The Fuzzy Supersphere

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative $Z_{2}$-graded algebras tending in a suitable limit to a dense subalgebra of the $Z_{2}$-graded algebra of ${\cal H}^{\infty}$-functions on the $(2| 2)$-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the "fuzzy level".Comment: 33 pages, LaTeX, some typos correcte

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

Finite dimensional representations of Uq(C(n+1))U_q(C(n+1)) at arbitrary qq

A method is developed to construct irreducible representations(irreps) of the quantum supergroup $U_q(C(n+1))$ in a systematic fashion. It is shown that every finite dimensional irrep of this quantum supergroup at generic $q$ is a deformation of a finite dimensional irrep of its underlying Lie superalgebra $C(n+1)$, and is essentially uniquely characterized by a highest weight. The character of the irrep is given. When $q$ is a root of unity, all irreps of $U_q(C(n+1))$ are finite dimensional; multiply atypical highest weight irreps and (semi)cyclic irreps also exist. As examples, all the highest weight and (semi)cyclic irreps of $U_q(C(2))$ are thoroughly studied.Comment: 21 page

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

Comments on Drinfeld Realization of Quantum Affine Superalgebra Uq[gl(m∣n)(1)]U_q[gl(m|n)^{(1)}] and its Hopf Algebra Structure

By generalizing the Reshetikhin and Semenov-Tian-Shansky construction to supersymmetric cases, we obtain Drinfeld current realization for quantum affine superalgebra $U_q[gl(m|n)^{(1)}]$. We find a simple coproduct for the quantum current generators and establish the Hopf algebra structure of this super current algebra.Comment: Some errors and misprints corrected and a remark in section 4 removed. 12 pages, Latex fil

Uqosp(2,2)U_q osp(2,2) Lattice Models

In this paper I construct lattice models with an underlying $U_q osp(2,2)$ superalgebra symmetry. I find new solutions to the graded Yang-Baxter equation. These {\it trigonometric} $R$-matrices depend on {\it three} continuous parameters, the spectral parameter, the deformation parameter $q$ and the $U(1)$ parameter, $b$, of the superalgebra. It must be emphasized that the parameter $q$ is generic and the parameter $b$ does not correspond to the `nilpotency' parameter of \cite{gs}. The rational limits are given; they also depend on the $U(1)$ parameter and this dependence cannot be rescaled away. I give the Bethe ansatz solution of the lattice models built from some of these $R$-matrices, while for other matrices, due to the particular nature of the representation theory of $osp(2,2)$, I conjecture the result. The parameter $b$ appears as a continuous generalized spin. Finally I briefly discuss the problem of finding the ground state of these models.Comment: 19 pages, plain LaTeX, no figures. Minor changes (version accepted for publication

Centre and Representations of U_q(sl(2|1)) at Roots of Unity

Quantum groups at roots of unity have the property that their centre is enlarged. Polynomial equations relate the standard deformed Casimir operators and the new central elements. These relations are important from a physical point of view since they correspond to relations among quantum expectation values of observables that have to be satisfied on all physical states. In this paper, we establish these relations in the case of the quantum Lie superalgebra U_q(sl(2|1)). In the course of the argument, we find and use a set of representations such that any relation satisfied on all the representations of the set is true in U_q(sl(2|1)). This set is a subset of the set of all the finite dimensional irreducible representations of U_q(sl(2|1)), that we classify and describe explicitly.Comment: Minor corrections, References added. LaTeX2e, 18 pages, also available at http://lapphp0.in2p3.fr/preplapp/psth/ENSLAPP583.ps.gz . To appear in J. Phys. A: Math. Ge

On representations of super coalgebras

The general structure of the representation theory of a $Z_2$-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantisations thereof as a special case. The general linear supergroups serve as an explicit illustration and the simplest example is carried out in detail.Comment: 18 pages, LaTeX, KCL-TH-94-

Conserved Charges in the Principal Chiral Model on a Supergroup

The classical principal chiral model in 1+1 dimensions with target space a compact Lie supergroup is investigated. It is shown how to construct a local conserved charge given an invariant tensor of the Lie superalgebra. We calculate the super-Poisson brackets of these currents and argue that they are finitely generated. We show how to derive an infinite number of local charges in involution. We demonstrate that these charges Poisson commute with the non-local charges of the model

Realizations of the Lie superalgebra q(2) and applications

The Lie superalgebra q(2) and its class of irreducible representations V_p of dimension 2p (p being a positive integer) are considered. The action of the q(2) generators on a basis of V_p is given explicitly, and from here two realizations of q(2) are determined. The q(2) generators are realized as differential operators in one variable x, and the basis vectors of V_p as 2-arrays of polynomials in x. Following such realizations, it is observed that the Hamiltonian of certain physical models can be written in terms of the q(2) generators. In particular, the models given here as an example are the sphaleron model, the Moszkowski model and the Jaynes-Cummings model. For each of these, it is shown how the q(2) realization of the Hamiltonian is helpful in determining the spectrum.Comment: LaTeX file, 15 pages. (further references added, minor changes in section 5