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Rota--Baxter and averaging operators on racks and rack algebras

V. G. Bardakov, V. A. Bovdi

Abstract

In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity to the rack algebra k[X]. On the other side, we have definitions of Rota--Baxter and averaging operators on arbitrary algebra. We find connections between these operators. In particular, we prove that if B : X --> X is an averaging operator on a rack, then its linear extension on a rack algebra k[X] gives an averaging operator.

Rota--Baxter and averaging operators on racks and rack algebras

Abstract

In the present article we define and investigate relative Rota--Baxter operators and relative averaging operators on racks and rack algebras. Also, if B is a Rota--Baxter or averaging operator on a rack X, then we can extend B by linearity to the rack algebra k[X]. On the other side, we have definitions of Rota--Baxter and averaging operators on arbitrary algebra. We find connections between these operators. In particular, we prove that if B : X --> X is an averaging operator on a rack, then its linear extension on a rack algebra k[X] gives an averaging operator.
Paper Structure (10 sections, 18 theorems, 77 equations)

This paper contains 10 sections, 18 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative Rota--Baxter operators on racks
  4. Rack extensions and averaging operators on racks
  5. Properties of averaging operators on racks
  6. Rota--Baxter and averaging operators on rack algebras
  7. Some generalization of conjugation quandle and core quandle
  8. Open problems
  9. Other operators
  10. From Lie racks to Leibniz algebra

Key Result

Proposition 2.3

Let $(G, B)$ be a Rota--Baxter group. Define on $G$ the operation The following conditions hold:

Theorems & Definitions (53)

  • Definition 2.1: GLS
  • Example 2.2
  • Proposition 2.3: GLS
  • Definition 2.4: JSZ
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Remark 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 43 more