Simple Lie superalgebras and nonintegrable distributions in characteristic p

Abstract

Recently, Grozman and Leites returned to the original Cartan's description of Lie algebras to interpret the Melikyan algebras (for p ≤ 5) and several other little-known simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description was performed in terms of Cartan-Tanaka-Shchepochkina prolongs using Shchepochkina's algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras.Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over C; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one.1991 Mathematics Subject Classification. 17B50, 70F25. Key words and phrases. Cartan prolongation, nonholonomic manifold, Melikyan algebras, Lie superalgebras. We are thankful to P. Grozman and I. Shchepochkina for help; DL is thankful to MPIMiS, Leipzig, for financial support and most creative environment.

New simple modular Lie superalgebras as generalized prolongs

“…In [13], we returned tó E. Cartan's description of Z-graded Lie algebras as subalgebras of vectorial Lie algebras preserving certain nonintegrable distributions; we thus interpreted the "mysterious" exceptional examples of simple Lie algebras for p = 3 (the Brown, Frank, Ermolaev and Skryabin algebras), further elucidated Kuznetsov's interpretation [16] of Melikyan's algebras (as prolongs of the nonpositive part of the exceptional Lie algebra g(2) in one of its Z-gradings) and discovered three new series of simple Lie algebras. In [1], the same approach yielded bj, a simple super version of g(2), and Bj(1; N |7), a simple super version of the Melikyan algebra. Both bj and Bj(1; N |7) are indigenous to the case p = 3, where g(2) K is not simple.…”

“…It is based on an idea entirely different from that of the KSh-conjecture. In it, as well as in [13] and [1], CTS-prolongs play the main role. (In the same way, Cartan obtained all simple Z-graded Lie algebras of polynomial growth and finite depth-the Lie algebras of type (1)-at the time when the root technique was not developed yet.…”

“…In [1], we have singled out Bj(1; N |7)-a simple analog of Me(5; N ) for p = 3-as a partial CTSprolong of the pair (the negative part of the contact superalgebra k(1; N |7), Bj(1; N |7) 0 = pgl(3)), and bj was obtained as a simple analog of g(2) for p = 3; its nonpositive part is the same as that of Bj(1; N |7). Thus, bj and Bj(1; N |7) were obtained in [1] as results of a super analog of construction 2.…”

“…This Lie superalgebra has three ideals I 1 ⊂ I 2 ⊂ I 3 with coinciding nonpositive parts but different positive parts: sdim(I 1 ) = 10|14, sdim(I 2 ) = 11|14, and sdim(I 3 ) = 12|14. The ideal I 1 is our bj (see [1], [2]). …”

Derivations of the Odd Contact Lie Algebras in Prime Characteristic

“…In the super-case, the theory of modular Lie superalgebras has also obtained many interesting results in the past decade. For example, one can find work on the classification of classical modular Lie superalgebras [1,2] and on the structures and representations of modular Lie superalgebras of Cartan type [6,7,9,16,17,18]. Recently, one can also find work on the representations of the classical modular Lie superalgebras (see, for example, [14,15]).…”