Involutive and oriented dendriform algebras

Das, Apurba; Saha, Ripan

doi:10.1016/j.jalgebra.2021.04.012

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…In [3] Braun has shown that for involutive associative algebras, the ordinary Hochschild cohomology splits as a direct sum of involutive Hochschild cohomology and a skew-factor. This splitting theorem has been explicitly described in a recent paper by the present author [9] and further extended it to the dendriform context. Here we conclude a similar result for relative Rota-Baxter operators.…”

Section: 1mentioning

confidence: 68%

“…It has been shown in [9] that {iC • Hoch (A, M ), δ Hoch } is a subcomplex of the ordinary Hochschild complex and the cohomology of this subcomplex is called the Hochschild cohomology of the involutive algebra A with coefficients in the involutive bimodule M .…”

Section: (Relative) Rota-baxter Operators On Involutive Associative A...mentioning

confidence: 99%

“…An explicit description of the cohomology was given in [8]. Here we require the cohomology of involutive dendriform algebras given in [9]. Let C n be the set of first n natural numbers.…”

Section: 3mentioning

confidence: 99%

“…Then it has been shown in [9] that {iC • dend (D, D), δ π } is a subcomplex of the cochain complex {O(•), δ π }. The cohomology groups of this subcomplex are called the cohomology of the involutive dendriform algebra (D, ≺, ≻, * ) and they are denoted by iH • dend (D, D).…”

Section: 3mentioning

confidence: 99%

“…An interpretation of Braun's Hochschild cohomology is given by the authors in [11] using involutive Bar complex which led them to also introduce Hochschild homology of involutive associative algebras. Recently, with Saha, the present author gave a more explicit description of Hochschild cohomology of involutive associative algebras [9]. More precisely, they defined involutive dendriform algebras, their cohomology and find relations with the Hochschild cohomology of involutive associative algebras.…”

Section: Introductionmentioning

confidence: 99%

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Rota-Baxter operators on involutive associative algebras

Das¹

2020

Preprint

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In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras restricts to the involutive case.

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Section: 1mentioning

confidence: 68%

Section: (Relative) Rota-baxter Operators On Involutive Associative A...mentioning

confidence: 99%

“…An explicit description of the cohomology was given in [8]. Here we require the cohomology of involutive dendriform algebras given in [9]. Let C n be the set of first n natural numbers.…”

Section: 3mentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Rota-Baxter operators on involutive associative algebras

Das¹

2020

Preprint

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Cohomology and deformations of oriented dialgebras

Koam

Saha

2020

J. Homotopy Relat. Struct.

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Reflexive homology

Graves¹

2023

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$ -equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$ -equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$ -equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

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Involutive and oriented dendriform algebras

Cited by 5 publications

References 17 publications

Rota-Baxter operators on involutive associative algebras

Rota-Baxter operators on involutive associative algebras

Cohomology and deformations of oriented dialgebras

Reflexive homology

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