We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift No...

We consider two different constructions of higher brackets. First, based on a Grassmann-odd, nilpotent ∆ operator, we define a non-commutative generalization of the higher Koszul brackets, which are used in a generalized Batalin-Vilkovisky algebra, and we show that they form a homotopy Lie algebra. Secondly, we investigate higher, so-called derived brackets built from symmetrized, nested Lie br...

We give a combinatorial description of the finitely generated free weak nilpotent minimum algebras, and provide explicit constructions of normal forms.

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in...

We give a condition ensuring that the operators in nilpotent Lie algebra of linear on finite dimensional vector space have common eigenvector.

We define the fuzzy description logic ALCHNM based on the t-norm of Nilpotent Minimum and we prove its reduction to the crisp description logic ALCH. Hence we prove decidability results for ALCHNM.

For an admissible affine vertex algebra Vk(g) of type A, we describe a new family of relaxed highest weight representations of Vk(g). They are simple quotients of representations of the affine Kac-Moody algebra ĝ induced from the following g-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from sl2; 2) all Gelfand-Tsetlin modul...

Let G be a connected linear semisimple Lie group with Lie algebra g, and let K C → Aut(p C ) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. Suppose that Ω is a nilpotent G-orbit in g and O is the nilpotent K C -orbit in p C associated to Ω by the Kostant-Sekiguchi correspondence. We show that the complexity of O as a K C variety measures ...

The main purpose of this paper is to prove that the extensions of a nilpotent block algebra and its Glauberman correspondent block algebra are Morita equivalent under an additional group-theoretic condition (see Theorem 1.6); in particular, Harris and Linckelman’s theorem and Koshitani and Michler’s theorem are covered (see Theorems 7.5 and 7.6). The ingredient to carry out our purpose is the t...