Poisson structures on the Lorentz group

Poisson-Lie structures on the Lorentz group are completely classified. A method applicable to an arbitrary semisimple complex Lie group (treated as real) is developed.     

Geometric crystals on flag varieties and unipotent subgroups of classical groups

For a classical simple algebraic group G$G$, we obtain the affirmative answer for the conjecture in Nakashima (2005) [3] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the maximal unipotent subgroup U⁻$U^{-}$.     

On the quantum group and quantum algebra approach toq-special functions

Two interpretations ofq-special functions based on quantum groups and algebras have been presented in the literature. The connection between these approaches is explained using as an example the case whereU _q (sl(2)) is the basic structure.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

A candidate for a noncompact quantum group

A previous letter (Bidegain, F. and Pinczon, G:Lett. Math. Phys. 33 (1995), 231–240) established that the star-product approach of a quantum group introduced by Bonneau et al. can be extended to a connected locally compact semisimple real Lie group. The aim of the present Letter is to give an example of what a noncompact quantum group could be. From half of the discrete series ofSL(2, ), a new type of quantum group is explicitly constructed.     

Noncommutative symmetric functions and laplace operators for classical Lie algebras

New systems of Laplace (Casimir) operators for the orthogonal and symplectic Lie algebras are constructed. The operators are expressed in terms of paths in graphs related to matrices formed by the generators of these Lie algebras with the use of some properties of the noncommutative symmetric functions associated with a matrix. The decomposition of the Sklyanin determinant into a product of quasi-determinants play the main role in the construction. Analogous decomposition for the quantum determinant provides an alternative proof of the known construction for the Lie algebra gl(N).     

Cohomology and deformation of Leibniz pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebraA together with a Lie algebraL mapped into the derivations ofA. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.     

Metric on quantum spaces

We introduce the analogue of the metric tensor in the case ofq-deformed differential calculus. We analyse the consequences of the existence of the metric, showing that this enforces severe restrictions on the parameters of the theory. We discuss in detail examples of the Manin plane and theq-deformation of SU(2). Finally we touch the topic of relations with Connes' approach.Partially supported by KBN grant 2P 302 168 4.     

A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.     

Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Covariant differential calculi on the quantum space for the quantum group SL _q (2) are classified. Our main assumptions are thatq is not a root of unity and that the differentials de _j of the generators of form a free right module basis for the first-order forms. Our result says, in particular, that apart from the two casesc =c(3), there exists a unique differential calculus with the above properties on the space which corresponds to Podles' quantum sphereS _qc ^/2 .     

Integrable deformations of nilpotent color Lie superalgebras

In this paper we study the infinitesimal deformations of the _Z3${\mathbb{Z}}_3$-color Lie superalgebra L^n,m,p$L^{n,m,p}$. By means of these deformations all filiform _Z3${\mathbb{Z}}_3$-color Lie superalgebras can be obtained. In particular, we give a method that will allow us to determine the dimension of the subspaces that are composed by linearly integrable deformations.     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).     

Deformations of quantum hyperplanes

We consider the quantum hyperplanex ⁱ x ^j =q _ij x ^j x ⁱ (i,j = 1..n) and define and consider deformations of the formx ⁱ x ^j =q _ij x ^j x ⁱ + _{k k} ^ij x ^k + ^ij , where _k ^ij and ^ij are complex numbers. We prove that for genericq _ij no nontrivial deformations exist forn 3.     

Modular Classes of Poisson–Nijenhuis Lie Algebroids

The modular vector field of a Poisson–Nijenhuis Lie algebroid A is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian A-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson–Nijenhuis structure.      

Deformation program for principal series representations

We suggest a realization of the deformation program for principal series representation of a semisimple Lie group by use of a generalized Weyl correspondence.     

Differential geometry on linear quantum groups

An exterior derivative, inner derivation, and Lie derivative are introduced on the quantum group GL _q (N). SL _q (N) is then found by constructing matrices with determnant unity, and the induced calculus is found.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 PHY90-21139.     

On a class of representations of the Virasoro algebra and a conjecture of Kac

We consider a class of representations of the Virasoro algebra that we call bounded admissible representations. For this class, we prove a conjecture of Victor Kac concerning the irreducibility of these representations. Results concerning the center and dimensions of weight spaces are also obtained.