Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

Abstract

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras.

26 pages. arXiv admin note: text overlap with arXiv:2008.06714

References in corpus (3)

• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• $\lambda$-Differential operators and $\lambda$-differential modules for the Virasoro algebra

Cited by in corpus (4)

• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights
• Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras
• Deformations and homotopy theory of Rota-Baxter algebras of any weight