Representations and Modules of Rota-Baxter Algebras

Abstract

We give a broad study of representation and module theory of Rota-Baxter algebras, motivated by Rota-Baxter matrix representations in the renormalization of quantum field theory and by geometric connections. Regular-singular decompositions of Rota-Baxter algebras and Rota-Baxter modules are obtained under the condition of quasi-idempotency. Representations of an Rota-Baxter algebra are shown to be equivalent to the representations of the ring of Rota-Baxter operators whose categorical properties are obtained and explicit constructions are provided. Representations from coalgebras are investigated and their algebraic Birkhoff factorization is given. Representations of Rota-Baxter algebras in the Lie algebra and tensor category contexts are also formulated. Contents 1. Introduction 2. Rota-Baxter modules and their regular singular decompositions 2.1. Rota-Baxter modules 2.2. Regular-singular decompositions of Rota-Baxter modules 2.3. Dual modules and derived modules of Rota-Baxter modules 2.4. Product of Rota-Baxter algebras 3. The ring of Rota-Baxter operators and Rota-Baxter modules 3.1. Ring of Rota-Baxter operators 3.2. Categorical properties 4. Construction of the ring of Rota-Baxter operators 4.1. The general construction 4.2. Special cases and examples 5. Endomorphism Rota-Baxter algebras 5.1. Endomorphism Rota-Baxter algebras from coalgebras 5.2. Bifunctors and schemes from Rota-Baxter algebras 5.3. Algebraic Birkhoff factorization for Rota-Baxter representations 6. Rota-Baxter representations for Lie algebras and tensor categories 6.1. Representations of Rota-Baxter Lie algebras 6.2. Tensor category setting of Rota-Baxter modules References

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

“…It is obvious that a Rota-Baxter associative algebra gives rise to a Rota-Baxter Lie algebra (A, [•, •], T ), where the Lie bracket [•, •] is the commutator Lie bracket. Furthermore, it is straightforward to deduce that a left Rota-Baxter module (V, T ) gives rise to a representation ρ of the Rota-Baxter Lie algebra (A, [•, •], T ) on V with respect to T : V → V, where ρ(x)u = xu, for all x ∈ A, u ∈ V. Thus the definition of representations of Rota-Baxter Lie algebras is consistent with representations of Rota-Baxter associative algebras given in [17].…”

“…More precisely, we use the second cohomology group to classify abelian extensions of relative Rota-Baxter Lie algebras, and use the third cohomology group to classify relative Rota-Baxter Lie 2-algebras, which was introduced in [26] under the terminology of O-operators on Lie 2-algebras in the study of solutions of 2-graded Yang-Baxter equations. Note that in the associative algebra context, the notion of a representation (module) over a Rota-Baxter associative algebra was given in [17], and further studied in [20,23,24,30]. The representation of a relative Rota-Baxter Lie algebra introduced in this paper is consistent with the representation of Rota-Baxter associative algebras, namely a representation of a Rota-Baxter associative algebra naturally gives rise to a representation of the corresponding Rota-Baxter Lie algebra.…”

“…In [17], the authors defined a left Rota-Baxter module over (A, T ) to be a pair (V, T ) with a left A-module V and a linear map…”

Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras

“…For the purpose of defining matched pairs of Rota-Baxter Lie algebras, we introduce the notion of representations of Rota-Baxter Lie algebras of weight λ. Note that the concept of representations of Rota-Baxter Lie algebras of weight 0 was already given in [21] in the study of cohomologies of Rota-Baxter Lie algebras, and the notion of representations of Rota-Baxter associative algebras was introduced in [23], and further studied in [31]. Definition 3.6.…”

Rota–Baxter paired comodules and Rota–Baxter paired Hopf modules

“…To study the representations of Rota-Baxter algebras, the authors in [7] introduced the conception of Rota-Baxter modules related to the ring of Rota-Baxter operators. By the definition, a Rota-Baxter module over a Rota-Baxter algebra (A, P ) is a pair (M, T ) where M is a (left) of Rota-Baxter paired comodules, which is dual to the definition of Rota-Baxter paired modules in [14].…”