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.     

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra $\text{E}$ (G) is called strongly positive if f(x) ? 0 for all x ? $\text{E}$(G) which are mapped on positive operators for all G-integrable irreducible representations of $\text{E}$(G). We prove that for each real connected Lie group G ≠ R₁ there are positive, not strongly positive, linear functionals on $\text{E}$(G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.     

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.     

Complex quantum groups and their quantum enveloping algebras

We construct complexified versions of the quantum groups associated with the Lie algebras of typeA _n?1 ,B _n ,C _n , andD _n . Following the ideas of Faddeev, Reshetikhin and Takhtajan we obtain the Hopf algebras of regular functionals U_? on these complexified quantum groups. In the special exampleA ₁ we derive theq-deformed enveloping algebraU _q (sl(2, ?)). In the limitq→1 the undeformedU _q (sl(2, ?)) is recovered.     

Complex quantum enveloping algebras as twisted tensor products

We introduce a *-structure on the quantum double and its dual in order to make contact with various approaches to the enveloping algebras of complex quantum groups. Furthermore, we introduce a canonical basis in the quantum double, its universalR-matrices and give its relation to subgroups in the dual Hopf algebra.     

Casimir invariants for quantized affine Lie algebras

Casimir invariants for quantized affine Lie algebras are constructed and their eigenvalues computed in any irreducible highest-weight representation.     

UniversalR-matrix for quantized (super)algebras

For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.     

Automorphism modular invariants of current algebras

We consider those two-dimensional rational conformal field theories (RCFTs) whose chiral algebras, when maximally extended, are isomorphic to the current algebra formed from some untwisted affine Lie algebra at fixed level. In this case the partition function is specified by an automorphism of the fusion ring and corresponding symmetry of the Kac-Peterson modular matrices. We classify all such partition functions when the underlying finite-dimensional Lie algebra is simple. This gives all possible spectra for this class of RCFTs. While accomplishing this, we also find the primary fields with second smallest quantum dimension.Supported in part by NSERC.     

Generalized enveloping algebras and quantum kinematic coherent states of noncompact Lie groups

A new concept of generalized enveloping algebra is introduced by means of the generalized Heisenberg commutation relations of non-Abelian quantum kinematics. This concept is examined within the quantum-kinematic formalism of some noncompact Lie groups of a special kind. The well known Gel'fand theorem (which relates the center of the traditional enveloping algebra with the adjoint representation) is then extended to the generalized enveloping algebra of the group. In this way, the isomorphism of the generalized left-center and the traditional right-center of the corresponding enveloping algebras is proved within the left regular representation of noncompact Lie groups of the chosen kind. As an interesting application of generalized enveloping algebras, this paper contains a brief discussion of quantum-kinematic (boson) ladder operators for non-Abelian noncompact finite Lie groups and of their corresponding coherent states.     

The universal structure of local algebras

It is shown that a few physically significant conditions fix the global structure of the local algebras appearing in quantum field theory: it is isomorphic to that of where is the unique hyperfinite factor of typeIII ₁ and the center of the respective algebra. The argument is based on results in [1, 2] relating to the type of the local algebras and an improvement of an argument in [3] concerning the split property.Heisenberg fellowResearch supported by Ministro della Pubblica Istruzione and CNR-GNAFA     

Q-analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras

We introduceq-analogues of Clifford and Weyl algebras. Using these, we construct spinor and oscillator representations of quantum enveloping algebras of typeA _N–1 ,B _N ,C _N ,D _N andA _N–1 ⁽¹⁾ . Also we discuss the irreducibility and the unitarity of these representations.     

Real complexified quantum groups,their vector fields and quantum enveloping algebras

We construct complex quantum groups associated with the Lie algebras of typeA _n–1 ,B _n ,C _n andD _n which are considered as real algebras. Following the ideas of Faddeev, Reshetikhin and Takhtayan, we obtain the Hopf algebras of regular functionalsU _R , on these real complexified quantum groups. Theq-analogues of the left invariant vector fields of the quantum enveloping algebras are defined. These quantum vector fields are functionals over the corresponding real formA of the complex quantum groupA. The equivalence of the Hopf algebra of regular functionals and the algebra of complex quantum vector fields is shown 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. In the special exampleA ₁ , we derive theq-deformed real complexified enveloping algebraU _q sl(2, ) with six generators.Presented at the Colloquium on the Quantum Groups, Prague, 18–20 June, 1992.Based on the papers: [i]Drabant B., Schlieker M., Weich W., and Zumino B.: PreprintLMU-TPW 1991-5 (to appear in Commun. Math. Phys.) [ii]Chryssomalakos C., Drabant B., Schlieker M., Weich W., and Zumino B.: Preprint UCB 92/03 (to appear in Commun. Math. Phys.) [iii]Drabant B., Juro B., Schlieker M., Weich W., and Zumino B.: Preprint MPI-Ph/92-39 (submitted to Lett. Math. Phys.)     

On the classification of modular fusion algebras

We introduce the notion of (nondegenerate) strongly-modular fusion algebras. Here strongly-modular means that the fusion algebra is induced via Verlinde's formula by a representation of the modular group =SL(2,) whose kernel contains a congruence subgroup. Furthermore, nondegenerate means that the conformal dimensions of possibly underlying rational conformal field theories do not differ by integers. Our main result is the classification of all strongly-modular fusion algebras of dimension two, three and four and the classification of all nondegenerate strongly-modular fusion algebras of dimension less than 24. We use the classification of the irreducible representations of the finite groups , wherep is a prime and a positive integer. Finally, we give polynomial realizations and fusion graphs for all simple nondegenerate strongly-modular fusion algebras of dimension less than 24.     

Symmetries and half-sided modular inclusions of von Neumann algebras

LetN, be a von Neumann algebras on a Hilbert space , a common cyclic and separating vector. Assume to be cyclic and also separating forN . Denote by , _N , _N the modular operators to (, ), (N, ), resp (N , ). Assume now ^-it N ^it N for allt 0. (Such type of inclusions ((N U, ) , ) are called half-sided modular.) Then the modular groups ^it , _N ^ir , _N ^is ,t, r, s generate a unitary representation of the group S1(2, )/Z ₂ of positive energy.Another result is related to two half-sided modular inclusions (₁ , ) and (₂ , ). Under proper conditions the three modular groups ^it , ₁ ^ir , ₂ ^is ,t, r, s generate the three-dimensional subgroup of O(2, 1) of two commuting translations and the Lorentz transformation.Partly supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

On universalR-matrix for quantized nontwisted rank 3 affine KM algebras

Explicit formulas of the universalR-matrix are given for all quantized nontwisted rank 3 affine KM algebras U _q (A ₂ ⁽¹⁾ ), U _q (C ₂ ⁽¹⁾ ) and U _q (G ₂ ⁽¹⁾ ).     

On the modular structure of local algebras of observables

An asymptotic expression for the modular operators of local algebras of observables in quantum field theory is given. In an asymptotically scale invariant theory this leads to an identification of the spectra of all modular operators with ₊. So in this case the local algebras are algebras of type III, and only factors of type III₁ can occur in a central decomposition.Heisenberg fellow     

Liapunov theorem for modular functions

We extend Liapunov Theorem to modular functions on complemented lattices.