3,855 research outputs found

    q and q,t-Analogs of Non-commutative Symmetric Functions

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    We introduce two families of non-commutative symmetric functions that have analogous properties to the Hall-Littlewood and Macdonald symmetric functions.Comment: Different from analogues in math.CO/0106191 - v2: 26 pages - added a definition in terms of triangularity/scalar product relations - to be submitted FPSAC'0

    From Reflection Equation Algebra to Braided Yangians

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    In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra. In this paper we analyse its specific properties, which distinguish it from other quantum matrix algebras (in first turn, from the RTT one). Thus, we exhibit a specific form of the Cayley-Hamilton identity for its generating matrix, which in a limit turns into the Cayley-Hamilton identity for the generating matrix of the enveloping algebra U(gl(m)). Also, we consider some specific properties of the braided Yangians, recently introduced by the authors. In particular, we establish an analog of the Cayley-Hamilton identity for the generating matrix of such a braided Yangian. Besides, by passing to a limit of the braided Yangian, we get a Lie algebra similar to that entering the construction of the rational Gaudin model. In its enveloping algebra we construct a Bethe subalgebra by the method due to D.Talalaev

    Commutative combinatorial Hopf algebras

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    We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245

    Examples of noncommutative manifolds: complex tori and spherical manifolds

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    We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the differential geometry and the algebraic geometry of these spaces.Comment: Survey article. Final version. To appear in the proceedings volume of the "International Workshop on Noncommutative Geometry", IPM, Tehran 200

    Braided Weyl algebras and differential calculus on U(u(2))

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    On any Reflection Equation algebra corresponding to a skew-invertible Hecke symmetry (i.e. a special type solution of the Quantum Yang-Baxter Equation) we define analogs of the partial derivatives. Together with elements of the initial Reflection Equation algebra they generate a "braided analog" of the Weyl algebra. When q→1q\to 1, the braided Weyl algebra corresponding to the Quantum Group Uq(sl(2))U_q(sl(2)) goes to the Weyl algebra defined on the algebra \Sym((u(2)) or that U(u(2))U(u(2)) depending on the way of passing to the limit. Thus, we define partial derivatives on the algebra U(u(2))U(u(2)), find their "eigenfunctions", and introduce an analog of the Laplace operator on this algebra. Also, we define the "radial part" of this operator, express it in terms of "quantum eigenvalues", and sketch an analog of the de Rham complex on the algebra U(u(2))U(u(2)). Eventual applications of our approach are discussed.Comment: LaTex, 18 pages. Accepted in Journal of Geometry and Physic
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