Extensions, deformation and categorification of $\text{AssDer}$ pairs

Das, Apurba; Mandal, Ashis

doi:10.48550/arxiv.2002.11415

Cited by 7 publications

(8 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [9], one constructs a deformation theory of a coalgebra morphism. Recently, deformations of algebraic derivations are devoloped [6,22]. Deformations of derivations are more difficult to describe than that of the algebraic objects themselves.…”

Section: Introductionmentioning

confidence: 99%

Cohomologies and deformations of coassociative coderivations

Du¹,

Ma²,

Xv³

et al. 2023

Preprint

View full text Add to dashboard Cite

The motivation of this paper is to construct a deformation theory of coderivations of coassociative coalgebras. We introduce a notion of a Coder pair, that is, a coassociative coalgebra with a coderivation. Then we define a proper corepresentation of a Coder pair and study the corresponding cohomology. Finally, we show that a Coder pair is rigid, provided that the second cohomology group is trivial and point out that a deformation of finite order is extensible to higher order deformations if the obstruction class, which is defined to be in the third cohomology group, is trivial.

show abstract

Section: Introductionmentioning

confidence: 99%

Cohomologies and deformations of coassociative coderivations

Du¹,

Ma²,

Xv³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Cohomological and deformation theories of differential Lie algebras with weight zero were studied in [32]. These results have been extended to the case of associative algebras, Leibniz algebras, Pre-Lie algebras, Lie triple systems and n-Lie algebras [7,8,28,29]. In [24], the cohomologies, extensions and deformations of differential algebras with any weight were developed.…”

Section: Introductionmentioning

confidence: 99%

Representations and cohomologies of differential 3-Lie algebras with any weight

Sun¹,

Chen²

2022

Preprint

View full text Add to dashboard Cite

The purpose of the present paper is to study cohomologies of differential 3-Lie algebras with any weight. We introduce the representation of a differential 3-Lie algebra. Moreover, we develop cohomology theory of a differential 3-Lie algebra. We also depict the relationship between the cohomologies of a differential 3-Lie algebra and its associated differential Leibniz algebra with weight zero. Formal deformations, abelian extensions and skeletal differential 3-Lie 2-algebras are characterized in terms of cohomology groups

show abstract

“…A difference operator can be viewed as a generalization of a derivation, and a difference Lie algebra can be viewed as a generalization of a LieDer pair introduced in [25], which consists of a Lie algebra and a derivation on it. Note that the deformation theory and the cohomology theory of LieDer pairs were studied in [25], and generalized to other algebraic structures [3,4]. In [6,16], the authors also study associative algebras with derivations from the operadic point of view.…”

Section: Introductionmentioning

confidence: 99%

Deformations, cohomologies and integrations of relative difference Lie algebras

Jiang¹,

Sheng²

2022

Preprint

View full text Add to dashboard Cite

In this paper, first using the higher derived brackets, we give the controlling algebra of relative difference Lie algebras, which are also called crossed homomorphisms or differential Lie algebras of weight 1 when the action is the adjoint action. Then using Getzler's twisted L ∞ -algebra, we define the cohomology of relative difference Lie algebras. In particular, we define the regular cohomology of difference Lie algebras by which infinitesimal deformations of difference Lie algebras are classified. We also define the cohomology of difference Lie algebras with coefficients in arbitrary representations, and using the second cohomology group to classify abelian extensions of difference Lie algebras. Finally, we show that any relative difference Lie algebra can be integrated to a relative difference Lie group in a functorial way. Contents 1. Introduction 1 2. The controlling algebras and cohomologies of LieAct triples and relative difference operators 3 2.1. The controlling algebra and cohomologies of LieAct triples 3 2.2. The controlling algebra and cohomologies of relative difference operators 4 3. Deformations and cohomologies of relative difference Lie algebras 6 3.1. L ∞ -algebras and the higher derived brackets 6 3.2. Deformations of relative difference Lie algebras 7 3.3. Cohomologies of relative difference Lie algebras 9 4. Cohomologies of difference Lie algebras and applications 11 4.1. Regular cohomologies of difference Lie algebras and infinitesimal deformations 11 4.2. Classification of abelian extensions of difference Lie algebras 13 5. Integrations of relative difference Lie algebras 18 References 20

show abstract

Extensions, deformation and categorification of $\text{AssDer}$ pairs

Cited by 7 publications

References 22 publications

Cohomologies and deformations of coassociative coderivations

Cohomologies and deformations of coassociative coderivations

Representations and cohomologies of differential 3-Lie algebras with any weight

Deformations, cohomologies and integrations of relative difference Lie algebras

Contact Info

Product

Resources

About