Universal enveloping Hom-algebras of regular Hom-Poisson algebras

<abstract><p>In this paper, we introduce universal enveloping Hom-algebras of Hom-Poisson algebras. Some properties of universal enveloping Hom-algebras of regular Hom-Poisson algebras are discussed. Furthermore, in the involutive case, it is proved that the category of involutive Hom-Poisson modules over an involutive Hom-Poisson algebra $ A $ is equivalent to the category of involutive Hom-associative modules over its universal enveloping Hom-algebra $ U_{eh}(A) $.</p></abstract>

Admissible Poisson bialgebras

An admissible Poisson algebra (or briefly, an adm-Poisson algebra) gives an equivalent presentation with only one operation for a Poisson algebra. We establish a bialgebra theory for adm-Poisson algebras independently and systematically, including but beyond the corresponding results on Poisson bialgebras given in [27]. Explicitly, we introduce the notion of adm-Poisson bialgebras which are equivalent to Manin triples of adm-Poisson algebras as well as Poisson bialgebras. The direct correspondence between adm-Poisson bialgebras with one comultiplication and Poisson bialgebras with one cocommutative and one anti-cocommutative comultiplications generalizes and illustrates the polarization–depolarization process in the context of bialgebras. The study of a special class of adm-Poisson bialgebras which include the known coboundary Poisson bialgebras in [27] as a proper subclass in general, illustrating an advantage in terms of the presentation with one operation, leads to the introduction of adm-Poisson Yang–Baxter equation in an adm-Poisson algebra. It is an unexpected consequence that both the adm-Poisson Yang–Baxter equation and the associative Yang–Baxter equation have the same form and thus it motivates and simplifies the involved study from the study of the associative Yang–Baxter equation, which is another advantage in terms of the presentation with one operation. A skew-symmetric solution of adm-Poisson Yang–Baxter equation gives an adm-Poisson bialgebra. Finally, the notions of an [Formula: see text]-operator of an adm-Poisson algebra and a pre-adm-Poisson algebra are introduced to construct skew-symmetric solutions of adm-Poisson Yang–Baxter equation and hence adm-Poisson bialgebras. Note that a pre-adm-Poisson algebra gives an equivalent presentation for a pre-Poisson algebra introduced by Aguiar.

On analogs of some classical group-theoretic results in Poisson algebras

We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.