A family of simple Lie algebras in characteristic two

Abstract

In this paper a new two-parameter family of simple Lie algebras defined over fields of characteristic two is described. They are used to construct loop algebras which are central to the classification of graded Lie algebras with maximal class and characteristic two.  2004 Elsevier Inc. All rights reserved.

(Finite) presentations of Bi-Zassenhaus loop algebras

“…The author is grateful for his help to A. Caranti, advisor of the doctoral dissertation [Jur98] this work is based on. He is also grateful to M.F.…”

Graded Lie algebras of maximal class, III

“…The author is grateful for help to A. Caranti, advisor of the doctoral dissertation [9] this work is based on. The author is also grateful to M.F.…”

“…These new algebras can be obtained, like the AFS ones, via a loop algebra construction from some finite-dimensional simple Lie algebras, called bi-Zassenhaus algebras, by means of a non-singular outer derivation. We will denote respectively by B and B l the simple and the loop structure: their construction and properties are described in [11]. C. Carrara proved in [6,7] that AFS-algebras can be characterized by a suitable finite quotient: this result was essential to prove the classification theorem for the odd case; the same property holds for B l -algebras as well (see also [10]).…”

“…We call them the bi-Zassenhaus loop algebras B l (g, h). They depend on two integer parameters g 2 and h 1, and they are described in [11]. They have only two distinct two-step centralizers and, in the notation described at the end of Section 2 in II, their sequence of constituent lengths reads as…”

Towards classification of simple finite dimensional modular Lie superalgebras

“…closed fields K of characteristic p > 3, and also gave an overview of the "mysterious" examples (due to Brown, Frank, Ermolaev and Skryabin) of simple finite dimensional Lie algebras for p = 3 with no counterparts for p > 3. Several researchers started afresh to work on the cases where p = 2 and 3, and new examples of simple Lie algebras with no counterparts for p = 2, 3 started to appear ( [J,GL4,Leb1], observe that the examples of [GG,Lin1] are erroneous as observed in MathRevies and [Leb2], respectively). The "mysterious" examples of simple Lie algebras for p = 3 were interpreted as vectorial Lie algebras preserving certain distributions ([GL4]).…”