A Note on Restricted Representations of the Witt Superalgebras

Let F be an algebraically closed field of prime characteristic p 〉 3, and W（n） the Witt superalgebra over F, which is the Lie superalgebra of superderivations of the Grassmann algebra in n indeterminates. The dimensions of simple atypical modules in the restricted supermodule category for W（n） are precisely calculated in this paper, and thereby the dimensions of all simple modules can be precisely given. Moreover, the restricted supermodule category for W（n） is proved to have one block.     

Axiomatic Aspects of N = 2 Vertex Superalgebras with Odd Formal Variables

We formulate the notion of “N = 2 vertex superalgebra with two odd formal variables” using a Jacobi identity with odd formal variables in which an N = 2 superconformal shift is incorporated into the usual Jacobi identity for a vertex superalgebra. It is shown that as a consequence of these axioms, the N = 2 vertex superalgebra is naturally a representation of the Lie superalgebra isomorphic to the three-dimensional algebra of superderivations with basis consisting of the usual conformal operator and the two N = 2 superconformal operators. In addition, this superconformal shift in the Jacobi identity dictates the form of the odd formal variable components of the vertex operators, and allows one to easily derive the useful formulas in the theory. The notion of N = 2 Neveu–Schwarz vertex operator superalgebra with two odd formal variables is introduced, and consequences of this notion are derived. In particular, we develop the duality properties which are necessary for a rigorous treatment of the correspondence with the underlying supergeometry. Various other formulations of the notion of N = 2 (Neveu–Schwarz) vertex (operator) superalgebra appearing in the mathematics and physics literature are discussed, and several mistakes in the literature are noted and corrected.     

Representations of the Restricted Cartan Type Lie Superalgebra <Emphasis Type="Italic">W (m,n, 1)</Emphasis>

Simple modules for the restricted Witt superalgebra W(m,n,1) are considered. Conditions are provided for the restricted and nonrestricted Kac modules to be simple.     

Simple Special Jordan Superalgebras with Associative Even Part

We describe the simple special unital Jordan superalgebras with associative even part A whose odd part M is an associative module over A. We prove that each of these superalgebras, not isomorphic to a superalgebra of nondegenerate bilinear superform, is isomorphically embedded into a twisted Jordan superalgebra of vector type. We exhibit a new example of a simple special Jordan superalgebra. We also describe the superalgebras such that M [A,M] 0.     

Complete hom-Lie superalgebras

Abstract

In this article, we investigate some basic results on complete hom-Lie superalgebras. In particular, we present some sufficient conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra. Making use of these conditions, we build a family of complete hom-Lie superalgebras.     

Representations of Toroidal General Linear Superalgebra

We consider general linear superalgebra (type A) and tensor with Laurent polynomial ring in several variables. We then consider the universal central extension of this Lie superalgebra which we call toroidal superalgebra. We give a faithful representation of toroidal superalgebra using vertex operators and bosons.     

Actions of Lie Superalgebras on Reduced Rings

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is Theorem 13 If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then . We can then use Theorem 13 to prove Corollary 15 Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R ^H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN, where N is the dimension of H. Presented by Donald S. Passman.     

The eigenvalues of four-dimensional rotations

Let X be an orthosymplectie Lie superalgebra of type B or D. The weight structure of the tensor product moduleW=? ^M V, of M-copies of the natural representation. V, of X is studied from a partition point of view. A combinatorial characterization of the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules is given. This partition point of view allows us to prove that the dominant weights of W and the weights of W which are highest weights for the finite dimensional irreducible modules stabilize as the rank of X gets large.     

A class of generalized odd Hamiltonian Lie superalgebras

We study class of finite-dimensional Cantan-type Lie superalgebras HO（m） over a field of prime characteristic, which can be regarded as extensions of odd Hamiltonian superalgebra HO. And we also determine the derivation superalgebras of Lie superalgebras HO（m）.     

Quadratic Lie Superalgebras Generalized by Balinsky–Novikov Superalgebras

Balinsky–Novikov superalgebras were introduced by Balinsky for constructing super-Virasoro type Lie superalgebras. In this paper, we give sufficient and necessary conditions for a Lie superalgebra generalized by a Balinsky–Novikov superalgebra with dimension 2|2 to be a quadratic Lie superalgebra.     

The Automorphism Group Functor of the N = 4 Lie Conformal Superalgebra

In this article, we study the structure and representability of the automorphism group functor of the N = 4 Lie conformal superalgebra over an algebraically closed field k of characteristic zero.     

Annihilation Theorem and Separation Theorem for basic classical Lie superalgebras

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

     

The Maximal Graded Left Quotient Algebra of a Graded Algebra1)

We construct the maximal graded left quotient algebra of every graded algebra A without homogeneous total right zero divisors as the direct limit of graded homomorphisms （of left A-modules） from graded dense left ideals of A into a graded left quotient algebra of A. In the case of a superalgebra, and with some extra hypothesis, we prove that the component in the neutral element of the group of the maximal graded left quotient algebra coincides with the maximal left quotient algebra of the component in the neutral element of the group of the superalgebra.     

Quantizations of the W-Algebra W（2, 2）

We quantize the W-algebra W(2, 2), whose Verma modules, Harish-Chandra modules, irreducible weight modules and Lie bialgebra structures have been investigated and determined in a series of papers recently.     

Differential algebras and simple Jordan superalgebras

In [14], a new example is constructed of a unital simple special Jordan superalgebra J over the field of reals. It turns out that J is a subsuperalgebra of a Jordan superalgebra of vector type but it cannot be isomorphic to a superalgebra of such a type. Moreover, the superalgebra of fractions of J is isomorphic to a Jordan superalgebra of vector type. In the present article, we find a similar example of a Jordan superalgebra. It is constructed over a field of characteristic 0 in which the equation t ² + 1 = 0 has no solutions.     

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type Ⅰ.     

Hochschild cohomologies and deformations of the pointwise superproduct

We consider the associative superalgebra of smooth compactly supported functions on ℝ ⁿ taking values in a Grassmann algebra. We find the lower Hochschild cohomologies and also the deformations of this superalgebra under certain additional conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 323–346, March, 2009.