The aim of this paper is to introduce and study quadratic Hom–Lie algebras, which are Hom–Lie algebras equipped with symmetric invariant nondegenerate bilinear forms. We provide several constructions ...leading to examples and extend the Double Extension Theory to this class of nonassociative algebras. Elements of Representation Theory for Hom–Lie algebras, including adjoint and coadjoint representations, are supplied with application to quadratic Hom–Lie algebras. Centerless involutive quadratic Hom–Lie algebras are characterized. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. Also, we establish a correspondence between the class of involutive quadratic Hom–Lie algebras and quadratic simple Lie algebras with symmetric involution.
We introduce a dual notion of the Poisson algebra, called the transposed Poisson algebra, by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. The ...transposed Poisson algebra shares common properties of the Poisson algebra and arises naturally from a Novikov-Poisson algebra by taking the commutator Lie algebra of the Novikov algebra. Consequently, the classic construction of a Poisson algebra from a commutative algebra with two commuting derivations similarly applies to a transposed Poisson algebra. More broadly, the transposed Poisson algebra captures the algebraic structures when the commutator is taken in pre-Lie Poisson algebras and two other Poisson type algebras. Furthermore, the transposed Poisson algebra improves two processes that produce 3-Lie algebras from Poisson algebras with a strongness condition.
The notion of a transposed Poisson $ n $-Lie algebra has been developed as a natural generalization of a transposed Poisson algebra. It was conjectured that a transposed Poisson $ n $-Lie algebra ...with a derivation gives rise to a transposed Poisson $ (n+1) $-Lie algebra. In this paper, we focus on transposed Poisson $ n $-Lie algebras. We have obtained a rich family of identities for these algebras. As an application of these formulas, we provide a construction of $ (n+1) $-Lie algebras from transposed Poisson $ n $-Lie algebras with derivations under a certain strong condition, and we prove the conjecture in these cases.
A thin Lie algebra is a Lie algebra L, graded over the positive integers, with its first homogeneous component L
1
of dimension two and generating L, and such that each nonzero ideal of L lies ...between consecutive terms of its lower central series. All homogeneous components of a thin Lie algebra have dimension one or two, and the two-dimensional components are called diamonds. If L
1
is the only diamond, then L is a graded Lie algebra of maximal class. We present simpler proofs of some fundamental facts on graded Lie algebras of maximal class, and on thin Lie algebras, based on a uniform method, with emphasis on a polynomial interpretation. Among else, we determine the possible values for the most fundamental parameter of such algebras, which is one less than the dimension of their largest metabelian quotient.
The algebras of the title are infinite-dimensional graded Lie algebras L=⨁i=1∞Li, over a field of positive characteristic p, which are generated by an element of degree 1 and an element of degree p, ...and satisfy Li,L1=Li+1 for i≥p. In case p=2 such algebras were classified by Caranti and Vaughan-Lee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof.
Let n>1 be an integer. The algebras of the title, which we abbreviate as algebras of type n, are infinite-dimensional graded Lie algebras L=⨁i=1∞Li, which are generated by an element of degree 1 and ...an element of degree n, and satisfy Li,L1=Li+1 for i≥n. Algebras of type 2 were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type n, over fields of sufficiently large characteristic relative to n. Our main result describes precisely all possibilities for the first constituent length of an algebra of type n, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.
We show that for a given nilpotent Lie algebra g with Z(g)⊆g,g all commutative post-Lie algebra structures, or CPA-structures, on g are complete. This means that all left and all right multiplication ...operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras Fg,c and discover a strong relationship to solving systems of linear equations of type x,u+y,v=0 for generator pairs x,y∈Fg,c. We use results of Remeslennikov and Stöhr concerning these equations to prove that, for certain g and c, the free-nilpotent Lie algebra Fg,c has only central CPA-structures.
In the paper we describe the class of all solvable extensions of an infinite-dimensional filiform Leibniz algebra. The filiform Leibniz algebra is taken as a maximal pro-nilpotent ideal of a ...residually solvable Leibniz algebra. It is proven that the second cohomology group of the extension is trivial.
Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product Ib=b⋉b⁎, where the dual b⁎ of b is equipped with the coadjoint action of b and is considered as an ...abelian ideal of Ib. We describe the automorphism group Aut(Ib) of the Lie algebra Ib. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type An, the dihedral subgroup was recently proved to be contained in Aut(Ib) by Dror Bar-Natan and Roland van der Veen in 1 (where Ib is denoted by Iun). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for Ib‾=b⋉n⁎ that is both an Inönü-Wigner contraction of g and the quotient of Ib by its center.
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