It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....

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A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the Hopf fibration S 2 → RP 2. A certain class of strong Uq(2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang–Mills action on a trivial Uq(2)bundle is investigated. It is proved to coincide with the Yang–Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.

Abstract: 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 S ±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 Uq(sl2) 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 Since the seminal paper of Atiyah [A] on the abstract definition of a topological quantum field theory (TQFT) much progress has been made in finding non trivial examples and

. This is a survey of interpretations of q-hypergeometric orthogonal polynomials on quantum groups. The first half of the paper gives general background on Hopf algebras and quantum groups. The emphasis in the rest of the paper is on the SU(2) quantum group. An interpretation of little q-Jacobi polynomials as matrix elements of its irreducible representations is presented. In the last two sections new results by the author on interpretations of Askey-Wilson polynomials are discussed. NOTE. Last modified: May 23, 1995 (a few references updated). The paper appeared earlier in: P. Nevai (ed.), Orthogonal Polynomials: Theory and Practice, 257--292. NATO ASI Series C, Vol.294, Kluwer Academic Publishers, 1990. PRESENT ADDRESS. University of Amsterdam, Department of Mathematics, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. email: thk@fwi.uva.nl 1. Introduction Quantum groups, recently introduced by Drinfeld [11], Woronowicz [49] and Jimbo [16], are fascinating objects, wher...

This paper begins a study of one and two variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebras considered are the quantum algebra Uq (su2 ) and a q-analog of the oscillator algebra (not a quantum algebra). In each case we use a simple one variable model of the positive discrete series of finite and infinite dimensional irreducible representations to compute the Clebsch-Gordan coefficients. We show that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the "group operators" on these representation spaces. We show that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. We also demonstrate that general q-hypergeometric functions can occur as basis functions in two variable models, in contrast with the very restricted parameter values for the q-hypergeometric functions arising as matrix elements in the theory of quantum groups. PACS: 02.20.+b, 03.65.Fd Running Title: Representations of Unitary and Oscillator q-Algebras 1.

This paper discusses various deformations of free associative algebras and of their convolution algebras. Our main examples are deformations of noncommutative symmetric functions related to families of idempotents in descent algebras, and a simple q-analogue of the shuffle product, which has unexpected connections with quantum groups, hyperplane arrangements, and certain questions in theoretical physics (the quon algebra).

The subject of this paper are two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second group is the set of formal diffeomorphisms with the group law being composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in

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