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Skew-symmetric solutions of the classical Yang-Baxter equation and $\mathcal{O}$-operators of Malcev algebras

Shan Ren, Runxuan Zhang

Abstract

We study connections between skew-symmetric solutions of the classical Yang-Baxter equation (CYBE) and $\mathcal{O}$-operators of Malcev algebras. We prove that a skew-symmetric solution of the CYBE on a Malcev algebra can be interpreted as an $\mathcal{O}$-operator associated to the coadjoint representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that $\mathcal{O}$-operators associated to a general representation could give skew-symmetric solutions of the CYBE on certain semi-direct products of Malcev algebras. We reveal the relationship between invertible $\mathcal{O}$-operators and compatible pre-Malcev algebra structures on a Malcev algebra. We finally obtain several analogous results on connections between the CYBE and $\mathcal{O}$-operators in the case of pre-Malcev algebras.

Skew-symmetric solutions of the classical Yang-Baxter equation and $\mathcal{O}$-operators of Malcev algebras

Abstract

We study connections between skew-symmetric solutions of the classical Yang-Baxter equation (CYBE) and -operators of Malcev algebras. We prove that a skew-symmetric solution of the CYBE on a Malcev algebra can be interpreted as an -operator associated to the coadjoint representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that -operators associated to a general representation could give skew-symmetric solutions of the CYBE on certain semi-direct products of Malcev algebras. We reveal the relationship between invertible -operators and compatible pre-Malcev algebra structures on a Malcev algebra. We finally obtain several analogous results on connections between the CYBE and -operators in the case of pre-Malcev algebras.
Paper Structure (11 sections, 17 theorems, 63 equations)

This paper contains 11 sections, 17 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Malcev Algebras and $\mathcal{O}$-operators
  3. Representations of Malcev algebras
  4. $\mathcal{O}$-operators of Malcev algebras
  5. Bilinear Forms, the CYBE and Semi-direct Products
  6. Invariant bilinear forms
  7. Symplectic forms
  8. The CYBE and semi-direct products
  9. $\mathcal{O}$-operators and the CYBE on Pre-Malcev Algebras
  10. $\mathcal{O}$-operators of Malcev algebras and Pre-Malcev algebras
  11. The CYBE on pre-Malcev algebras

Key Result

Theorem 1.1

Let $A$ be a finite-dimensional Malcev algebra over a field $\mathbb{F}$ of characteristic zero and $r$ be a skew-symmetric element in $A\otimes A$. Then $r\in \mathcal{Sol}(A)$ if and only if $T_r\in \mathcal{O}_A(A^*,\mathop{\mathrm{ad}}\nolimits^*)$, where $(A^*,\mathop{\mathrm{ad}}\nolimits^*)$

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Lemma 2.6
  • ...and 32 more