Novikov-Poisson algebras and associative commutative derivation algebras

Abstract

We describe Novikov-Poisson algebras in which a Novikov algebra is not simple while its corresponding associative commutative derivation algebra is differentially simple. In particular, it is proved that a Novikov algebra is simple over a field of characteristic not 2 iff its associative commutative derivation algebra is differentially simple. The relationship is established between Novikov-Poisson algebras and Jordan superalgebras.

Novikov–Poisson Algebras and Superalgebras of Jordan Brackets

“…Then by [13] the algebra A · is differentially simple. By Corollary 1 from [11], the Novikov algebra A is simple. The last equality holds by Proposition 1.…”

“…In this section, we will show that A · is always unital, and therefore, [11,Theorem 4] holds in general. Indeed we will show that the unit of A · is a consequence of J A being simple, and of A being simple together with AA = A.…”

“…N. Zhelyabin and A. S. Tikhov considered Novikov-Poisson in [11] algebras with associative commutative unit. They proved that for all nontrivial Novikov-Poisson algebras with associative commutative unit the Novikov algebra is simple if and only the associative commutative algebra is differential simple relative to a certain derivation.…”

The quadratic Poisson structures and related nonassociative noncommutative Zinbiel type algebras

“…V.N. Zhelyabin and A.S. Tikhov [38], asked: is true that an associative commutative algebra ( , , ) A   with a derivation  is  -simple in the usual sense if and only if its corresponding Balinsky-Novikov algebra ( , , ) A   is simple? In the next we need the following .…”

On free Gelfand–Dorfman–Novikov–Poisson algebras and a PBW theorem

“…There are also papers about embedding on GDN-Poisson algebras, for example [21]; papers on GDN-Poisson algebras and associative commutative derivation algebras [25].…”