Super Korteweg-de Vries equations associated to super extensions of the Virasoro algebra

All Lie superalgebras whose even part is the Virasoro algebra are found. There are three types of such superalgebras: the standard Neveu-Schwarz-Ramond superalgebra, the Ramond-Schwarz superalgebra, and a new series with continuous parameters. To each such superalgebra is attached an infinite hierarchy of integrable bi-hamiltonian superextensions of the Korteweg-de Vries hierarchy.     

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.     

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.     

Gelfand–Tsetlin Pattern and Strict Partitions

We give a representation-theoretical interpretation of the Gelfand–Tsetlin pattern for strict partitions. Using the Howe duality involving a pair of the queer Lie superalgebras and an analogue of the Littlewood–Richardson rule for Schur Q-functions, we show that such patterns give the branching rule for the irreducible tensor representations of the queer Lie superalgebra.     

Contraction of graded su(2) algebra

The Inönu-Wigner contraction scheme is extended to Lie superalgebras. The structure and representations of extended BRS algebra are obtained from contraction of the graded su(2) algebra. From cohomological consideration, we demonstrate that the graded su(2) algebra is the only superalgebra which, on contraction, yields the full BRS algebra.     

sl(1|2) Super-Toda Fields

Explicit exact solution of supersymmetric Toda fields associated with the Lie superalgebra sl(2|1) is constructed. The approach used is a super extension of Leznov-Saveliev algebraic analysis, which is based on a pair of chiral and antichiral Drienfeld-Sokolov systems. Though such approach is well understood for Toda field theories associated with ordinary Lie algebras, its super analogue was only successful in the super Liouville case with the underlying Lie superalgebra osp(1|2). The problem lies in that a key step in the construction makes use of the tensor product decomposition of the highest weight representations of the underlying Lie superalgebra, which is not clear until recently. So our construction made in this paper presents a first explicit example of Leznov-Saveliev analysis for super Toda systems associated with underlying Lie superalgebras of the rank higher than 1.     

Integrable Highest Weight Modules over Affine Superalgebras and Appell's Function

We classify integrable irreducible highest weight representations of non-twisted affine Lie superalgebras. We give a free field construction in the level 1 case. The analysis of this construction shows, in particular, that in the simplest case of the sℓ (2|1) level 1 affine superalgebra the characters are expressed in terms of the Appell elliptic function. Our results demonstrate that the representation theory of affine Lie superalgebras is quite different from that of affine Lie algebras. Received: 17 April 2000 / Accepted: 7 July 2000     

Conformal Lie superalgebras and moduli spaces

A conformal Lie superalgebra is a superextension of the centerless Virasoro algebra W—the Lie algebra of complex vector fields on the circle. The algebras of Ramond and Neveu-Schwarz are not the only examples of such superalgebras. All known superconformal algebras can be obtained as comlexifications of Lie superalgebras of vector fields on a supercircle with an additional structure. For every such superalgebra

a class of geometric objects—complex

— is defined. For the superalgebras of Neveu-Schwarz and Ramond they are super Riemann surfaces with punctures of different kinds. We construct moduli superspaces for compact

, and show that the superalgebra

acts infinitesimally on the corresponding moduli space.     

The algebraic structure of BRST quantization

It is shown that a system of first-class bosonic constraints obeying a Lie algebra has associated with it a natural superalgebra. BRST quantization arises as a non-linear representation of this superalgebra. Two distinct superalgebras are explicitly constructed and their associated BRST quantizations presented. The first BRST quantization is the canonical one with the BRST charge a grassmannian scalar. The second is new — the BRST charge is a grassmannian spinor transforming in the fundamental representation of the appropriate superalgebra. Generalizations are briefly discussed.     

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.     

Cyclic Representations of Quantum (Super) Algebras At qp=1

Cyclic representations of quantum (super) algebras are studied at q^p=1 using two methods:the quotient module method and the q-boson realization method.For the quantum algebras associated with any finite dimensional simple Lie algebra the general theory of two methods is given,and is generated to the quantum superalgebra U_qosp(1.2).By constructing the cyclic representation of q-Heisenberg-Wey1 superalgebras the q-boson realization method is generated to construction of cyclic representations of some high-rank quantum superalgebras.     

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.     

Classification of Nilpotent Lie Superalgebras of Dimension Five. II

In this paper a classification is made for allnilpotent Lie superalgebras (graded Lie algebras) ofmaximum dimension five. The superversion of the Kirillovlemma for nilpotent Lie superalgebra is given with its application to thisclassification.     

Superstrings from Hamiltonian reduction

In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with N − 2 supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras osp(N|2). The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted N = 4 superconformal algebra.     

Canonical Basis for Quantum {\mathfrak{osp}(1|2)}

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra ${\mathfrak{osp}(1|2)}$ . Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for ${\mathfrak{sl}(2)}$ . This provides a first example of canonical basis for quantum superalgebras.     

Expansions of Algebras and Superalgebras and Some Applications

After reviewing the three well-known methods to obtain Lie algebras and superalgebras from given ones, namely, contractions, deformations and extensions, we describe a fourth method recently introduced, the expansion of Lie (super)algebras. Expanded (super)algebras have, in general, larger dimensions than the original algebra, but also include the ?nönü–Wigner and generalized IW contractions as a particular case. As an example of a physical application of expansions, we discuss the relation between the possible underlying gauge symmetry of eleven-dimensional supergravity and the superalgebra osp(1|32).     

W-algebras and superalgebras from constrained WZW models: A group theoretical classification

We present a classification ofW algebras and superalgebras arising in Abelian as well as non Abelian Toda theories. Each model, obtained from a constrained WZW action, is related with anSl(2) subalgebra (resp.OSp(1/2) superalgebra) of a simple Lie algebra (resp. superalgebra)G. However, the determination of anU(1) _Y factor, commuting withSl(2) (resp.OSp(1/2)), appears, when it exists, particularly useful to characterize the correspondingW algebra. The (super) conformal spin contents of eachW (super) algebra is performed. The class of all the superconformal algebras (i.e. with conformal spinss<=2) is easily obtained as a byproduct of our general results.