<abstract><p>The main purpose of this paper is to provide a full cohomology Hom-pre-Lie algebra with coefficients in given representation. This new type exploits strongly the Hom-type structure and fits perfectly simultaneous deformations multiplication homomorphism defining algebra. Moreover, we show that its second group classifies abelian extensions by representation.</p>&l...

Let M be a von Neumann algebra and ρ : M → M be a ∗-homomorphism. Then ρ is called a centrally extendable ∗-homomorphism (CEH) if there is a maximal abelian subalgebra (masa) M of the commutant M of M and a surjective ∗-homomorphism φ : M → M such that φ(Z) = ρ(Z) for all Z in the center of M. A ∗-ρderivation δ : M → M is called a centrally extendable ∗-ρ-derivation (CED) if there is a masa M o...

We prove that a discrete group G is amenable iff it is strongly unitarizable in the following sense: every unitarizable representation π on G can be unitarized by an invertible chosen in the von Neumann algebra generated by the range of π. Analogously a C *-algebra A is nuclear iff any bounded homomorphism u : A → B(H) is strongly similar to a *-homomorphism in the sense that there is an invert...

The notion of a Poisson algebra was probably introduced in the first time by A.M. Vinogradov and J. S. Krasil’shchik 1975 under name “canonical algebra” Braconnier his short note “Algèbres de Poisson” (Comptes rendus Ac.Sci) 1977.

We prove a result that can be applied to determine the finitedimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to th...

Abstract We prove that every Novikov–Poisson algebra over a field of zero characteristic can be embedded into commutative conformal with derivation. As corollary, we show commutator Gelfand–Dorfman obtained from is special, i.e., embeddable differential Poisson algebra.

The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.

Let B be a strictly real commutative real Banach algebra with the carrier space Φ B. If A is a commutative real Banach algebra, then we give a representation of a ring homomorphism ρ : A → B, which needs not be linear nor continuous. If A is a commutative complex Banach algebra, then ρ(A) is contained in the radical of B. 1. Introduction and results. Ring homomorphisms are mappings between two ...

In this paper we consider deformations of finite or infinite dimensional Lie algebras over a field of characteristic 0. By “deformations of a Lie algebra” we mean the (affine algebraic) manifold of all Lie brackets. Consider the quotient of this variety by the action of the group GL. It is well-known (see [Hart]) that in the category of algebraic varieties the quotient by a group action does no...