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arXiv1995

Affine Hecke algebras associated to Kac-Moody groups

In this paper we give a geometric construction of Cherednik's double affine Hecke algebra. We construct the algebra as the equivariant $K$-theory of the Lagrangian subvariety of the cotangent variety of the square of the flag variety of $G$, the variety being given by the union of the conormal bundles to the $G$-orbits on $\flag\times\flag$. This is a generalisation of work of Kazhdan and Lusztig to the Kac-Moody case, and is suitable for describing a certain class of modules for this algebra. In a paper in preparation we will do this, in the case $G$ is affine (Cherednik's case).

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arXiv1995

Chern-Simons theory on a lattice and a new description of 3-manifolds invariants

A new approach to the quantization of Chern-Simons theory has been developed in recent papers of the author. It uses a "simulation" of the moduli space of flat connections modulo the gauge group which reveals to be related to a lattice gauge theory based on a quantum group. After a generalization of the formalism of q-deformed gauge theory to the case of root of unity, we compute explicitely the correlation functions associated to Wilson loops (and more generally to graphs) on a surface with punctures, which are the interesting quantity in the study of moduli space. We then give a new description of Chern-Simons three manifolds invariants based on a description in terms of the mapping class group of a surface. At last we introduce a three dimensional lattice gauge theory based on a quantum group which is a lattice regularization of Chern-Simons theory.

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arXiv1995

Matched differential calculus on the quantum groups $GL_q(2,C),SL_q(2,C),C_q(2|0)$

We proposed the construction of the differential calculus on the quantum group and its subgroup with the property of the natural reduction: the differential calculus on the quantum group $GL_q(2,C)$ has to contain the differential calculus on the quantum subgroup $SL_q(2,C)$ and quantum plane $C_q(2|0)$ (''quantum matrjoshka''). We found, that there are two differential calculi, associated to the left differential Maurer--Cartan 1-forms and to the right differential 1-forms. Matched reduction take the degeneracy between the left and right differentials. The classical limit ($q\to 1$) of the ''left'' differential calculus and of the ''right'' differential calculus is undeformed differential calculus. The condition ${\cal D}_qG=1$ gives the differential calculus on $SL_q(2,C)$, which contains the differential calculus on the quantum plane $C_q(2|0)$.

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arXiv1995

Generalization and Deformations of Quantum Groups; Quantization of All Simple Lie Bi-Algebras

A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Special values of the parameters are characterized by the appearance of certain ideals; in this case the universal R-matrix exists on the associated algebraic quotient. In special cases the quotient is a "standard" quantum group; all familiar quantum groups including twisted ones are obtained in this way. In other special cases one finds new types of coboundary bi-algebras. A large class of first order deformations of all these standard bi-algebras is investigated and the associated deformed universal R-matrices have been calculated. One obtains, in particular, universal R-matrices associated with all simple, complex Lie algebras (classification by Belavin and Drinfeld) to first order in the deformation parameter.

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arXiv1995

Vassiliev invariants I: Braid groups and rational homotopy theory

We get a detailed account of Vassiliev type invariants starting with Chen's theory of iterated integrals and Malcev's completion of discrete groups. The canonical injection of the group of pure braids into its completion is identified with the universal Kontsevich-Vassiliev invariant.Further we discuss the extension of this morphism to the whole braid group and the multiplication law for the last one.

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arXiv1995

Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories

Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. As a consequence a generalization of the Radford-Majid criterion for a braided Hopf algebra to be a cross product is obtained. The results of this paper turn out to be fundamental for the construction of (bicovariant) differential calculi on braided Hopf algebras.

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arXiv1995

On Reduced Q-Functions

Schur's $Q$-functions with reduced variables are discussed by employing a combinatorics of strict partitions. They are called reduced $Q$-functions. We give a description of the linear relations among reduced $Q$-functions.

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arXiv1995

The Use of Quantum Groups in Nuclear Structure Problems

Various applications of quantum algebraic techniques in nuclear structure physics, such as the su$_q$(2) rotator model and its extensions, the use of deformed bosons in the description of pairing correlations, and the construction of deformed exactly soluble models (Interacting Boson Model, Moszkowski model) are briefly reviewed. Emphasis is put in the study of the symmetries of the anisotropic quantum harmonic oscillator with rational ratios of frequencies, which underly the structure of superdeformed and hyperdeformed nuclei, the Bloch--Brink $α$-cluster model and possibly the shell structure in deformed atomic clusters.

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arXiv1995

Double-Bosonisation and the Construction of {$U_q(g)$}

We introduce a quasitriangular Hopf algebra or `quantum group' $U(B)$, the {\em double-bosonisation}, associated to every braided group $B$ in the category of $H$-modules over a quasitriangular Hopf algebra $H$, such that $B$ appears as the `positive root space', $H$ as the `Cartan subalgebra' and the dual braided group $B^*$ as the `negative root space' of $U(B)$. The choice $B=f$ recovers Lusztig's construction of $U_q(g)$, where $f$ is Lusztig's algebra associated to a Cartan datum; other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct $U_q(sl_3)$ from $U_q(sl_2)$ by this method, extending it by the quantum-braided plane $A_q^2$. We provide a fundamental representation of $U(B)$ in $B$. A projection from the quantum double, a theory of double biproducts and a Tannaka-Krein reconstruction point of view are also provided.

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arXiv1995

Modular functor and representation theory of \hat{sl}_{2} at a rational level

We define a new modular functor based on Kac-Wakimoto admissible representations and the corresponding D-module on the moduli space of rank 2 vector bundles with parabolic structure. A new fusion functor arises which is related to representation theory of the pair "osp(1|2),sl(2)" in the same way as the fusion functor for the Virasoro algebra is related to representation theory of the pair "sl(2),sl(2)"

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arXiv1995

Quantum hyperboloid and braided modules

When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, in the usual manner, the following problem arises: what is a ``representation theory'' of this algebra? We construct the series of all spin representations of $\ahqc$, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules. A braided trace and a braided involution are discussed as well.

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arXiv1995

Unitary operator bases and q-deformed algebras

Starting from the Schwinger unitary operator bases formalism constructed out of a finite dimensional state space, the well-known q-deformed commutation relation is shown to emerge in a natural way, when the deformation parameter is a root of unity.

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arXiv1995

Dilute Algebras and Solvable Lattice Models

The definition of a dilute braid-monoid algebra is briefly reviewed. The construction of solvable vertex and interaction-round-a-face models built on representations of the dilute Temperley-Lieb and Birman-Wenzl-Murakami algebras is discussed.

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arXiv1995

Twining characters, orbit Lie algebras, and fixed point resolution

We describe the resolution of field identification fixed points in coset conformal field theories in terms of representation spaces of the coset chiral algebra. A necessary ingredient from the representation theory of Kac Moody algebras is the recently developed theory of twining characters and orbit Lie algebras, as applied to automorphisms representing identification currents.

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arXiv1995

Some aspects of noncommutative differential geometry

We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework.

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arXiv1995

On a Universal Invariant of 3-Manifolds

We construct an invariant of 3-manifolds using a modification of the Kontsevich integral and Kirby's calculus. This invariant, as expected in perturbative Chern-Simon theory, takes values in the algebra of oriented 3-valent graphs. This algebra is a Hopf algebra, graded by half the number of vertices in 3-valent graphs. The degree 1 term of the invariant coincides with Casson-Walker-Lescop invariant. The degree $n$ term is constructed out of the universal Vassiliev invariant of links of degree less than or equal to $(l+1)n$ where $l$ is the number of link components.

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arXiv1995

Geometrical Quantization in Fock Space

We investigate an infinite dimensional analog of the theory of Lagrangian manifolds with complex germs. To such a manifold we assign a canonical operator that depends on creation and annihilation operators. This operator is by definition the geometrical quantization for these isotropic manifolds with complex germs. We prove that for secondary quantized equations this quantization is the asymptotics for the Cauchy problem. Results of Berezin are used thouroughly in the construction of the canonical operator and in proofs of the theorems.

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arXiv1995

Tensor operators in R-matrix approach

The definitions and some properties (e.g. the Wigner-Eckart theorem, the fusion procedure) of covariant and contravariant q-tensor operators for quasitriangular quantum Lie algebras are formulated in the R-matrix language. The case of U_q(sl(n)) (in particular, for n=2) is discussed in more detail.

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