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Induced structures of operated algebras with applications to multi-Novikov algebras

Li Guo, Xiaoyan Wang, Huhu Zhang

Abstract

We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.

Induced structures of operated algebras with applications to multi-Novikov algebras

Abstract

We provide a general notion of induced structures of operated algebras in the context of unary-binary operads. This notion fully captures the binary quadratic relations encoded by a unary-binary operad, thereby unifying and formalizing the various constructions that have appeared in the literature under the informal term of ``induced structures''. As an application, we show that the Novikov algebra and the recently introduced multi-Novikov algebra are the induced structures of the differential commutative algebra and the multi-differential commutative algebra respectively. We also explicitly determine the induced structure of the noncommuting multi-differential commutative algebra.
Paper Structure (10 sections, 6 theorems, 57 equations)

This paper contains 10 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Operated algebras
  3. Induced structures from operated algebras
  4. Layout of the paper
  5. Induced structures of operated algebras and operated operads
  6. Background and notions on operads
  7. The induced structure of an operated operad
  8. Various multi-differential algebras and their operads
  9. The pre-induced structure of commuting multi-differential commutative algebras
  10. The pre-induced structure of noncommuting multi-differential commutative algebras

Key Result

Theorem 1.1

$($Gelfand's Construction$)$ Let $(R,\cdot)$ be a commutative associative algebra equipped with a derivation $D$, that is $(A,\cdot,D)$ is a commutative differential algebra. Define Then $(R,\rhd)$ is a Novikov algebra.

Theorems & Definitions (21)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 11 more