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Rota-Baxter operators on $ω$-Lie algebras

Yin Chen, Shan Ren, Jiawen Shan, Runxuan Zhang

Abstract

This article explores Rota-Baxter operators on finite-dimensional $ω$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $ω$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $ω$-Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight $0$ and isometric Rota-Baxter operators of weight $1$ over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight $1$ on any finite-dimensional non-Lie complex simple $ω$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $ω$-Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight $0$ such that the induced Hom-Lie algebra is nonabelian but solvable.

Rota-Baxter operators on $ω$-Lie algebras

Abstract

This article explores Rota-Baxter operators on finite-dimensional -Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, -Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given -Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight and isometric Rota-Baxter operators of weight over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight on any finite-dimensional non-Lie complex simple -Lie algebra is -dimensional. Furthermore, we show that for every -dimensional non-Lie complex -Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight such that the induced Hom-Lie algebra is nonabelian but solvable.
Paper Structure (9 sections, 19 theorems, 87 equations)

This paper contains 9 sections, 19 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. General Rota-Baxter Operators
  3. Rota-Baxter operators and invertible derivations
  4. Rota-Baxter operators and left-symmetric algebras
  5. Rota-Baxter operators and new $\upomega$-Lie algebras
  6. Rota-Baxter operators and Hom-Lie algebras
  7. Rota-Baxter operators and representations of $\upomega$-Lie algebras
  8. Rota-Baxter Operators on $3$-dimensional $\upomega$-Lie Algebras
  9. Rota-Baxter Operators and Hom-Lie Algebras

Key Result

Proposition 2.1

Suppose $L$ is finite-dimensional. Then $\mathcal{B}(L)\cap \mathop{\mathrm{Aut}}\nolimits(L)\neq \emptyset$ if and only if $\mathop{\mathrm{Der}}\nolimits(L)\cap \mathop{\mathrm{Aut}}\nolimits(L)\neq \emptyset$. Moreover, if $\mathcal{B}(L)\cap \mathop{\mathrm{Aut}}\nolimits(L)\neq \emptyset$, then

Theorems & Definitions (43)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Example 2.5
  • Remark 2.6
  • Theorem 2.7
  • proof
  • ...and 33 more