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Universal enveloping algebras of weighted differential Poisson algebras

Ying Chen, Chuangchuang Kang, Jiafeng Lü

Abstract

The $λ$-differential operators and modified $λ$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $λ$-differential Poisson ($λ$-DP for short) algebras and modified $λ$-differential Poisson ($λ$-mDP for short) algebras as generalizations of differential Poisson algebras. The $λ$-DP algebra is proved to be closed under tensor product, and a $λ$-DP algebra structure is provided on the cohomology algebra of the $λ$-DP algebra. These conclusions are also applied to $λ$-mDP algebras and their modules. Finally, the universal enveloping algebras of $λ$-DP algebras are generalized by constructing a $\mathcal{P}$-triple. Three isomorphisms among opposite algebras, tensor algebras and the universal enveloping algebras of $λ$-DP algebras are obtained.

Universal enveloping algebras of weighted differential Poisson algebras

Abstract

The -differential operators and modified -differential operators are generalizations of classical differential operators. This paper introduces the notions of -differential Poisson (-DP for short) algebras and modified -differential Poisson (-mDP for short) algebras as generalizations of differential Poisson algebras. The -DP algebra is proved to be closed under tensor product, and a -DP algebra structure is provided on the cohomology algebra of the -DP algebra. These conclusions are also applied to -mDP algebras and their modules. Finally, the universal enveloping algebras of -DP algebras are generalized by constructing a -triple. Three isomorphisms among opposite algebras, tensor algebras and the universal enveloping algebras of -DP algebras are obtained.

Paper Structure

This paper contains 11 sections, 16 theorems, 72 equations.

Table of Contents

  1. Introduction and Main Statements
  2. Differential algebras
  3. $\lambda$-differential algebras and $\lambda$-differential Lie algebras
  4. Modified $\lambda$-Rota-Baxter algebras and modified $\lambda$-differential algebras
  5. Universal enveloping algebras of Poisson algebras
  6. Motivations
  7. Outline of this paper
  8. $\lambda$-DP algebras, $\lambda$-mDP algebras and their modules
  9. $\lambda$-DP algebras and $\lambda$-mDP algebras
  10. The modules of $\lambda$-DP algebras and $\lambda$-mDP algebras
  11. Universal enveloping algebras of $\lambda$-DP (or $\lambda$-mDP) algebras

Key Result

Proposition 2.6

Let $(A,\cdot,\{\cdot,\cdot\},d)$ be a $\lambda$-DP (or $\lambda$-mDP) algebra, then $(A^{op},\cdot_{op},\{\cdot,\cdot\}_{op},d_{op})$ is a $\lambda$-DP (or $\lambda$-mDP) algebra, where for all $a,b \in A$,

Theorems & Definitions (43)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 2.1
  • Example 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • ...and 33 more