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.

arXiv comment26 pages. arXiv admin note: text overlap with arXiv:2008.06714

References in corpus (15)

• Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
• Higher derived brackets and homotopy algebras
• What a classical r-matrix really is
• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Simple finite-dimensional double algebras
• Weak composition quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras
• $\lambda$-Differential operators and $\lambda$-differential modules for the Virasoro algebra
• Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
• Representations of Rota-Baxter algebras and regular singular decompositions
• Representations and Modules of Rota-Baxter Algebras
• Rota-Baxter modules toward derived functors
• Extensions of Operators, Liftings of Monads and Distributive Laws
• Classification of operator extensions, monad liftings and distributive laws for differential algebras and Rota-Baxter algebras

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