Deformations of Loday-type algebras and their morphisms

Das, Apurba

doi:10.1016/j.jpaa.2020.106599

Cited by 13 publications

(18 citation statements)

References 16 publications

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“…There are other Loday-type algebras (di-and triassociative algebras, dendriform trialgebras, cubical algebras [18]) which also has the same feature [23]. In [4] we study a formal deformation theory of multiplications in an operad, which gives deformation theory of all Loday-type algebras.…”

Section: 10mentioning

confidence: 99%

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Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebras

Das¹

2019

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A dendriform algebra is an associative algebra whose product splits into two binary operations and the associativity splits into three new identities. These algebras arise naturally from some combinatorial objects and through Rota-Baxter operators. In this paper, we start by defining cohomology of dendriform algebras with coefficient in a representation. The deformation of a dendriform algebra A is governed by the cohomology of A with coefficient in itself. Next we study Dend∞-algebras (dendriform algebras up to homotopy) in which the dendriform identities hold up to certain homotopy. They are certain splitting of A∞-algebras. We define Rota-Baxter operator on A∞-algebras which naturally gives rise to Dend∞-algebras. Finally, we classify skeletal and strict Dend∞-algebras.

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Section: 10mentioning

confidence: 99%

“…Strongly homotopy dendriform algebras. (Contents of section 4,5,6) The notion of A ∞space first arise in the work of Stasheff in the recognition of loop spaces [22]. The algebraic analogue of an A ∞ -space is called an A ∞ -algebra.…”

Section: Introductionmentioning

confidence: 99%

Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebras

Das¹

2019

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“…Formal deformation theory of associative algebras was first introduced by Gerstenhaber [7] and then extend to several other algebraic structures, such as Lie algebras, Leibniz algebras and many others [2,4,14,17]. In [8], Gerstenhaber and Schack defined the same theory for associative coalgebras.…”

Section: Introductionmentioning

confidence: 99%

Cohomology and deformations of dendriform coalgebras

Das¹

2019

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Dendriform coalgebras are the dual notion of dendriform algebras and are splitting of associative coalgebras. In this paper, we define a cohomology theory for dendriform coalgebras and show that the cohomology with self coefficients govern the formal deformation of the structure. We also relate this cohomology with the cohomology of dendriform algebras and coHochschild (Cartier) cohomology of associative coalgebras. Finally, we introduce and study a homotopy analogue of dendriform coalgebras.

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“…Deformations of associative algebras can be generalized to operads with multiplications [5]. Various Loday-type algebras (e.g.…”

Section: Discussionmentioning

confidence: 99%

Extensions and deformations of algebras with higher derivations

Das¹

2020

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Higher derivations on an associative algebra generalizes higher order derivatives. We call a tuple consisting of an algebra and a higher derivation on it by an AssHDer pair. We define a cohomology for AssHDer pairs with coefficients in a representation. Next, we study central extensions, abelian extensions of AssHDer pairs and relate them to the second cohomology group with appropriate coefficients. Deformations of AssHDer pairs are also considered which are governed by the cohomology with self coefficient.

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Deformations of Loday-type algebras and their morphisms

Cited by 13 publications

References 16 publications

Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebras

Cohomology and deformations of dendriform algebras, and $\mathrm{Dend}_\infty$-algebras

Cohomology and deformations of dendriform coalgebras

Extensions and deformations of algebras with higher derivations

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