Some cohomologically rigid solvable Leibniz algebras

Abstract

In this paper we describe solvable Leibniz algebras whose quotient algebra by one-dimensional ideal is a Lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical. We prove that such Leibniz algebra is unique and centerless. Also it is proved that the first and the second cohomology groups of the algebra with coefficients in itself is trivial.

Abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras

“…Hence these last cases do not occur. Since the grading (9) has three homogeneous components while the remaining non-trivial gradings have only two homogeneous components we have that the grading ( 9)is not equivalent to any grading in ( 6)- (8). The grading (6) is not equivalent to grading (8).…”

“…Since in (8) any grading has e 1 and e 2 in the same homogeneous component, we also have that ( 9) is not equivalent to (8).…”

“…Since the grading (9) has three homogeneous components while the remaining non-trivial gradings have only two homogeneous components we have that the grading ( 9)is not equivalent to any grading in ( 6)- (8). The grading (6) is not equivalent to grading (8). Indeed, in the opposite case we would have an automorphism φ of A satisfying φ( e 2 ) = e 3 and φ( e 1 , e 3 ) = e 2 , e 3 .…”

“…Proposition 18. Any abelian group grading of a one-parametric filiform Leibniz algebra A (with a nonzero parameter) of dimension 3 is equivalent to the trivial grading or to one of the gradings ( 6), (7), (8) or (9) if θ / ∈ {±1}; or to the trivial grading or to one of the gradings (6), (8) or (9) if θ ∈ {±1}.…”

“…In the past years, Leibniz algebras have been under active research (see, for example, [1,3,[8][9][10][25][26][27]). The main result on the structure of finite-dimensional Leibniz algebras asserts that a Leibniz algebra decomposes into a semidirect sum of a solvable radical and a semisimple Lie algebra [3].…”