Quantum Lie algebras; their existence, uniqueness and qq-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 $q$-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 gl(m∣n)↓osp(m∣n)gl(m|n)\downarrow osp(m|n) Branching Rules

The graded-fermion algebra and quasi-spin formalism are introduced and applied to obtain the $gl(m|n)\downarrow osp(m|n)$ branching rules for the "two-column" tensor irreducible representations of gl(m|n), for the case $m\leq n (n > 2)$. 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 $Z_2$ 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 $B_{q,\lambda}(G)$ and the vertex type $A_{q,p}[\hat{sl(n|n)}]$ (and $A_{q,p}[\hat{gl(n|n)}]$), where $G$ is any Kac-Moody superalgebra with symmetrizable generalized Cartan matrix. It appears that the vertex type twistor can be constructed only for $U_q[\hat{sl(n|n)}]$ 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 $U_q({\cal G}^{(1)})$ be the quantized nontwisted affine Lie algebra and $U_q({\cal G})$ be the corresponding quantum simple Lie algebra. Using the previous obtained universal $R$-matrix for $U_q(A_1^{(1)})$ and $U_q(A_2^{(1)})$, we determine the explicitly spectral-dependent universal $R$-matrix for $U_q(A_1)$ and $U_q(A_2)$. We apply these spectral-dependent universal $R$-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 $R$-matrix for the adjoint representation of $U_q(A_2)$, 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 $gl(2|2)$ 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 $gl(2|2)$ are constructed explicitly through the explicit construction of all $gl(2)\oplus gl(2)$ 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 $gl(2|2)$ 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