Leibniz algebras as non-associative algebras

Abstract

In this paper we define the basic concepts for left or right Leibniz algebras and prove some of the main results. Our proofs are often variations of the known proofs but several results seem to be new.

Spectral sequences for commutative Lie algebras

“…the cokernel of m • 1 (up to a degree shift). We then have a long exact sequence in cohomology from the short exact sequence of complexes like in Proposition 2.1 in [2].…”

“…For this filtration/spectral sequence associated to C • rel,Λ (g, M ), the arguments of Section 2 in [2] go through word by word in order to show the following theorem (this is Theorem A of Pirashvili [7]):…”

“…[x, x] = 0 for all x ∈ g). The next step is to alter the arguments in Section 2 in [2] to apply to the filtration/spectral sequence associated to C • rel,Λ,S (g, M ). We introduce a filtration as follows.…”

“…whose associated spectral sequence converges thus in the strong (i.e., finite) sense to H n rel,Λ,S (g, M ). Again like in Section 2 of [2], we have a product map…”

“…the cokernel of m 2 (up to a degree shift). We then have a long exact sequence in cohomology from the short exact sequence of complexes like in Proposition 2.1 in [2]. For this filtration/spectral sequence associated to C • rel,Λ,S (g, M ), the arguments of Section 2 in [2] still go through in order to show the following theorem:…”