The second cohomology of sl(m|1) with coefficients in its enveloping algebra is trivial

Using techniques developed in a recent article by the authors, it is proved that the 2-cohomology of the Lie superalgebra sl(m|1); m > 1, with coefficients in its enveloping algebra is trivial. The obstacles in solving the analogous problem for sl(3|2) are also discussed.Comment: 15 pages, Latex, no figure

Finite Chains with Quantum Affine Symmetries

We consider an extension of the (t-U) Hubbard model taking into account new interactions between the numbers of up and down electrons. We confine ourselves to a one-dimensional open chain with L sites (4^L states) and derive the effective Hamiltonian in the strong repulsion (large U) regime. This Hamiltonian acts on 3^L states. We show that the spectrum of the latter Hamiltonian (not the degeneracies) coincides with the spectrum of the anisotropic Heisenberg chain (XXZ model) in the presence of a Z field (2^L states). The wave functions of the 3^L-state system are obtained explicitly from those of the 2^L-state system, and the degeneracies can be understood in terms of irreducible representations of U_q(\hat{sl(2)}).Comment: 31pp, Latex, CERN-TH.6935/93. To app. in Int. Jour. Mod. Phys. A. (The title of the paper is changed. This is the ONLY change. Previous title was: Hubbard-Like Models in the Infinite Repulsion Limit and Finite-Dimensional Representations of the Affine Algebra U_q(\hat{sl(2)}).

Invariant integration on classical and quantum Lie supergroups

Invariant integrals on Hopf superalgebras, in particular, the classical and quantum Lie supergroups, are studied. The uniqueness ~up to scalar multiples! of a left integral is proved, and a Z2-graded version of Maschke’s theorem is discussed. A construction of left integrals is developed for classical and quantum Lie supergroups. Applied to several classes of examples the construction yields the left integrals in explicit form

Classification of N=6 superconformal theories of ABJM type

Studying the supersymmetry enhancement mechanism of Aharony, Bergman, Jafferis and Maldacena, we find a simple condition on the gauge group generators for the matter fields. We analyze all possible compact Lie groups and their representations. The only allowed gauge groups leading to the manifest N=6 supersymmetry are, up to discrete quotients, SU(n) x U(1), Sp(n) x U(1), SU(n) x SU(n), and SU(n) x SU(m) x U(1) with possibly additional U(1)'s. Matter representations are restricted to be the (bi)fundamentals. As a byproduct we obtain another proof of the complete classification of the three algebras considered by Bagger and Lambert.Comment: 18 page

Tensor operators and Wigner-Eckart theorem for the quantum superalgebra U_{q}[osp(1\mid 2)]

Tensor operators in graded representations of Z_{2}-graded Hopf algebras are defined and their elementary properties are derived. Wigner-Eckart theorem for irreducible tensor operators for U_{q}[osp(1\mid 2)] is proven. Examples of tensor operators in the irreducible representation space of Hopf algebra U_{q}[osp(1\mid 2)] are considered. The reduced matrix elements for the irreducible tensor operators are calculated. A construction of some elements of the center of U_{q}[osp(1\mid 2)] is given.Comment: 16 pages, Late

Gel'fand-Zetlin Basis and Clebsch-Gordan Coefficients for Covariant Representations of the Lie superalgebra gl(m|n)

A Gel'fand-Zetlin basis is introduced for the irreducible covariant tensor representations of the Lie superalgebra gl(m|n). Explicit expressions for the generators of the Lie superalgebra acting on this basis are determined. Furthermore, Clebsch-Gordan coefficients corresponding to the tensor product of any covariant tensor representation of gl(m|n) with the natural representation V ([1,0,...,0]) of gl(m|n) with highest weight (1,0,. . . ,0) are computed. Both results are steps for the explicit construction of the parastatistics Fock space.Comment: 16 page

On the Two-Point Correlation Function for the Uq[SU(2)]U_q[SU(2)] Invariant Spin One-Half Heisenberg Chain at Roots of Unity

Using $U_q[SU(2)]$ tensor calculus we compute the two-point scalar operators (TPSO), their averages on the ground-state give the two-point correlation functions. The TPSOs are identified as elements of the Temperley-Lieb algebra and a recurrence relation is given for them. We have not tempted to derive the analytic expressions for the correlation functions in the general case but got some partial results. For $q=e^{i \pi/3}$, all correlation functions are (trivially) zero, for $q=e^{i \pi/4}$, they are related in the continuum to the correlation functions of left-handed and right-handed Majorana fields in the half plane coupled by the boundary condition. In the case $q=e^{i \pi/6}$, one gets the correlation functions of Mittag's and Stephen's parafermions for the three-state Potts model. A diagrammatic approach to compute correlation functions is also presented.Comment: 19 pages, LaTeX, BONN-HE-93-3