q-Oscillators and Quantum Lie superalgebras

With the help of the q-delormed bosonic and fermionic oscillation opcrators, which can be constructed from the ordinary ones, the quantum enveloping algebras of basic Lie super-algebras (BLS) B(m, n), B(0, n), C(1+n)and D(m,n)iu Ihcir Boson-Fermion oscillator representation are written down explicitly.     

Filiform color Lie superalgebras

The aim of this work is to generalize nilpotent Lie superalgebras of a very important type, i.e. filiform Lie superalgebras, obtaining the notion of filiform color Lie superalgebras. We have proved the existence of an adapted basis for any G$G$-grading of the filiform color superalgebras. Also we have proved that in order to obtain the whole class of filiform G$G$-color Lie superalgebras it is only necessary to determine the infinitesimal deformations of the associated model color superalgebra.     

Some remarks on Lie superalgebras

We propose an approach to the theory of Lie superalgebras based on what we call a Lie algebra square root. Every Lie algebra square root has a Lie algebra as its square, but many different Lie algebra square roots may have the same square.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

Orthosymplectic representations of Lie superalgebras

We investigate when an irreducible finite-dimensional representation of a Lie superalgebra is orthosymplectic. Then we turn to basic classical Lie superalgebras and give the conditions for orthosymplecticity in terms of Kac-Dynkin labels.     

Integrable deformations of nilpotent color Lie superalgebras

In this paper we study the infinitesimal deformations of the _Z3${\mathbb{Z}}_3$-color Lie superalgebra L^n,m,p$L^{n,m,p}$. By means of these deformations all filiform _Z3${\mathbb{Z}}_3$-color Lie superalgebras can be obtained. In particular, we give a method that will allow us to determine the dimension of the subspaces that are composed by linearly integrable deformations.     

Unitary representations of basic classical Lie superalgebras

We have obtained all the finite-dimensional unitary irreps of gl(m|n) and C(n), which also exhaust such irreps of all the basic classical Lie superalgebras. The lowest weights of such irreps are worked out explicitly. It is also shown that the contravariant and covariant tensor irreps of gl(m|n) are unitary irreps of type (1) and type (2) respectively, explaining the applicability of the Young diagram method to these two types of tensor irreps.     

Serre-type relations for special linear Lie superalgebras

It is pointed out that, for m, n 2, the naive Serre presentation corresponding to the simplest Cartan matrix of sl(m, n) does not define the Lie superalgebra sl(m, n) but a larger algebra s(m, n) of which sl(m, n) is a nontrivial quotient. The supplementary relations for the generators are found and the definition of the q-deformed universal enveloping algebra of sl(m, n) is modified accordingly.     

Conformal Einstein spaces

We study conformal transformations in four-dimensional manifolds. In particular, we present a new set of two necessary and sufficient conditions for a space to be conformal to an Einstein space. The first condition defines the class of spaces conformal to C spaces, whereas the last one (the vanishing of the Bach tensor) gives the particular subclass ofC spaces which are conformally related to Einstein spaces.This work has been partly supported bym a grand from the National Science Foundation.     

Bundle gerbes and moduli spaces

In this paper, we construct the index bundle gerbe of a family of self-adjoint Dirac-type operators, refining a construction of Segal. In a special case, we construct a geometric bundle gerbe called the caloron bundle gerbe, which comes with a natural connection and curving, and show that it is isomorphic to the analytically constructed index bundle gerbe. We apply these constructions to certain moduli spaces associated to compact Riemann surfaces, constructing on these moduli spaces, natural bundle gerbes with connection and curving, whose 3-curvature represent Dixmier-Douady classes that are generators of the third de Rham cohomology groups of these moduli spaces.     

Extensions of Lie superalgebras and supersymmetric abelian gauge fields

We construct an extension of Lie superalgebras which allows an algebraic descriprion of the supersymmetric Hopf fibration of the supersymmetric two-sphere. Moreover, we construct a supersymmetic generalization of electromagnetism.     

Structure of basic Lie superalgebras and of their affine extensions

We generalize to the case of superalgebras several properties of simple Lie algebras involving the use of Dynkin diagrams. If to a simple Lie algebra can be associated one Dynkin diagram, it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra (or B.S.A.). The knowledge of these diagrams, which can be obtained for each B.S.A. in a systematic way, allows us to deduce the regular subsuperalgebras of a B.S.A. The symmetries of the Dynkin diagrams are related to outer automorphisms of B.S.A. and lead to some singular subsuperalgebras. Finally we consider the extended Dynkin diagrams in order to classify the affine B.S.A. and use their symmetries to construct the twisted basic superalgebras.     

Lie Conformal Algebra Cohomology and the Variational Complex

We find an interpretation of the complex of variational calculus in terms of the Lie conformal algebra cohomology theory. This leads to a better understanding of both theories. In particular, we give an explicit construction of the Lie conformal algebra cohomology complex, and endow it with a structure of a \mathfrakg{\mathfrak{g}}-complex. On the other hand, we give an explicit construction of the complex of variational calculus in terms of skew-symmetric poly-differential operators.     

On moduli spaces of graded bundles

This is a review of the main topological properties of moduli spaces of graded bundles. These spaces consist of such elements of the cohomology H ¹(M, GL(n, E)), where E»M is a vector bundle, which are identical when restricted to ⁰ M=M × . Some explicit formulae in the case when M = P ^m and M is a torus are quoted. Applications to instantons and supergravity are discussed.     

The minimal dimensions of faithful representations for Heisenberg Lie superalgebras

This paper aims to determine the minimal dimensions and super-dimensions of faithful representations for Heisenberg Lie superalgebras over an algebraically closed field of characteristic zero.