Quantum Algebra And Topology

We prove the Cauchy type identities for the universal double Schubert polynomials, introduced recently by W. Fulton. As a corollary, the determinantal formulae for some specializations of the universal double Schubert polynomials corresponding to the Grassmannian permutations are obtained. We also introduce and study the universal Schur functions and multiparameter deformation of Schubert polynomials.

We study a finite-dimensional quotient of the Hecke algebra of type H n for general n , using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring.

For each of the classical Lie algebras g(n)=o(2n+1),sp(2n),o(2n) of type B, C, D we consider the centralizer of the subalgebra g(n−m) in the universal enveloping algebra U(g(n)) . We show that the n th centralizer algebra can be naturally projected onto the (n−1) th one, so that one can form the projective limit of the centralizer algebras as n→∞ with m fixed. The main result of the paper is a precise description of this limit (or stable) centralizer algebra, denoted by A m . We explicitly construct an algebra isomorphism A m =Z⊗ Y m , where Z is a commutative algebra and Y m is the so-called twisted Yangian associated to the rank m classical Lie algebra of type B, C, or D. The algebra Z may be viewed as the algebra of virtual Laplace operators; it is isomorphic to the algebra of polynomials with countably many indeterminates. The twisted Yangian Y m (and hence the algebra A m ) can be described in terms of a system of generators with quadratic and linear defining relations which are conveniently presented in R-matrix form involving the so-called reflection equation. This extends the earlier work on the type A case by the second author.

The Lie algebra gl(lambda) dependent on the complex parameter lambda is a continuous version of the Lie algebra gl(inf) of infinite matrices with only finite number of nonzero entries. The gl(lambda) was first introduced by B.L.Feigin in [1] in connection with the Lie algebra cohomologies of the differential operators on the complex line. The paper is devoted to the representation theory of the gl(lambda). The Shapovalov's form determinant is calculated; it depends polynomially on lambda and the parameters of the representation. It is natural to include the Hamiltonian Lie algebra of functions on the hyperboloid to the range of gl(lambda), relating it to the value lambda = infinity. Thus, every family of representations of gl(lambda) depending holomorphically on lambda is associated with a holomorphic vector bundle on the Riemann sphere. The determinant of the Shapovalov's form thus determines the corresponding line bundle. The Chern class of this bundle is calculated in two different ways (for every level on the representation). The comparison of the two formulas yields new combinatorial identities with power series. The results of Chapter 1 about the Lie algebra gl(inf) may have some independent interest.

We study the path realization of Demazure crystals related to solvable lattice models in statistical mechanics. Various characters are represented in a unified way as the sums over one dimensional configurations which we call unrestricted, classically restricted and restricted paths. As an application characters of Demazure modules are obtained in terms of q -multinomial coefficients for several level 1 modules of classical affine algebras.

This paper contains a proof that chromatic weight systems, introduced by Chmutov, Duzhin and Lando, can be expressed in terms of weight systems associated with direct sums of the Lie algebras gl_n and so_n. As a consequence the Vassiliev invariants of knots corresponding to the chromatic weight systems distinguish exactly the same knots as a one variable specialisation Y of the Homfly and Kauffman polynomial.

The classical limit of the scaled elliptic algebra A ℏ,η (s l 2 ) is investigated. The limiting Lie algebra is described in two equivalent ways: as a central extension of the algebra of generalized automorphic s l 2 valued functions on a strip and as an extended algebra of decreasing automorphic s l 2 valued functions on the real line. A bialgebra structure and an infinite-dimensional representation in the Fock space are studied. The classical limit of elliptic algebra A q,p (s l 2 ) is also briefly presented.

We present a derivation of the dynamical r-matrices of the Calogero-Moser models using the Hamiltonian reduction procedure to get general formulae. We describe the dynamical r-matrices thus found for spin Calogero-Moser models and relativistic Ruijsenaars-Schneider models

We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dim V 2 . The differential calculi on a finite group algebra CG are also classified and shown to be in correspondence with pairs consisting of an irreducible representation V and a continuous parameter in C P dimV−1 . They have dimension dimV . For a classical Lie group we obtain an infinite family of non-standard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.

For transcendental values of q all bicovariant first order differential calculi on the coordinate Hopf algebras of the quantum groups S L q (n+1) and S p q (2n) are classified. It is shown that the irreducible bicovariant first order calculi are determined by an irreducible corepresentation of the quantum group and a complex number ζ such that ζ n+1 =1 for S L q (n+1) and ζ 2 =1 for S p q (2n) . Any bicovariant calculus is inner and its quantum Lie algebra is generated by a central element. The main technical ingredient is a result of the Hopf algebra R( G q ) 0 for arbitrary simple Lie algebras.