Representations of Rota-Baxter algebras and regular singular decompositions

Lin, Zongzhu; Qiao, Li

doi:10.48550/arxiv.1603.05912

Cited by 6 publications

(16 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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.…”

Section: Introductionmentioning

confidence: 69%

“…Then G is a skeletal Lie 2algebra. Therefore, we have the following equalities 0 = l 2 (l 2 (x, y), z) + l 2 (l 2 (y, z), x) + l 2 (l 2 (z, x), y), (18) 0 = l 2 (l 2 (x, y), α) + l 2 (l 2 (y, α), x) + l 2 (l 2 (α, x), y), (19) and 0 = −l 3 (l 2 (x, y), z, w) − l 3 (l 2 (y, z), x, w) − l 3 (l 2 (z, w), x, y) − l 3 (l 2 (x, w), y, z) +l 3 (l 2 (y, w), x, z) + l 3 (l 2 (x, z), y, w) + l 2 (l 3 (x, y, z), w) (20) −l 2 (l 3 (y, z, w), x) + l 2 (l 3 (x, z, w), y) − l 2 (l 3 (x, y, w), z), for all x, y, z, w ∈ g 0 , α ∈ g 1 . Now we are ready to give the main result in this section.…”

Section: Classification Of Skeletal Relative Rota-baxter Lie 2-algebrasmentioning

confidence: 99%

See 1 more Smart Citation

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

Jiang,

Sheng

2021

Preprint

View full text Add to dashboard Cite

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. Contents 1. Introduction 1 2. Representations of relative Rota-Baxter Lie algebras 3 3. Cohomology of relative Rota-Baxter Lie algebras 7 4. Classification of abelian extensions of relative Rota-Baxter Lie algebras 13 5. Classification of skeletal relative Rota-Baxter Lie 2-algebras 18 6. Representations and cohomologies of Rota-Baxter Lie algebras 21 References 25and cohomologies of Rota-Baxter Lie algebras using the above general framework of relative Rota-Baxter Lie algebras and give various applications.Acknowledgements. This research is supported by NSFC (11922110). Representations of relative Rota-Baxter Lie algebrasIn this section, we introduce the notion of a representation of a relative Rota-Baxter Lie algebra. We give the semidirect products characterization of representations of relative Rota-Baxter Lie algebras. The dual representation is given. A representation of a relative Rota-Baxter Lie algebra also induces a representation of the underlying pre-Lie algebra.Definition 2.1.Lie algebra, ρ : g → gl(V) is a representation of g on a vector space V and T : V → g is a relative Rota-Baxter operator, i.e.Note that a Rota-Baxter operator on a Lie algebra is a relative Rota-Baxter operator with respect to the adjoint representation. Definition 2.2. Let ((g, [•, •] g ), (V, ρ), T ) and ((g ′ , [•, •] g ′ ), (V ′ , ρ ′ ), T ′ ) be two relative Rota-Baxter Lie algebras. A homomorphism from ((g, [•, •] g ), (V, ρ), T ) to ((g ′ , [•, •] g ′ ), (V ′ , ρ ′ ), T ′ ) consists of a Lie algebra homomorphism φ : g → g ′ and a linear map ϕ : V → V ′ such that

show abstract

Section: Introductionmentioning

confidence: 69%

Section: Classification Of Skeletal Relative Rota-baxter Lie 2-algebrasmentioning

confidence: 99%

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

Jiang,

Sheng

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…If p is quasi-idempotent, then we have the decomposition M = M −λ ⊕ M 0 which will be called the regular-singular decomposition. A more detailed study in this case can be found in [22]. From Corollary 2.8 we know that…”

Section: Let the Matrices Of τ And P I Corresponding To The Basismentioning

confidence: 91%

“…The concept of modules (or representations) of Rota-Baxter algebras was introduced in [17]. Further studies in this direction were pursued in [20][21][22] on regular-singular decompositions, geometric representations and derived functors of Rota-Baxter modules, especially those over the Rota-Baxter algebras of Laurent series and polynomials.…”

Section: Introductionmentioning

confidence: 99%

Modules of polynomial Rota-Baxter Algebras and matrix equations

Tang¹

2020

Preprint

View full text Add to dashboard Cite

The all Rota-Baxter algebra structures on the polynomial algebra R = k[x] are well known. We study the finite dimensional modules of polynomial Rota-Baxter algebras (k[x], P) or (xk[x], P) of weight nonzero since some cases of weight zero have been studied. The main result shows that every module over the polynomial Rota-Baxter algebra (k[x], P) or (xk[x], P) is equivalent to the modules over a plane k x, y /I where I is some ideal of free algebra k x, y . Furthermore, we provide the classification of modules of polynomial Rota-Baxter algebras of weight nonzero through solution to some matrix equation.

show abstract

“…One the other hand, we try to understand further the algebra framework that leads to the matrix representation of Rota-Baxter algebras that arise from the aforementioned applications. As related studies 1 in [32], representations of the Rota-Baxter algebra of Laurent series algebra were discussed and one finds interesting connections with class numbers in algebraic number theory. A similar approach to the Rota-Baxter algebra of polynomial algebra is taken in [36].…”

Section: Introductionmentioning

confidence: 97%

Representations and Modules of Rota-Baxter Algebras

Guo¹,

Lin²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

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

show abstract

Representations of Rota-Baxter algebras and regular singular decompositions

Cited by 6 publications

References 1 publication

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

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

Modules of polynomial Rota-Baxter Algebras and matrix equations

Representations and Modules of Rota-Baxter Algebras

Contact Info

Product

Resources

About