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     

Skew-symmetric identities of octonions

Quadratic alternative superalgebras are introduced and their super-identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classified.     

Super-biderivations of classical simple Lie superalgebras

In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field \(\mathbb {C}\). Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras.     

Maximal subalgebras of Jordan superalgebras

The maximal subalgebras of the finite-dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by the same authors about maximal subalgebras of simple associative superalgebras, which is instrumental here.     

On herstein's constructions relating jordan and associative superalgebras

We investigate the Jordan structure of a prime associative superalgebra and the Jordan structure of the symmetric elements of a *-prime associative superalgebra with superinvolution.     

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 .     

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.     

A Lie grading which is not a semigroup grading

This note is devoted to the construction of a graded Lie algebra, whose grading is not given by a semigroup; thus providing a counterexample to an assertion by Patera and Zassenhaus.     

Representations of Lie superalgebras in prime characteristic II: The queer series

The modular representation theory of the queer Lie superalgebra q(n) over characteristic p>2 is developed. We obtain a criterion for the irreducibility of baby Verma modules with semisimple p-characters χ and a criterion for the semisimplicity of the corresponding reduced enveloping algebras U_χ(q(n)). A (2p)-power divisibility of dimensions of q(n)-modules with nilpotent p-characters is established. The representation theory of q(2) is treated in detail. We formulate a Morita super-equivalence conjecture for q(n) with general p-characters which is verified for n=2.     

Simple finite-dimensional modular noncommutative Jordan superalgebras

We classify the central simple finite-dimensional noncommutative Jordan superalgebras over an algebraically closed field of characteristic $p>2$. The case of characteristic 0 was considered by the authors in the previous paper [21]. In particular, we describe Leibniz brackets on all finite dimensional central simple Jordan superalgebras except mixed (nor vector neither Poisson) Kantor doubles of the supercommutative superalgebra $B(m,n)$.     

P.I. envelopes of classical simple Lie superalgebras

Let be a classical simple Lie superalgebra. We describe the prime ideals in the enveloping algebra such that satisfies a polynomial identity. If the factor algebra is not artinian, then it is an order in a matrix algebra over .     

A Kronecker factorization theorem for the exceptional Jordan superalgebra

We prove that a Jordan superalgebra J containing the 10-dimensional exceptional Kac superalgebra K₁₀ is isomorphic to (K₁₀⊗_FS)⊕J′, where S is an associative commutative algebra.     

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.     

Quantum Schur superalgebras and Kazhdan-Lusztig combinatorics

We introduce the notion of quantum Schur (or q-Schur) superalgebras. These algebras share certain nice properties with q-Schur algebras such as the base change property, the existence of canonical Z[v,v⁻¹]-bases, the duality relation with Manin’s quantum matrix superalgebra A(m|n), and the bridging role between quantum enveloping superalgebras of gl(m|n) and the Hecke algebras of type A. We also construct a cellular -basis and determine its associated cells, called supercells, in terms of a Robinson-Schensted-Knuth supercorrespondence. In this way, we classify all irreducible representations over via supercell modules.