arXiv subjects

Anita Naolekar

Publications and source records attributed to Anita Naolekar.

9 recordsLinked to original sources

arXiv2024

On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras

In this paper, we use the language of monads, comonads and Eilenberg-Moore categories to describe a categorical framework for $A_\infty$-algebras and $A_\infty$-coalgebras, as well as $A_\infty$-modules and $A_\infty$-comodules over them respectively. The resulting formalism leads us to investigate relations between representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras. In particular, we relate $A_\infty$-comodules and $A_\infty$-modules by considering a rational pairing between an $A_\infty$-coalgebra $C$ and an $A_\infty$-algebra $A$. The categorical framework also motivates us to introduce $A_\infty$-contramodules over an $A_\infty$-coalgebra $C$.

math.CT↗

arXiv2022

Adjunctions between Eilenberg-Moore categories and a PBW-type theorem

Recently, Dotsenko and Tamaroff have shown that a morphism of $T\longrightarrow S$ of monads over a category $\mathscr C$ satisfies the PBW-property if and only if it makes $S$ into a free right $T$-module. We consider an adjunction $Ψ=(G,F)$ between categories $\mathscr C$, $\mathscr D$, a monad $S$ on $\mathscr C$ and a monad $T$ on $\mathscr D$. We show that a morphism $ϕ:(\mathscr C,S)\longrightarrow (\mathscr D,T)$ that is well behaved with respect to the adjunction $Ψ$ has a PBW-property if and only if it makes $S$ satisfy a certain freeness condition with respect to $T$-modules with values in $\mathscr C$.

math.CT↗

arXiv2021

Cohomology and deformations of Filippov algebroids

In this article, we study the deformations of Filippov algebroids. We define a differential graded Lie algebra (in short DGLA) for a Filippov algebroid by introducing the notion of Filippov multiderivations for a vector bundle. Later on, we discuss deformations of a Filippov algebroid in terms of low-dimensional cohomology associated to this DGLA. We define Nijenhuis operators on Filippov algebroids and characterize trivial deformations of Filippov algebroids in terms of these operators. In the end, we define finite order deformations and discuss the problem of extending a given finite order deformation to a deformation of a higher order.

math.DG↗

arXiv2021

Noncommutative Differential Calculus Structure on Secondary Hochschild (co)homology

Let $B$ be a commutative algebra and $A$ be a $B$-algebra (determined by an algebra homomorphism $\varepsilon:B\rightarrow A$). M. D. Staic introduced a Hochschild like cohomology $H^{\bullet}((A,B,\varepsilon);A)$ called secondary Hochschild cohomology, to describe the non-trivial $B$-algebra deformations of $A$. J. Laubacher et al later obtained a natural construction of a new chain (and cochain) complex $\overline{C}_{\bullet}(A,B,\varepsilon)$ (resp. $\overline{C}^{\bullet}(A,B,\varepsilon)$) in the process of introducing the secondary cyclic (co)homology. It turns out that unlike the classical case of associative algebras (over a field), there exist different (co)chain complexes for the $B$-algebra $A$. In this paper, we establish a connection between the two (co)homology theories for $B$-algebra $A$. We show that the pair $\big(H^{\bullet}((A,B,\varepsilon);A),HH_{\bullet}(A,B,\varepsilon)\big)$ forms a non-commutative differential calculus, where $HH_{\bullet}(A,B,\varepsilon)$ denotes the homology of the complex $\overline{C}_{\bullet}(A,B,\varepsilon)$.

math.RA↗

arXiv2021

Weak comp algebras and cup products in secondary Hochschild cohomology of entwining structures

We define the secondary Hochschild complex for an entwining structure over a commutative $k$-algebra $B$. We show that this complex carries the structure of a weak comp algebra. We obtain two distinct cup product structures for the secondary cohomology groups. We also consider a subcomplex on which the two cup products coincide and which satisfies the axioms for being a comp algebra. The cohomology of this subcomplex then forms a Gerstenhaber algebra. We also construct a bicomplex that controls the deformations of the entwining structure over $B$.

math.RA↗

arXiv2020

$\mathcal{O}$-Operators on Hom-Lie algebras

$\mathcal{O}$-operators (also known as relative Rota-Baxter operators) on Lie algebras have several applications in integrable systems and the classical Yang-Baxter equations. In this article, we study $\mathcal{O}$-operators on hom-Lie algebras. We define cochain complex for $\mathcal{O}$-operators on hom-Lie algebras with respect to a representation. Any $\mathcal{O}$-operator induces a hom-pre-Lie algebra structure. We express the cochain complex of an $\mathcal{O}$-operator in terms of certain hom-Lie algebra cochain complex of the sub-adjacent hom-Lie algebra associated with the induced hom-pre-Lie algebra. If the structure maps in a hom-Lie algebra and its representation are invertible, then we can extend the above cochain complex to a deformation complex for $\mathcal{O}$-operators by adding the space of zero cochains. Subsequently, we study linear and formal deformations of $\mathcal{O}$-operators on hom-Lie algebras in terms of the deformation cohomology. In the end, we deduce deformations of $s$-Rota-Baxter operators (of weight 0) and skew-symmetric $r$-matrices on hom-Lie algebras as particular cases of $\mathcal{O}$-operators on hom-Lie algebras.

math.RA↗

arXiv2020

BV-operators and secondary Hochschild complex

We introduce the notion of a BV-operator $Δ=\{Δ^n:V^n\longrightarrow V^{n-1}\}_{n\geq 0}$ on a homotopy $G$-algebra $V^\bullet$ such that the Gerstenhaber bracket on $H(V^\bullet)$ is determined by $Δ$ in a manner similar to the BV-formalism. As an application, we produce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra $A$ over a commutative algebra $B$.

math.RA↗

arXiv2018

On n-Hom-Leibniz algebras and cohomology

The purpose of this paper is to provide a cohomology of $n$-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

math.RA↗

arXiv2013

Versal deformation theory of algebras over a quadratic operad

We develop deformation theory of algebras over quadratic operads where the parameter space is a commutative local algebra. We also give a construction of a distinguised deformation of an algebra over a quadratic operad with a complete local algebra as its base--the so called `versal deformation'--which induces all other deformations of the given algebra.

math.KT↗