897 research outputs found
Quantum Lie algebras; their existence, uniqueness and -antisymmetry
Quantum Lie algebras are generalizations of Lie algebras which have the
quantum parameter h built into their structure. They have been defined
concretely as certain submodules of the quantized enveloping algebras. On them
the quantum Lie bracket is given by the quantum adjoint action.
Here we define for any finite-dimensional simple complex Lie algebra g an
abstract quantum Lie algebra g_h independent of any concrete realization. Its
h-dependent structure constants are given in terms of inverse quantum
Clebsch-Gordan coefficients. We then show that all concrete quantum Lie
algebras are isomorphic to an abstract quantum Lie algebra g_h.
In this way we prove two important properties of quantum Lie algebras: 1) all
quantum Lie algebras associated to the same g are isomorphic, 2) the quantum
Lie bracket of any quantum Lie algebra is -antisymmetric. We also describe a
construction of quantum Lie algebras which establishes their existence.Comment: 18 pages, amslatex. Files also available from
http://www.mth.kcl.ac.uk/~delius/q-lie/qlie_biblio/qlieuniq.htm
R-matrices and Tensor Product Graph Method
A systematic method for constructing trigonometric R-matrices corresponding
to the (multiplicity-free) tensor product of any two affinizable
representations of a quantum algebra or superalgebra has been developed by the
Brisbane group and its collaborators. This method has been referred to as the
Tensor Product Graph Method. Here we describe applications of this method to
untwisted and twisted quantum affine superalgebras.Comment: LaTex 7 pages. Contribution to the APCTP-Nankai Joint Symposium on
"Lattice Statistics and Mathematical Physics", 8-10 October 2001, Tianjin,
Chin
Quasi-Spin Graded-Fermion Formalism and Branching Rules
The graded-fermion algebra and quasi-spin formalism are introduced and
applied to obtain the branching rules for the
"two-column" tensor irreducible representations of gl(m|n), for the case . In the case m < n, all such irreducible representations of gl(m|n)
are shown to be completely reducible as representations of osp(m|n). This is
also shown to be true for the case m=n except for the "spin-singlet"
representations which contain an indecomposable representation of osp(m|n) with
composition length 3. These branching rules are given in fully explicit form.Comment: 19 pages, Latex fil
Quasi-Hopf Superalgebras and Elliptic Quantum Supergroups
We introduce the quasi-Hopf superalgebras which are graded versions of
Drinfeld's quasi-Hopf algebras. We describe the realization of elliptic quantum
supergroups as quasi-triangular quasi-Hopf superalgebras obtained from twisting
the normal quantum supergroups by twistors which satisfy the graded shifted
cocycle condition, thus generalizing the quasi-Hopf twisting procedure to the
supersymmetric case. Two types of elliptic quantum supergroups are defined,
that is the face type and the vertex type
(and ), where is any
Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears
that the vertex type twistor can be constructed only for
in a non-standard system of simple roots, all of which are fermionic.Comment: 22 pages, Latex fil
Quantum affine algebras and universal R-matrix with spectral parameter, II
This paper is an extended version of our previous short letter \cite{ZG2} and
is attempted to give a detailed account for the results presented in that
paper. Let be the quantized nontwisted affine Lie algebra
and be the corresponding quantum simple Lie algebra. Using the
previous obtained universal -matrix for and
, we determine the explicitly spectral-dependent universal
-matrix for and . We apply these spectral-dependent
universal -matrix to some concrete representations. We then reproduce the
well-known results for the fundamental representations and we are also able to
derive for the first time the extreamly explicit and compact formula of the
spectral-dependent -matrix for the adjoint representation of , the
simplest nontrival case when the tensor product of the representations is {\em
not} multiplicity-free.Comment: 22 page
A Unified and Complete Construction of All Finite Dimensional Irreducible Representations of gl(2|2)
Representations of the non-semisimple superalgebra in the standard
basis are investigated by means of the vector coherent state method and
boson-fermion realization. All finite-dimensional irreducible typical and
atypical representations and lowest weight (indecomposable) Kac modules of
are constructed explicitly through the explicit construction of all
particle states (multiplets) in terms of boson and fermion
creation operators in the super-Fock space. This gives a unified and complete
treatment of finite-dimensional representations of in explicit form,
essential for the construction of primary fields of the corresponding current
superalgebra at arbitrary level.Comment: LaTex file, 23 pages, two references and a comment added, to appear
in J. Math. Phy
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