Nilpotent Lie superalgebras
by
Marc Gilg
Universite de Haute-Alsace, Mulhouse, France

Nilpotent Lie superalgebras For a Lie superalgebra G over an algebraic closed field of carateristic 0 we define the lower central series C⁰(G)=G, Cⁱ⁺¹(G)=[G, Cⁱ(G)]. A Lie superalgebra G is nilpotent if there exist an integer n such that Cⁿ(G)={0}.

This definition is not easy to use. We define for a Lie superalgebra G=G₀\oplusG₁ two sequences : C⁰(G₀)=G₀, Cⁱ⁺¹(G₀)=[G₀, Cⁱ(G₀)] and C⁰(G₁)=G₁, Cⁱ⁺¹(G₁)=[G₀, Cⁱ(G₁)] It is easy to see that if G is nilpotent, there exist (p, q) such that : C^p(G₀)={0} and C^q(G₁)={0}.

Theorem Let G=G₀\oplusG₁ be a Lie superalgebras. Then G is nilpotent if and only if there exist (p, q) such that C^p(G₀)={0} and C^q(G₁)={0}.

Definition Let \G be a nilpotent Lie superalgebra, the super-nilindex of \G is the pair (p, q) such that : C^p(G₀)={0}, C^p-1(G₀) =/= {0} and C^q(G₁)={0}, C^q-1(G₁) =/= {0}. It is and invariant up to ismoporphism.

Filiform Lie superalgebras Let G=G₀\oplusG₁ be a nilpotent Lie superalgebra with dimG₀=n+1 and dimG₁=m. \G is called filiform if it's super-nilindex is (n, m).

The set of filiform Lie superalgebras is an open set of the variety of nilpotent Lie superalgebras' laws.

Classifications of filiforms over C in lower dimensions Using the bases of the cocycles and adapted chages of bases, like it was done for Lie algebras we have classifications of filiform Lie superalgebras.

The next table give use for the dimensions of \G₀ and \G₁ the number of nonisomorphic filiform Lie algebras and Lie superalgebras. The descritions of this superalgebras can be found in [1].

References

[1] M. Gilg Super-algèbres de Lie nilpotentes
Thèse, Université de Haute-Alsace, 2000

Date received: May 31, 2000