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Averaging operators on groups, racks and Leibniz algebras

Apurba Das

Abstract

This paper considers averaging operators on various algebraic structures and studies the induced structures. We first introduce the notion of an averaging operator on a group $G$ and show that it induces a rack structure. Moreover, the given group structure and the induced rack structure form a group-rack. We observe that any pointed group-rack can be embedded into an averaging group. We show that the differentiation of a smooth pointed averaging operator on a Lie group gives rise to an averaging operator on the corresponding Lie algebra. Next, we consider averaging operators on a rack that induces a hierarchy of new rack structures. Moreover, any two racks with increasing hierarchy levels form a rack-pairing, a structure that is related to two-sided skew braces by conjugation. We also consider averaging operators on cocommutative Hopf algebras and braided vector spaces in relations to averaging operators on groups, Lie algebras and racks. In the end, we define averaging operators on a Leibniz algebra, find the induced structure and show that the differentiation of a smooth pointed averaging operator on a pointed Lie rack yields an averaging operator on the corresponding Leibniz algebra.

Averaging operators on groups, racks and Leibniz algebras

Abstract

This paper considers averaging operators on various algebraic structures and studies the induced structures. We first introduce the notion of an averaging operator on a group and show that it induces a rack structure. Moreover, the given group structure and the induced rack structure form a group-rack. We observe that any pointed group-rack can be embedded into an averaging group. We show that the differentiation of a smooth pointed averaging operator on a Lie group gives rise to an averaging operator on the corresponding Lie algebra. Next, we consider averaging operators on a rack that induces a hierarchy of new rack structures. Moreover, any two racks with increasing hierarchy levels form a rack-pairing, a structure that is related to two-sided skew braces by conjugation. We also consider averaging operators on cocommutative Hopf algebras and braided vector spaces in relations to averaging operators on groups, Lie algebras and racks. In the end, we define averaging operators on a Leibniz algebra, find the induced structure and show that the differentiation of a smooth pointed averaging operator on a pointed Lie rack yields an averaging operator on the corresponding Leibniz algebra.
Paper Structure (14 sections, 45 theorems, 159 equations)

This paper contains 14 sections, 45 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Averaging algebras
  3. Rota-Baxter operators and averaging operators on groups
  4. Averaging operators on racks and rack-pairings
  5. Averaging operators on cocommutative Hopf algebras and braided vector spaces
  6. Averaging operators on Leibniz algebras
  7. Organization of the paper
  8. Averaging operators on groups
  9. Averaging operators on racks
  10. Rack-pairings and di-racks
  11. Averaging operators on cocommutative Hopf algebras and braided vector spaces
  12. Di-Leibniz algebras and averaging operators on Leibniz algebras
  13. Some Lie functors
  14. Some generalizations and future works

Key Result

Proposition 2.10

Let $\mathcal{A}: G \rightarrow G$ be an averaging operator on $G$. Then for each $n \in \mathbb{N}$, the map $\mathcal{A}^n: G \rightarrow G$ is an averaging operator.

Theorems & Definitions (152)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Proposition 2.10
  • ...and 142 more