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arXiv1995

Geometry of Quantum Principal Bundles I

A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential forms on the base manifold with an appropriate differential calculus on the structure quantum group. Relations between the calculus on the group and the calculus on the bundle are investigated. A concept of (pseudo)tensoriality is formulated. The formalism of connections is developed. In particular, operators of horizontal projection, covariant derivative and curvature are constructed and analyzed. Generalizations of the first structure equation and of the Bianchi identity are found. Illustrative examples are presented.

q-alg↗
arXiv1995

Simple currents and extensions of vertex operator algebras

We consider how a vertex operator algebra can be extended to an abelian intertwining algebra by a family of weak twisted modules which are {\em simple currents} associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except for $E_8$) and to get various extensions of the vertex operator algebras associated with integrable representations.

q-alg↗
arXiv1995

Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory

We define and study the properties of observables associated to any link in $Σ\times {\bf R}$ (where $Σ$ is a compact surface) using the combinatorial quantization of hamiltonian Chern-Simons theory. These observables are traces of holonomies in a non commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non commutative algebra, the so called Moduli Algebra. When $Σ=S^2$ these link invariants are pure numbers and are equal to Reshetikhin-Turaev link invariants.

q-alg↗
arXiv1995

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.

q-alg↗
arXiv1997

Spiders for rank 2 Lie algebras

A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. We define certain combinatorial spiders by generators and relations that are isomorphic to the representation theories of the three rank two simple Lie algebras, namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider for A1. Among other things, they yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants.

q-alg↗
arXiv1995

Solutions of Klein--Gordon and Dirac equations on quantum Minkowski spaces

Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein--Gordon and Dirac equations are found. The Fock space construction is sketched.

q-alg↗
arXiv1996

Spectral Decomposition of Path Space in Solvable Lattice Model

We give the {\it spectral decomposition} of the path space of the $U_q(\hatsl)$ vertex model with respect to the local energy functions. The result suggests the hidden Yangian module structure on the $\hatsl$ level $l$ integrable modules, which is consistent with the earlier work [1] in the level one case. Also we prove the fermionic character formula of the $\hatsl$ level $l$ integrable representations in consequence.

q-alg↗
arXiv1994

Vertex--IRF correspondence and factorized L-operators for an elliptic R-operator

As for an elliptic $R$-operator which satisfies the Yang--Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex--IRF correspondence for the elliptic $R$-operator is obtained. The vertex--IRF correspondence implies that the Boltzmann weights of the IRF model satisfy the star--triangle relation. By means of these intertwining vectors, the factorized L-operators for the elliptic $R$-operator are also constructed. The vertex--IRF correspondence and the factorized L-operators for Belavin's $R$-matrix are reproduced from those of the elliptic $R$-operator.

q-alg↗
arXiv1995

On the classification of quantum Poincaré groups

Using the general theory of [10] ( hep-th 9412058 ), quantum Poincaré groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most cases we solve them and give the classification of quantum Poincaré groups. Each of them corresponds to exactly one quantum Minkowski space. The Poincaré series of these objects are the same as in the classical case. We also classify possible $R$-matrices for the fundamental representation of the group.

hep-th↗
arXiv1995

Quantum Affine Algebras and Deformations of the Virasoro and W-algebras

We generalize some results concerning affine algebras at the critical level to the corresponding quantum algebras. In particular, we show that the Wakimoto realization provides a homomorphism of Poisson algebras from the center of a quantum affine algebra to a Heisenberg-Poisson algebra. This homomorphism is a q-deformation of the Miura transformation. It is given by the same formulas as the spectra of transfer-matrices of the corresponding quantum integrable models. The image of the center in the Heisenberg-Poisson algebra is a Poisson subalgebra of the latter, which is a q-deformation of the corresponding classical W-algebra. The relations in this Poisson algebra are explicitly computed in the case of sl(n)^.

q-alg↗
arXiv1995

Quantum W-algebras and Elliptic Algebras

We define quantum W-algebras generalizing the results of Reshetikhin and the second author, and Shiraishi-Kubo-Awata-Odake. The quantum W-algebra associated to sl_N is an associative algebra depending on two parameters. For special values of parameters it becomes the ordinary W-algebra of sl_N, or the q-deformed classical W-algebra of sl_N. We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U_q(n^).

q-alg↗
arXiv1995

Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4ν)

We prove polynomial identities for the N=1 superconformal model SM(2,4ν) which generalize and extend the known Fermi/Bose character identities. Our proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic side and a recently introduced very general method of producing recursion relations for q-series on the fermionic side. We use these polynomials to demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4ν) and M(2ν-1,4ν). We also introduce a generalization of the Witten index which is expressible in terms of the Rogers false theta functions.

hep-th↗
arXiv1997

Fourier Duality as a Quantization Principle

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no more complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.

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