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Register a new userOn the algebraic variety of Hom-Lie algebras
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The set HLie(n) of the n-dimensional Hom-Lie algebras over an algebraically closed field of characteristic zero is provided with a structure of algebraic subvariety of the affine plane of dimension n^2(n-1)/2}. For n=3, these two sets coincide, for n=4 it is an hypersurface in K^{24}. For n>4, we describe the scheme of polynomial equations which define HLie(n). We determine also what are the classes of Hom-Lie algebras which are P-algebras where P is a binary quadratic operads.
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We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an $A$-split abelian extension and the equivalence classes of $A$-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
We describe Hom-Lie structures on affine Kac-Moody and related Lie algebras, and discuss the question when they form a Jordan algebra.
After endowing with a 3-Lie-Rinehart structure on Hom 3-Lie algebras, we obtain a class of special Hom 3-Lie algebras, which have close relationships with representations of commutative associative algebras. We provide a special class of Hom 3-Lie-Rinehart algebras, called split regular Hom 3-Lie-Rinehart algebras, and we then characterize their structures by means of root systems and weight systems associated to a splitting Cartan subalgebra.
The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+sum_{[gamma]inGamma/thicksim}I_{[gamma]}$ with $U$ a vector space complement in $H$ and $I_{[gamma]}$ are well described ideals of $L $ satisfying $I_{[gamma]},I_{[delta]}=0$ if $I_{[gamma]} eq I_{[delta]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.
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