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Register a new userDegenerations of Leibniz and anticommutative algebras
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We describe all degenerations of three dimensional anticommutative algebras $mathfrak{Acom}_3$ and of three dimensional Leibniz algebras $mathfrak{Leib}_3$ over $mathbb{C}.$ In particular, we describe all irreducible components and rigid algebras in the corresponding varieties
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We consider the variety of Filippov ($n$-Lie) algebra structures on an $(n+1)$-dimensional vector space. The group $GL_n(K)$ acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex $(n+1)$-dimensional Filippov $n$-ary algebras.
The present paper is devoted to the description of rigid solvable Leibniz algebras. In particular, we prove that solvable Leibniz algebras under some conditions on the nilradical are rigid and we describe four-dimensional solvable Leibniz algebras with three-dimensional rigid nilradical. We show that the Grunewald-OHallorans conjecture any $n$-dimensional nilpotent Lie algebra is a degeneration of some algebra of the same dimension holds for Lie algebras of dimensions less than six and for Leibniz algebras of dimensions less than four. The algebra of level one, which is omitted in the 1991 Gorbatsevichs paper, is indicated.
In this paper, we introduce twisted relative Rota-Baxter operators on a Leibniz algebra as a generalization of twisted Poisson structures. We define the cohomology of a twisted relative Rota-Baxter operator $K$ as the Loday-Pirashvili cohomology of a certain Leibniz algebra induced by $K$ with coefficients in a suitable representation. Then we consider formal deformations of twisted relative Rota-Baxter operators from cohomological points of view. Finally, we introduce and study NS-Leibniz algebras as the underlying structure of twisted relative Rota-Baxter operators.
Anticommutative Engel algebras of the first five degeneration levels are classified. All algebras appearing in this classification are nilpotent Malcev algebras.
We describe infinite-dimensional Leibniz algebras whose associated Lie algebra is the Witt algebra and we prove the triviality of low-dimensional Leibniz cohomology groups of the Witt algebra with the coefficients in itself.
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