREM FOR VECTORVALUED FUNCTIONS ON QUANTUM GROUPS
"... Abstract. We generalize Chevalley’s theorem about restriction Res: C[g] g → C[h] W to the case when a semisimple Lie algebra g is replaced by a quantum group and the target space C of the polynomial maps is replaced by a finite dimensional representation V of this quantum group. We prove that the re ..."
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Abstract. We generalize Chevalley’s theorem about restriction Res: C[g] g → C[h] W to the case when a semisimple Lie algebra g is replaced by a quantum group and the target space C of the polynomial maps is replaced by a finite dimensional representation V of this quantum group. We prove that the restriction map Res: (Oq(G) ⊗ V) Uq(g) → O(H) ⊗ V is injective and describe the image. 1.
DIFFERENTIAL OPERATORS ON G/U AND THE AFFINE GRASSMANNIAN
, 2013
"... ABSTRACT. We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of Dmodules of the basic affine space, ..."
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ABSTRACT. We describe the equivariant cohomology of cofibers of spherical perverse sheaves on the affine Grassmannian of a reductive algebraic group in terms of the geometry of the Langlands dual group. In fact we give two equivalent descriptions: one in terms of Dmodules of the basic affine space, and one in terms of intertwining operators for universal Verma modules. We also construct natural collections of isomorphisms parametrized by the Weyl group in these three contexts, and prove that they are compatible with our isomorphisms. As applications we reprove some results of the first author and of Braverman–Finkelberg.
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"... Concentration inequalities and martingale inequalities: a survey. (English summary) Internet Math. 3 (2006), no. 1, 79–127. This survey presents extensions and generalizations of concentration inequalities and martingale inequalities. In this way a rigorous analysis for random graphs with general de ..."
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Concentration inequalities and martingale inequalities: a survey. (English summary) Internet Math. 3 (2006), no. 1, 79–127. This survey presents extensions and generalizations of concentration inequalities and martingale inequalities. In this way a rigorous analysis for random graphs with general degree distributions can be carried out. As an application, concentration of the power law distribution for the infinite Pólya process is analysed. The socalled preferential attachment scheme can be rewritten as a variation of this Pólya process. Reviewed by Dominique Lépingle
MATHEMATICAL RESEARCHES OF D. P. ZHELOBENKO
, 2009
"... Abstract. This is a brief overview of researches of Dmitry Petrovich Zhelobenko (1934–2006). He is the best known for his book ”Compact Lie groups and their representations ” and for the classification of all irreducible representations of complex semisimple Lie groups. We tell also on other his wor ..."
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Abstract. This is a brief overview of researches of Dmitry Petrovich Zhelobenko (1934–2006). He is the best known for his book ”Compact Lie groups and their representations ” and for the classification of all irreducible representations of complex semisimple Lie groups. We tell also on other his works, especially on the spectral analysis of representations. 1. Brief personal data. The mathematical researches of Dmitriî Petrovich Zhelobenko (Ul’yanovsk, 1934 Moscow, 2006) are mainly devoted to the representation theory of semisimple Lie groups and Lie algebras and to the noncommutative harmonic analysis. He graduated from the Physical Department of the Moscow State University (1958) and took his postgraduate program at the Steklov Mathematical Institute under the supervision of S. V.Fomin and of M. A.Naimark (1961). He has defended his PhD thesis ”Harmonic Analysis on the Lorentz group and some questions of the theory of linear representations ” in 1962. The doctoral thesis ”Harmonic analysis of functions on semisimple Lie groups and its applications to the representation theory ” was defended in the Steklov Institute in 1972 (official opponents: