29 research outputs found
An Extension of the Character Ring of sl(3) and Its Quantisation
We construct a commutative ring with identity which extends the ring of
characters of finite dimensional representations of sl(3). It is generated by
characters with values in the group ring of the extended affine
Weyl group of at . The `quantised' version at
rational level realises the fusion rules of a WZW conformal field
theory based on admissible representations of .Comment: contains two TeX files: main file using harvmac.tex, amssym.def,
amssym.tex, 35p.; file with figures using XY-pic package, 4p; v2: minor
corrections, Note adde
S_4-symmetry of 6j-symbols and Frobenius-Schur indicators in rigid monoidal C^*-categories
We show that a left-rigid monoidal C^*-category with irreducible monoidal
unit is also a sovereign and spherical category. Defining a Frobenius-Schur
type indicator we obtain selection rules for the fusion coefficients of
irreducible objects. As a main result we prove S_4-invariance of 6j-symbols in
such a category.Comment: 21 pages + 16 pages with figures; LaTeX2e plus macro package XYpic;
file with included pictures available as
http://www.desy.de/~jfuchs/s4/s4.ps.g
Parafermionic algebras, their modules and cohomologies
We explore the Fock spaces of the parafermionic algebra introduced by H.S.
Green. Each parafermionic Fock space allows for a free minimal resolution by
graded modules of the graded 2-step nilpotent subalgebra of the parafermionic
creation operators. Such a free resolution is constructed with the help of a
classical Kostant's theorem computing Lie algebra cohomologies of the nilpotent
subalgebra with values in the parafermionic Fock space. The Euler-Poincar\'e
characteristics of the parafermionic Fock space free resolution yields some
interesting identities between Schur polynomials. Finally we briefly comment on
parabosonic and general parastatistics Fock spaces.Comment: 10 pages, talk presented at the International Workshop "Lie theory
and its applications in Physics" (17-23 June 2013, Varna, Bulgaria
Microscopic and macroscopic properties of A-superstatistics
The microscopic and the macroscopic properties of A-superstatistics, related
to the class A(0,n-1)\equiv sl(1|n) of simple Lie superalgebras are
investigated. The algebra sl(1|n) is described in terms of generators f_1^\pm,
>..., f_n^\pm, which satisfy certain triple relations and are called Jacobson
generators. The Fock spaces of A-superstatistics are investigated and the Pauli
principle of the corresponding statistics is formulated. Some thermal
properties of A-superstatistics are constructed under the assumption that the
particles interact only via statistical interaction imposed by the Pauli
principle. The grand partition function and the average number of particles are
written down explicitly in the general case and in two particular examples: 1)
the particles have one and the same energy and chemical potential; 2) the
energy spectrum of the orbitals is equidistant.Comment: 26 pages, 3 figure
Hopf algebraic structure of the parabosonic and parafermionic algebras and paraparticle generalization of the Jordan Schwinger map
The aim of this paper is to show that there is a Hopf structure of the
parabosonic and parafermionic algebras and this Hopf structure can generate the
well known Hopf algebraic structure of the Lie algebras, through a realization
of Lie algebras using the parabosonic (and parafermionic) extension of the
Jordan Schwinger map. The differences between the Hopf algebraic and the graded
Hopf superalgebraic structure on the parabosonic algebra are discussed.Comment: 11 pages, LaTex2e fil
The quantum superalgebra : deformed para-Bose operators and root of unity representations
We recall the relation between the Lie superalgebra and para-Bose
operators. The quantum superalgebra , defined as usual in terms
of its Chevalley generators, is shown to be isomorphic to an associative
algebra generated by so-called pre-oscillator operators satisfying a number of
relations. From these relations, and the analogue with the non-deformed case,
one can interpret these pre-oscillator operators as deformed para-Bose
operators. Some consequences for (Cartan-Weyl basis,
Poincar\'e-Birkhoff-Witt basis) and its Hopf subalgebra are
pointed out. Finally, using a realization in terms of ``-commuting''
-bosons, we construct an irreducible finite-dimensional unitary Fock
representation of and its decomposition in terms of
representations when is a root of unity.Comment: 15 pages, LaTeX (latex twice), no figure