Simple Vectorial Lie Algebras in Characteristic 2 and their Superizations

Abstract

We overview the classifications of simple finite-dimensional modular Lie algebras. In characteristic 2, their list is wider than that in other characteristics; e.g., it contains desuperizations of modular analogs of complex simple vectorial Lie superalgebras. We consider odd parameters of deformations. For all 15 Weisfeiler gradings of the 5 exceptional families, and one Weisfeiler grading for each of 2 serial simple complex Lie superalgebras (with 2 exceptional subseries), we describe their characteristic-2 analogs-new simple Lie algebras. Descriptions of several of these analogs, and of their desuperizations, are far from obvious. One of the exceptional simple vectorial Lie algebras is a previously unknown deform (the result of a deformation) of the characteristic-2 version of the Lie algebra of divergencefree vector fields; this is a new simple Lie algebra with no analogs in characteristics distinct from 2. In characteristic 2, every simple Lie superalgebra can be obtained from a simple Lie algebra by one of the two methods described in arXiv:1407.1695. Most of the simple Lie superalgebras thus obtained from simple Lie algebras we describe here are new.

Hom-Lie Superalgebras in Characteristic 2

“…In this section, we define a cohomology theory of Lie superalgebras in characteristic 2. The first instances can be found in [26]. Let g be a Lie superalgebra in characteristic 2 and M be a g-module.…”

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

“…For the most comprehensive background on specifics of Lie (super)algebras for p = 2, see [10]. For classification of isomorphism classes of orthogonal Lie (super)algebras without Cartan matrix, and their periplectic analogs preserving non-degenerate odd symmetric bilinear form for any p, and its numerous versions for p = 2, see [46] and Section 7 below.…”

“…Interestingly, the well-known example of "quantization" of the Poisson Lie algebra was not taken into account, although it manifestly shows that the bold idea to replace computation of H 2 (g; g) by that of H 2 (g − ; g) does not always work. For one more, very interesting exceptional example of Shchepochkina, see [10].…”

Support varieties for Lie superalgebras in characteristic 2

“…Furthermore, there are inverse bijections 9 Again, the results in [3] are stated in the language of finite group schemes, so the reader must translate to the category of rational G-modules, where k[G] = V (g) * . 10 Actually, this property holds without the assumption that M is compact, but we will find it convenient to use this more restricted formulation of the Tensor Product Property. 11 As used in algebraic geometry, the term quasi-compact means that every open cover has a finite subcover.…”

“…Indeed, it is sometimes assumed that super-mathematics over fields of characteristic 2 is uninteresting. Nevertheless, there is an extensive literature on the study of Lie superalgebras over fields of characteristic 2; see, for example, [10] and the references therein. The current paper hopefully adds additional evidence against that assumption.…”