Superderivations for Modular Graded Lie Superalgebras of Cartan-type

Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic $p>3$ are completely determined by a uniform approach: The infinite dimensional case is reduced to the finite dimensional case and the latter is further reduced to the restrictedness case, which proves to be far more manageable. In particular, the outer superderivation algebras of those Lie superalgebras are completely determined

Restricted infinitesimal deformations of restricted simple Lie algebras

We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic different from 2 and 3.Comment: 15 pages; final version, to appear in Journal of Algebra and Its Application

Laguerre polynomials of derivations

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation exp(x) · exp(y) = exp(x+y) for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case

Filtered multiplicative bases of restricted enveloping algebras

We study the problem of the existence of filtered multiplicative bases of a restricted enveloping algebra u(L), where L is a finite-dimensional and p-nilpotent restricted Lie algebra over a field of positive characteristic p

Geometric representation theory of restricted lie algebras

We modify the Hochschild -map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a goup scheme that leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotentp-character