Simple WZW superselection sectors

The superselection structure of simple currents of chiral Wess-Zumino-Witten theories, at arbitrary valuek of the corresponding affine Lie algebra, is described in terms of explicit localizable automorphisms of the affine algebra. These automorphisms are induced by certain Dynkin diagram automorphisms; under composition, they form an Abelian group isomorphic to the center of the relevant simply connected simple Lie group and, hence, reproduce the WZW fusion rules.     

Fusion structures from quantum groups: II. Why truncation is necessary

It is shown that a finite, reflection positive, and nontruncated fusion structure on an arbitrary Hopf algebra is trivial in the sense thatq-traces coincide with ordinary traces andq-dimensions coincide with ordinary dimensions. Thus, nontruncated fusion structures are ruled out to describe the fusion rules of quantum field theories with noninteger statistical dimensions and a finite number of superselection sectors.Work supported in part by DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Construction of a new quantum double and new solutions to the Yang-Baxter equation

A new quantum double is established from a new Hopf algebra and a new kind of quantum R-matrix is obtained.     

Lie groups whose coadjoint orbits are of dimension smaller or equal to two

We give a complete classification of the class of connected, simply connected Lie groups whose coadjoint orbits are of dimension smaller or equal to two.     

Quantum double and differential calculi

We show that bicovariant bimodules as defined by Woronowicz are in one-to-one correspondence with the Drinfeld quantum double representations. We then prove that a differential calculus associated to a bicovariant bimodule of dimension n is connected to the existence of a particular (n+1)-dimensional representation of the double. An example of bicovariant differential calculus on the nonquasitriangular quantum group E _q (2) is developed. The construction is studied in terms of Hochschild cohomology and a correspondence between differential calculi and 1-cocycles is proved. Some differences of calculi on quantum and finite groups with respect to Lie groups are stressed.     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

The double of a Jacobian quasi-bialgebra

We show that the construction of the double of a Lie bialgebra can be extended to the case where a vector space is only equipped with the structure of a Jacobian quasi-bialgebra (also called a Lie quasi-bialgebra). In this case, the double is itself a Jacobian quasi-bialgebra and it is quasi-triangular. The more general case of the double of a proto-Lie bialgebra is also discussed. In the first section, the notions of exact, strictly exact, quasi-triangular and triangular Jacobian quasi-bialgebras are defined and their equivalence classes under twisting are studied.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

On the universalR-matrix for Uq(241-1241-1241-1)

A compact form for the universalR-matrix of U _q (sl _n ) is derived and illustrated by simple applications.     

On solutions to the braid equation identified by Woronowicz

Solutions of the braid equation recently identified by Woronowicz are shown to be derived from subrepresentations of the regular representations of a quantum double Hopf algebra.     

Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

Solutions of the braid equation related to a Hopf algebra

Two solutionsT andT of the braid equation acting onA A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=. IfA is commutative, thenT= ( denotes the flip: (a b) =b a for anya,b A).Supported by a grant of the Ministry of Education of Poland.     

Reality in the differential calculus on q-Euclidean spaces

The nonlinear reality structure of the derivatives and the differentials for the Euclidean q-spaces are found. A real Laplacian is constructed and reality properties of the exterior derivative are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

Six generatorq-deformed Lorentz algebra

The six generator deformation of the Lorentz algebra is presented. The Hopf algebra structure and the reality conditions are found. The chiral decomposition of SL(2, C) is generalized to theq-case. Casimir operators for theq-Lorentz algebra are given.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098 and in part by the National Science Foundation under grant PHY-85-15857.     

SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

The hopf algebra of vector fields on complex quantum groups

We derive the equivalence of the complex quantum enveloping algebra and the algebra of complex quantum vector fields for the Lie algebra types A _n , B _n , C _n , and D _n by factorizing the vector fields uniquely into a triangular and a unitary part and identifying them with the corresponding elements of the algebra of regular functionals.Humboldt Fellow.     

Rational forms for twistings of enveloping algebras of simple Lie algebras

We give rational forms for twistings of classical enveloping algebras. We also remark a link with the generalized formalism of Gurevich, Manin, and Cartier.     

Differential operators on quantum spaces for GL q (n) and SO q (n)

We prove that the rings of q-differential operators on quantum planes of the GL _q (n) and SO _q (n) types are isomorphic to the rings of classical differential operators. Also, we construct decompositions of the rings of q-differential operators into tensor products of the rings of q-differential operators with less variables.     

Deformation of the Poisson-Lie structure in the SU(2) WZW model

A Lie bialgebra structure in the WZW model is considered. By the deformation of the Poisson structure in the SU(2) WZW model, a commutative noncocommutative Hopf algebra with nonlinear Poisson structure is obtained.