The Universal Vassiliev Invariant for the Lie Superalgebra

We compute the universal weight system for Vassiliev coming from the Lie superalgebra applying the construction of [13]. This weight system is a function from the space of chord diagrams to the center Z of the universal enveloping algebra of , and we find a combinatorial expression for it in terms of the standard generators of Z. The resulting knot invariants generalize the Alexander-Conway polynomial. Received: 4 July 1996 / Accepted: 21 August 1996     

Equivalence of Blocks for the General Linear Lie Superalgebra

We develop a reduction procedure which provides an equivalence (as highest weight categories) from an arbitrary block (defined in terms of the central character and the integral Weyl group) of the BGG category ${\mathcal{O}}$ for a general linear Lie superalgebra to an integral block of ${\mathcal{O}}$ for (possibly a direct sum of) general linear Lie superalgebras. We also establish indecomposability of blocks of ${\mathcal{O}}$ .     

Second Variational Formulae for Dimension Spectra

Consider basic set for Axiom A diffeomorphism on compact surface.We derive second variational formulae for the dimension spectra of equilibrium state on the basic set with respect to the perturbations of both the potential and\break the dynamical system.In particular we obtain a second variational formula for the Hausdorff dimension of the basic set.These results will find their use in the study of a quadratic extremal problem for multifrcatal analysis.     

Lie Superalgebra Valued Self-Dual Yang-Mills Fields and Symmetry Reduction

Abstract

Self-dual Yang-Mills fields with values in a Lie superalgebra on the four-dimensional Euclidean space and pseudo-Euclidean space of signature (2,2) can be reduced by subgroups of the corresponding conformal group to integrable systems with anticommuting degrees of freedom. Examples of reductions are presented.     

Extensions of the Neveu–Schwarz Lie Superalgebra

We consider the generalizations of the Virasoro Lie algebra constructed recently in [16] and we classify their superanalogues. The main result of this article is a series of Lie Superalgebras generalizing the well-known Neveu–Schwarz Superalgebra. Received: 26 May 1998 / Accepted: 8 April 1999     

Finite-Dimensional Half-Integer Weight Modules over Queer Lie Superalgebras

We give a new interpretation of representation theory of the finite-dimensional half-integer weight modules over the queer Lie superalgebra \({\mathfrak{q}(n)}\). It is given in terms of the Brundan’s work on finite-dimensional integer weight \({\mathfrak{q}(n)}\)-modules by means of Lusztig’s canonical basis. Using this viewpoint we compute the characters of the finite-dimensional half-integer weight irreducible modules. For a large class of irreducible modules whose highest weights are of special types (i.e., totally connected or totally disconnected) we derive closed-form character formulas that are reminiscent of the Kac–Wakimoto character formula for basic Lie superalgebras.     

Deforming the Lie Superalgebra of Contact Vector Fields on S1|1 Inside the Lie Superalgebra of Superpseudodifferential Operators on S1|1

Abstract

We classify nontrivial deformations of the standard embedding of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional supercircle into the Lie superalgebra of superpseudodifferential operators on the supercircle. This approach leads to the deformations of the central charge induced on K(1) by the canonical central extension of SΨDO.     

Representation Theory of the Affine Lie Superalgebra at Fractional Level

N= 2 noncritical strings are closely related to the Wess-Zumino-Novikov-Witten model, and there is much hope to further probe the former by using the algebraic apparatus provided by the latter. An important ingredient is the precise knowledge of the representation theory at fractional level. In this paper, the embedding diagrams of singular vectors appearing in Verma modules for fractional values of the level (, p and q coprime) are derived analytically. The nilpotency of the fermionic generators in $\hslc$ requires the introduction of a nontrivial generalisation of the MFF construction to relate singular vectors among themselves. The diagrams reveal a striking similarity with the degenerate representations of the N= 2 superconformal algebra. Received: 10 June 1996\,/\,Accepted: 8 October 1996     

An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

Abstract

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1), however in the latter, Bose operators have to be replaced with Fermi operators.     

Classification of Finite Irreducible Modules over the Lie Conformal Superalgebra CK 6

We classify all continuous degenerate irreducible modules over the exceptional linearly compact Lie superalgebra E(1, 6), and all finite degenerate irreducible modules over the exceptional Lie conformal superalgebra CK ₆, for which E(1, 6) is the annihilation superalgebra.     

Quantum Group Structure of Lie Superalgebra osp (1/2)

We quantize the enveloping Lie superalgebraU(osp(1/2)) in the nonstandard simple root system withtwo odd simple roots. The quantum supergroup structureassociated with the nonstandard simple root system is also given.     

Quantum Deformation of the Lie Superalgebra spl(2, 1)

In this paper we obtain a q-deformation of theLie superalgebra spl(2, 1), which is in correspondencewith its one-parameter quantum group. This proceduresuggests a two-parameter deformation of the Lie superalgebra which leads to the two-parameterquantum group.     

Coherent State Representations of Lie Superalgebra SU(2/1)

We present a kind of new coherent states associated with the Lie superalgebra SU(2/1), and discuss their properties in detail. We also evaluate the matrix elements of the SU(2/1) generators in the coherent state representations and obtain differential realizations of the SU(2/1) algebra in the coherent state space. The differential realizations may be useful for the study of the quasi-exactly solvable problems in the quantum mechanics.     

Highest Weight Modules Over Quantum Queer Superalgebra {U_q(\mathfrak {q}(n))}

In this paper, we investigate the structure of highest weight modules over the quantum queer superalgebra U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}. The key ingredients are the triangular decomposition of U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))} and the classification of finite dimensional irreducible modules over quantum Clifford superalgebras. The main results we prove are the classical limit theorem and the complete reducibility theorem for U_q(\mathfrak q(n)){U_q(\mathfrak {q}(n))}-modules in the category O_q ^{3 0}{\mathcal {O}_{q}^{\geq 0}}.     

Oscillatory Integral Formulae for Left-invariant Star Products on a Class of Lie Groups

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)     

Irreducible Representations of Quantum Lie Superalgebra SLq(2/1)

In this paper,some generators of SL_q(2/1),satisfying the Serre-like relations,can be considered as 1/2 rank tensor-like operators.The reduced matrix elements of them can be calculated by a set of recurrent formula.Thus th irreducible representations of SL_q(2/1) may be completely determined.