Coding the principal character formula for affine Kac-Moody lie algebras

In this paper, an algorithm for computing the principal character for affine Lie algebras is discussed and presented. The principal characters discovered using this program are given and/or proven. Results include level 2 and 3 character formulas in and the sole existence of the Rogers-Ramanujan products in , , , , , , .     

Linear maps preserving ideals of -algebras

We show that unital self-adjoint linear bijections of matrix algebras, type factors and abelian -algebras preserving maximal left ideals are isomorphisms and we show that a unital continuous linear map of a -algebra that maps the minimal left ideal into itself is the identity map.     

BRST Resolution for the Principally Graded Wakimoto Module of $$\widehat{\mathfrak{s}\mathfrak{l}}_2 $$

BRST resolution is studied for the principally graded Wakimoto module of recently found in math.QA/0005203. The submodule structure is completely determined and irreducible representations can be obtained as the zero-th cohomology group.     

Isometries of Quaternionic Lie Groups

We study connected Lie groups whose Lie algebra is obtained as the tensor product of a real associative algebra and the algebra of quaternions. It is proved that they carry a natural integrable -structure. We endow such quaternionic Lie groups with a left-invariant Hermitian metric and study the identity connected component of their isometry groups. The determination of such identity connected component is illustrated with a family of examples.     

Bi-Invariant Domains in Semisimple Groups: Nonprincipal Orbits and Boundaries

Let G be a complex semisimple group with real form G . For the action of G × G on G by left and right translation, we define and study an orbit-type stratification of G. In particular, we compute the dimension of an arbitrary stratum; we identify certain strata of low codimension in G, the union of whose closures contains every nonprincipal stratum; and we describe the boundaries and adjacency relations of the maximal connected domains in G that contain only principal orbits.     

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ), (R, K, )) of congruence classes of extensions of (R, K, ) by (U, Q, ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.2000 Mathematics Subject Classification: 17B56, 17B99, 18G99     

Higher order symmetric spaces and the roots of the identity in a Lie group

Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .

     

Crystal bases for the quantum superalgebra

A crystal base theory is introduced for the quantized enveloping algebra of the general linear Lie superalgebra , and an explicit realization of the crystal base is given in terms of semistandard tableaux.     

Extensions of Modules over Hopf Algebras Arising from Lie Algebras of Cartan Type

In this paper we prove that there are no self-extensions of simple modules over restricted Lie algebras of Cartan type. The proof given by Andersen for classical Lie algebras not only uses the representation theory of the Lie algebra, but also representations of the corresponding reductive algebraic group. The proof presented in the paper follows in the same spirit by using the construction of a infinite-dimensional Hopf algebra D(G) u( ) containing u( ) as a normal Hopf subalgebra, and the representation theory of this algebra developed in our previous work. Finite-dimensional hyperalgebra analogs D(G _r ) u( ) have also been constructed, and the results are stated in this setting.     

The atomic and molecular reaction statics

This work presents a new science called atomic and molecular reaction statics (AMRS). There are four parts for AMRS, i.e. the group theoretical derivation of molecular electronic states, the principle of microscopic reversibility, the principle of microscopic transitivity and the optimum energy process rule. AMRS has been developed for about twenty years.