5,224 research outputs found
The Hopf Algebra Structure of the Character Rings of Classical Groups
The character ring \CGL of covariant irreducible tensor representations of
the general linear group admits a Hopf algebra structure isomorphic to the Hopf
algebra \Sym$ of symmetric functions. Here we study the character rings \CO and
\CSp of the orthogonal and symplectic subgroups of the general linear group
within the same framework of symmetric functions. We show that \CO and \CSp
also admit natural Hopf algebra structures that are isomorphic to that of \CGL,
and hence to \Sym. The isomorphisms are determined explicitly, along with the
specification of standard bases for \CO and \CSp analogous to those used for
\Sym. A major structural change arising from the adoption of these bases is the
introduction of new orthogonal and symplectic Schur-Hall scalar products.
Significantly, the adjoint with respect to multiplication no longer coincides,
as it does in the \CGL case, with a Foulkes derivative or skew operation. The
adjoint and Foulkes derivative now require separate definitions, and their
properties are explored here in the orthogonal and symplectic cases. Moreover,
the Hopf algebras \CO and \CSp are not self-dual. The dual Hopf algebras \CO^*
and \CSp^* are identified. Finally, the Hopf algebra of the universal rational
character ring \CGLrat of mixed irreducible tensor representations of the
general linear group is introduced and its structure maps identified.Comment: 38 pages, uses pstricks; new version is a major update, new title,
new material on rational character
Gradings, Braidings, Representations, Paraparticles: some open problems
A long-term research proposal on the algebraic structure, the representations
and the possible applications of paraparticle algebras is structured in three
modules: The first part stems from an attempt to classify the inequivalent
gradings and braided group structures present in the various parastatistical
algebraic models. The second part of the proposal aims at refining and
utilizing a previously published methodology for the study of the Fock-like
representations of the parabosonic algebra, in such a way that it can also be
directly applied to the other parastatistics algebras. Finally, in the third
part, a couple of Hamiltonians is proposed, and their sutability for modeling
the radiation matter interaction via a parastatistical algebraic model is
discussed.Comment: 25 pages, some typos correcte
Topological Hopf algebras, quantum groups and deformation quantization
After a presentation of the context and a brief reminder of deformation
quantization, we indicate how the introduction of natural topological vector
space topologies on Hopf algebras associated with Poisson Lie groups, Lie
bialgebras and their doubles explains their dualities and provides a
comprehensive framework. Relations with deformation quantization and
applications to the deformation quantization of symmetric spaces are describedComment: 13 pages, to appear in the proceedings of the conference "Hopf
algebras in noncommutative geometry and physics" (VUB, Brussels, May 2002
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