Lie algebras with quadratic dimension equal to 2

The quadratic dimension of a Lie algebra is defined as the dimension of the linear space spanned by all its invariant non-degenerate symmetric bilinear forms. We prove that a quadratic Lie algebra with quadratic dimension equal to 2 is a local Lie algebra, this is to say, it admits a unique maximal ideal. We describe local quadratic Lie algebras using the notion of double extension and characterize those with quadratic dimension equal to 2 by the study of the centroid of such Lie algebras. We also give some necessary or sufficient conditions for a Lie algebra to have quadratic dimension equal to 2. Examples of local Lie algebras with quadratic dimension larger than 2 are given.     

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.     

Malcev superalgebras with trival nucleus

The nucleus of a Malcev superalgebra M measures how far it is from being a Lie superalgebraM being a Lie superalgebra if and only if its nucleus is the whole M. This paper is devoted to study Malcev superalgebras in the opposite direction, that is, with trivial nucleus. The odd part of any finite-dimensional Malcev superalgebra with trivial nucleus is shown to be contained in the solvable radical. For algebraically closed fields, any such superalgebra splits as the sum of its solvable radical and a semisimple Malcev algebra contained in the even part, which is a direct sum of copies of sl(2, F) and the seven-dimensional simple non-Lie Malcev algebra, obtained from the Cayley-Dickson algebra.     

Infinite-dimensional modular special odd contact superalgebras

The infinite-dimensional special odd contact Lie superalgebras SKO(n,n+1;c) over a field of positive characteristic are studied. In particular, we prove that Lie superalgebras SKO(n,n+1;c) are simple. Besides, the Weisfeiler filtration of SKO(n,n+1;c) is proved to be invariant under automorphisms by characterizing -nilpotent elements. Thereby, we obtain some properties of these Lie superalgebras.     

Whittaker Categories and Whittaker Modules for Lie Superalgebras

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, the construction of Whittaker modules from Whittaker modules for the even part.     

局限李超代数的局限表示

本文对如下结果给出一个构造性、初等的、简洁的证明:有限维局限李超代数的有限维局限表示均完全可约当且仅当它的奇部为0而偶部是一个环面.这使得李代数中相应结果成为特例.     

The extended Freudenthal Magic Square and Jordan algebras

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras. I. Cunha was supported by CMUC, Department of Mathematics, University of Coimbra. A. Elduque was supported by the Spanish Ministerio de Educación y Ciencia and FEDER (MTM 2004-081159-C04-02) and by the Diputación General de Aragón (Grupo de Investigación de álgebra).     

Shapovalov determinant for loop superalgebras

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 3, pp. 378–397, September, 2008.     

On restricted simple modules of contact Lie superalgebras of odd type

For contact Lie superalgebras of odd type over an algebraically closed field of characteristic , the composition factors of Kac modules and the character formulas for all the restricted simple modules are explicitly determined. And the dimensions of simple modules in the restricted supermodule category are precisely calculated, thereby dimensions of all the restricted simple modules are given.     

Lie superalgebras graded by the root system B(<Emphasis Type="Italic">m,n</Emphasis>)

We determine the Lie superalgebras that are graded by the root system B(m,n) of the orthosymplectic Lie superalgebra osp(2m + 1,2n). Mathematics Subject Classification (2000) Primary 17B70, Secondary 17A70     

Computation by computer of Lie superalgebra homology and cohomology

In this paper, we introduce a package to compute homology and cohomology spaces of Lie superalgebras. We describe most of its features and the implementation in REDUCE.     

关于完备限制李超代数

我们分别给出了完备限制李超代数和强完备限制李超代数的一个充分必要条件,同时得到了一些完备限制李超代数的重要性质.     

On the lie structure of the skew elements of a prime superalgebra with superinvolution

We investigate the Lie structure of the Lie superalgebra K of skew elements of a prime associative superalgebra A with superinvolution. It is proved that if A is not a central order in a Clifford superalgebra of dimension at most 16 over the center then any Lie ideal of K or [K,K] contains[J ∩K,K] for some nonzero ideal J of A or is contained in the even part of the center of A.     

Engel’s theorem for malcev superalgebras

A version of Engel’s theorem for Malcev superalgebras is proved in the spirit of theJacobson-Engel theorem for Lie algebras. Some consequences for the structure of Malcev superalgebras with trivial Lie nucleus are derived.     

Infinite Dimensional Lie and Jordan Algebras

The aim of this work is to provide a survey of some of the main structural results about graded algebras, in both, Lie and Jordan cases and relate them with some results about infinite dimensional superalgebras. Partially supported by MTM 2004 08115-C04-01 and FICYT IB05-186.     

Invariant forms of the elementary unitary Lie superalgebra

In this paper we describe the invariant forms of toral K-graded Lie superalgebras and, in particular, of the elementary unitary Lie superalgebra over a superring K containing .     

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.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.