Mapping Class Group Actions on Quantum Doubles

We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for the ${\cal S}^{\pm 1}$-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projective $SL(2, Z)$-action on the center of $U_q(sl_2)$ for $q$ an $l=2m+1$-st root of unity. It appears that the $3m+1$-dimensional representation decomposes into an $m+1$-dimensional finite representation and a $2m$-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of $SL(2, Z)$ and the finite, $m$-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category of $U_q(sl_2)\,$.Comment: 45 page

Convex Bases of PBW type for Quantum Affine Algebras

This note has two purposes. First we establish that the map defined in [L, $\S 40.2.5$ (a)] is an isomorphism for certain admissible sequences. Second we show the map gives rise to a convex basis of Poincar\'e--Birkhoff--Witt (PBW) type for \bup, an affine untwisted quantized enveloping algebra of Drinfel$'$d and Jimbo. The computations in this paper are made possible by extending the usual braid group action by certain outer automorphisms of the algebra.Comment: 7 pages, to appear in Comm. Math. Phy

Worldsheet boundary conditions in Poisson-Lie T-duality

We apply canonical Poisson-Lie T-duality transformations to bosonic open string worldsheet boundary conditions, showing that the form of these conditions is invariant at the classical level, and therefore they are compatible with Poisson-Lie T-duality. In particular the conditions for conformal invariance are automatically preserved, rendering also the dual model conformal. The boundary conditions are defined in terms of a gluing matrix which encodes the properties of D-branes, and we derive the duality map for this matrix. We demonstrate explicitly the implications of this map for D-branes in two non-Abelian Drinfel'd doubles.Comment: 20 pages, Latex; v2: typos and wording corrected, references added; v3: three-dimensional example added, reference added, discussion clarified, published versio

More on quantum groups from the the quantization point of view

Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex" quantum groups and bicovariant quantum Lie algebras are discused from this point of view. Further we discuss the quantization of the Poisson structure on symmetric algebra $S(g)$ leading to the quantized enveloping algebra $U_{h}(g)$ as an example of biquantization in the sense of Turaev. Description of $U_{h}(g)$ in terms of the generators of the bicovariant differential calculus on $F(G_q)$ is very convenient for this purpose. Finally we interpret in the deformation framework some well known properties of compact quantum groups as simple consequences of corresponding properties of classical compact Lie groups. An analogue of the classical Kirillov's universal character formula is given for the unitary irreducible representation in the compact case.Comment: 18 page

Generalized Drinfeld-Sokolov Hierarchies II: The Hamiltonian Structures

In this paper we examine the bi-Hamiltonian structure of the generalized KdV-hierarchies. We verify that both Hamiltonian structures take the form of Kirillov brackets on the Kac-Moody algebra, and that they define a coordinated system. Classical extended conformal algebras are obtained from the second Poisson bracket. In particular, we construct the $W_n^l$ algebras, first discussed for the case $n=3$ and $l=2$ by A. Polyakov and M. Bershadsky.Comment: 41 page

SUq(2)SU_q(2) Lattice Gauge Theory

We reformulate the Hamiltonian approach to lattice gauge theories such that, at the classical level, the gauge group does not act canonically, but instead as a Poisson-Lie group. At the quantum level, it then gets promoted to a quantum group gauge symmetry. The theory depends on two parameters - the deformation parameter $\lambda$ and the lattice spacing $a$. We show that the system of Kogut and Susskind is recovered when $\lambda \rightarrow 0$, while QCD is recovered in the continuum limit (for any $\lambda$). We thus have the possibility of having a two parameter regularization of QCD.Comment: 26 pages, LATEX fil

Combinatorial quantization of the Hamiltonian Chern-Simons theory I

Motivated by a recent paper of Fock and Rosly \cite{FoRo} we describe a mathematically precise quantization of the Hamiltonian Chern-Simons theory. We introduce the Chern-Simons theory on the lattice which is expected to reproduce the results of the continuous theory exactly. The lattice model enjoys the symmetry with respect to a quantum gauge group. Using this fact we construct the algebra of observables of the Hamiltonian Chern-Simons theory equipped with a *-operation and a positive inner product.Comment: 49 pages. Some minor corrections, discussion of positivity improved, a number of remarks and a reference added

Yangians, Integrable Quantum Systems and Dorey's rule

We study tensor products of fundamental representations of Yangians and show that the fundamental quotients of such tensor products are given by Dorey's rule.Comment: We have made corrections to the results for the Yangians associated to the non--simply laced algebra

Analytic Bethe Ansatz for Fundamental Representations of Yangians

We study the analytic Bethe ansatz in solvable vertex models associated with the Yangian $Y(X_r)$ or its quantum affine analogue $U_q(X^{(1)}_r)$ for $X_r = B_r, C_r$ and $D_r$. Eigenvalue formulas are proposed for the transfer matrices related to all the fundamental representations of $Y(X_r)$. Under the Bethe ansatz equation, we explicitly prove that they are pole-free, a crucial property in the ansatz. Conjectures are also given on higher representation cases by applying the $T$-system, the transfer matrix functional relations proposed recently. The eigenvalues are neatly described in terms of Yangian analogues of the semi-standard Young tableaux.Comment: 45 pages, Plain Te

A One-Parameter Family of Hamiltonian Structures for the KP Hierarchy and a Continuous Deformation of the Nonlinear \W_{\rm KP} Algebra

The KP hierarchy is hamiltonian relative to a one-parameter family of Poisson structures obtained from a generalized Adler map in the space of formal pseudodifferential symbols with noninteger powers. The resulting \W-algebra is a one-parameter deformation of \W_{\rm KP} admitting a central extension for generic values of the parameter, reducing naturally to \W_n for special values of the parameter, and contracting to the centrally extended \W_{1+\infty}, \W_\infty and further truncations. In the classical limit, all algebras in the one-parameter family are equivalent and isomorphic to \w_{\rm KP}. The reduction induced by setting the spin-one field to zero yields a one-parameter deformation of \widehat{\W}_\infty which contracts to a new nonlinear algebra of the \W_\infty-type.Comment: 31 pages, compressed uuencoded .dvi file, BONN-HE-92/20, US-FT-7/92, KUL-TF-92/20. [version just replaced was truncated by some mailer