Quantum Algebra And Topology

We present a quantum version of the construction of the KZ system of equations as a flat connection on the spaces of coinvariants of representations of tensor products of Kac-Moody algebras. We consider here representations of a tensor product of Yangian doubles and compute the coinvariants of a deformation of the subalgebra generated by the regular functions of a rational curve with marked points. We observe that Drinfeld's quantum Casimir element can be viewed as a deformation of the zero-mode of the Sugawara tensor in the Yangian double. These ingredients serve to define a compatible system of difference equations, which we identify with the quantum KZ equations introduced by I. Frenkel and N. Reshetikhin.

Quite recently, a ``coloured'' extension of the Yang-Baxter equation has appeared in the literature and various solutions of it have been proposed. In the present contribution, we introduce a generalization of Hopf algebras, to be referred to as coloured Hopf algebras, wherein the comultiplication, counit, and antipode maps are labelled by some colour parameters. The latter may take values in any finite, countably infinite, or uncountably infinite set. A straightforward extension of the quasitriangularity property involves a coloured universal R -matrix, satisfying the coloured Yang-Baxter equation. We show how coloured Hopf algebras can be constructed from standard ones by using an algebra isomorphism group, called colour group. Finally, we present two examples of coloured quantum universal enveloping algebras.

Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is also presented. The coloured two-parameter quantum universal enveloping algebra of gl(1/1) is considered as an example.

Coloured Hopf algebras, related to the coloured Yang-Baxter equation, are reviewed, as well as their duals. The special case of coloured quantum universal enveloping algebras provides a coloured extension of Drinfeld and Jimbo formalism. The universal T -matrix is then generalized to the coloured context, and shown to lead to an algebraic formulation of the coloured RTT-relations, previously proposed by Basu-Mallick as part of a coloured extension of Faddeev, Reshetikhin, and Takhtajan approach to quantum groups and quantum algebras.

We define some new algebraic structures, termed coloured Hopf algebras, by combining the coalgebra structures and antipodes of a standard Hopf algebra set H , corresponding to some parameter set Q , with the transformations of an algebra isomorphism group G , herein called colour group. Such transformations are labelled by some colour parameters, taking values in a colour set C . We show that various classes of Hopf algebras, such as almost cocommutative, coboundary, quasitriangular, and triangular ones, can be extended into corresponding coloured algebraic structures, and that coloured quasitriangular Hopf algebras, in particular, are characterized by the existence of a coloured universal R -matrix, satisfying the coloured Yang-Baxter equation. The present definitions extend those previously introduced by Ohtsuki, which correspond to some substructures in those cases where the colour group is abelian. We apply the new concepts to construct coloured quantum universal enveloping algebras of both semisimple and nonsemisimple Lie algebras, considering several examples with fixed or varying parameters. As a by-product, some of the matrix representations of coloured universal R -matrices, derived in the present paper, provide new solutions of the coloured Yang-Baxter equation, which might be of interest in the context of integrable models.

We review and motivate recently-observed relationships between exactly solvable lattice models and modular representations of Hecke algebras. Firstly, we describe how the set of n -regular partitions label both of the following classes of objects: 1. The spectrum of unrestricted solid-on-solid lattice models based on level-1 representations of the affine algebras $\sl_n$, 2. The irreducible representations of type-A Hecke algebras at roots of unity: H m ( 1 – √ n ) . Secondly, we show that a certain subset of the n -regular partitions label both of the following classes of objects: 1. The spectrum of restricted solid-on-solid lattice models based on cosets of affine algebras $(sl(n)^_1 \times sl(n)^_1)/ sl(n)^_2$. 2. Jantzen-Seitz (JS) representations of H m ( 1 – √ n ) : irreducible representations that remain irreducible under restriction to H m−1 ( 1 – √ n ) . Using the above relationships, we characterise the JS representations of H m ( 1 – √ n ) and show that the generating series that count them are branching functions of affine $\sl_n$.

By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain Drinfeld current realization for quantum affine superalgebra U q [gl(m|n ) (1) ] . We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.

We calculate commutation relations of vertex operators for the spin representation of U q ( D (1) n ) by using recursive formulae of R-matrices. In quantum symmetry approach, we obtain the energy and momentum spectrum of the quantum spin chain model related with the spin representation from these commutation relations.

Let V be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group G of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for G and the irreducible modules for V G which are contained in V where V G is the space of G -invariants of V. We also prove a concomitant Galois correspondence between vertex operator subalgebras of V which contain V G and closed Lie subgroups of G in the case that G is abelian.

While the problem of knot classification is far from solved, it is possible to create computer programs that can be used to tabulate knots up to a desired degree of complexity. Here we discuss the main ideas on which such programs can be based. We also present the actual results obtained after running a computer program on which knots are denoted through regular projections.