The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras

Abstract

In this paper, the so-called differential graded (DG for short) Poisson Hopf algebra is introduced, which can be considered as a natural extension of Poisson Hopf algebras in the differential graded setting. The structures on the universal enveloping algebras of differential graded Poisson Hopf algebras are discussed.

Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases

“…The differential algebra, introduced by Ritt [38], is an algebraic approach to differential equations replacing analytic notions like differential quotient by Leibniz rule. Later due to the work of Kolchin and many other mathematicians, the differential algebra has broad applications in mathematics and physics, such as algebraic group [33], category [36], Galois theory [44], operad [21], Poisson Hopf algebra [29] and representation theory [42].…”

The DG Poisson Adjoint Action and its Application

“…We know that, Poisson Hopf algebras arise naturally in Poisson geometry and quantum groups. Recently, Poisson Hopf algebras are studied by many authors from different perspectives [5,7,14,15]. In [5], the authors developed the theory of Poisson Hopf algebras, given the definition of a DG Poisson Hopf algebra A, and discussed the structures for the universal enveloping algebra of A.…”