Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal enveloping H-pseudoalgebras of DGP pseudoalgebras

Ying Chen, Jiafeng Lü, Jiaqun Wei

TL;DR

This work develops a framework for Poisson $H$-pseudoalgebras in the pseudotensor setting by introducing Poisson pseudo-Ore extensions and differential graded Poisson $H$-pseudoalgebras (DGP pseudoalgebras). It proves a main Poisson pseudo-Ore extension theorem and constructs current and annihilation algebra examples, then defines universal enveloping $H$-pseudoalgebras for DGP pseudoalgebras via a $\mathcal{P}$-triple, establishing a universal mapping property. The results extend classical Poisson-Ore theory and enveloping constructions to the realm of $H$-pseudoalgebras and their differential graded analogues. Overall, the paper provides foundational tools for Poisson type structures in pseudotensor categories with cocommutative Hopf algebras, with potential impact on representation theory and noncommutative geometry.

Abstract

The notions of Poisson $H$-pseudoalgebras are generalizations of Poisson algebras in a pseudotensor category $\mathcal{M}^{\ast}(H)$. This paper introduces an analogue of Poisson-Ore extension in Poisson $H$-pseudoalgebras. Poisson $H$-pseudoalgebras with the differential graded setting induces the notions of differential graded Poisson $H$-pseudoalgebras (DGP pseudoalgebras, for short). The DGP pseudoalgebra with some compatibility conditions is proved to be closed under tensor product. Furthermore, the universal enveloping $H$-pseudoalgebras of DGP pseudoalgebras are constructed by a $\mathcal{P}$-triple. A unique differential graded pseudoalgebra homomorphism between a universal enveloping $H$-pseudoalgebra of a DGP pseudoalgebra and a $\mathcal{P}$-triple of a DGP pseudoalgebra is obtained.

Universal enveloping H-pseudoalgebras of DGP pseudoalgebras

TL;DR

This work develops a framework for Poisson -pseudoalgebras in the pseudotensor setting by introducing Poisson pseudo-Ore extensions and differential graded Poisson -pseudoalgebras (DGP pseudoalgebras). It proves a main Poisson pseudo-Ore extension theorem and constructs current and annihilation algebra examples, then defines universal enveloping -pseudoalgebras for DGP pseudoalgebras via a -triple, establishing a universal mapping property. The results extend classical Poisson-Ore theory and enveloping constructions to the realm of -pseudoalgebras and their differential graded analogues. Overall, the paper provides foundational tools for Poisson type structures in pseudotensor categories with cocommutative Hopf algebras, with potential impact on representation theory and noncommutative geometry.

Abstract

The notions of Poisson -pseudoalgebras are generalizations of Poisson algebras in a pseudotensor category . This paper introduces an analogue of Poisson-Ore extension in Poisson -pseudoalgebras. Poisson -pseudoalgebras with the differential graded setting induces the notions of differential graded Poisson -pseudoalgebras (DGP pseudoalgebras, for short). The DGP pseudoalgebra with some compatibility conditions is proved to be closed under tensor product. Furthermore, the universal enveloping -pseudoalgebras of DGP pseudoalgebras are constructed by a -triple. A unique differential graded pseudoalgebra homomorphism between a universal enveloping -pseudoalgebra of a DGP pseudoalgebra and a -triple of a DGP pseudoalgebra is obtained.
Paper Structure (13 sections, 12 theorems, 98 equations)

This paper contains 13 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction and Main Statements
  2. Poisson-Ore extension
  3. differential graded Poisson algebras
  4. Hopf algebras
  5. Motivations
  6. Outline of this paper
  7. Poisson pseudo-Ore extension
  8. Some examples of Poisson $H$-pseudoalgebras
  9. current $H$-pseudoalgebras
  10. $H$-differential algebras and annihilation algebras
  11. DGP pseudoalgebras and their universal enveloping $H$-pseudoalgebras
  12. DGP pseudoalgebras
  13. Universal enveloping $H$-pseudoalgebras of DGP pseudoalgebras

Key Result

Theorem 1.3

$($Oh-2006-3$)$ Let $(A,\cdot,\{\cdot,\cdot\}_A)$ be a Poisson algebra and $\alpha, \delta:~A\rightarrow A$ are two linear maps. Then the polynomial ring $A[x]$ is a Poisson algebra with Poisson bracket for all $a,b\in A$ if and only if $\alpha$ is a Poisson derivation of $(A,\cdot,\{\cdot,\cdot\}_A)$ and $\delta$ is a Poisson $\alpha$-derivation of $(A,\cdot,\{\cdot,\cdot\}_A)$.

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • ...and 33 more