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General multi-Novikov algebras, multi-differential algebras and their free constructions

Xiaoyan Wang, Li Guo, Huhu Zhang

Abstract

Motivated by the recent development of noncommutative Novikov algebras and multi-Novikov algebras from the study of regularity structures of stochastic PDEs, this paper gives a general approach to study various multi-Novikov algebras and multi-differential algebras, with close connection with Poisson algebras. The construction of S. Gelfand of Novikov algebras from differential commutative algebras is generalized to this context. Free noncommuting multi-Novikov algebras are constructed from typed decorated rooted trees and from noncommuting multi-differential polynomials with populated conditions.

General multi-Novikov algebras, multi-differential algebras and their free constructions

Abstract

Motivated by the recent development of noncommutative Novikov algebras and multi-Novikov algebras from the study of regularity structures of stochastic PDEs, this paper gives a general approach to study various multi-Novikov algebras and multi-differential algebras, with close connection with Poisson algebras. The construction of S. Gelfand of Novikov algebras from differential commutative algebras is generalized to this context. Free noncommuting multi-Novikov algebras are constructed from typed decorated rooted trees and from noncommuting multi-differential polynomials with populated conditions.
Paper Structure (19 sections, 13 theorems, 102 equations, 1 figure, 1 table)

This paper contains 19 sections, 13 theorems, 102 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Novikov algebras and differential algebras
  3. Novikov algebras
  4. Novikov algebras from differential algebras
  5. Free Novikov algebras
  6. New notions related to Novikov algebras and differential algebras
  7. Layout of the paper
  8. Multi-differential algebras and their free objects
  9. Notions and examples of multi-differential algebras
  10. Free multi-differential algebras
  11. Variations of multi-Novikov algebras and their derivations from variations of multi-differential algebras
  12. Constructions of free noncommuting multi-Novikov algebras
  13. Free noncommuting multi-Novikov algebras from typed decorated rooted trees
  14. Statement of the main theorem
  15. Preparations
  16. ...and 4 more sections

Key Result

Proposition 2.4

Let $(A,\cdot)$ be a commutative algebra equipped with a Lie bracket $[\,,\,]$. Consider the adjoint action Then the triple $(A,\cdot,[\,,\,])$ is a Poisson algebra if and only if the triple $(A,\cdot,\mathrm{ad}(A))$ is a noncommuting multi-differential algebra.

Figures (1)

  • Figure 1:

Theorems & Definitions (37)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • proof
  • ...and 27 more