[Paper Review] A cohomological study of modified Rota-Baxter algebras

Apurba Das|arXiv (Cornell University)|Jul 5, 2022

Advanced Topics in Algebra4 citations

TL;DR

This paper introduces a cohomology theory for modified Rota-Baxter algebras, establishing a connection with Rota-Baxter algebra cohomology and applying it to classify formal deformations and abelian extensions. The key contribution is a cohomological framework that links modified Rota-Baxter algebras to associative algebra structures via an induced product $*_R$, enabling deformation and extension classification through Hochschild-type complexes.

ABSTRACT

A modified Rota-Baxter algebra is an algebra equipped with an operator that satisfies the modified Yang-Baxter equation. In this paper, we define the cohomology of a modified Rota-Baxter algebra with coefficients in a suitable bimodule. We relate our cohomology of a modified Rota-Baxter algebra with the known cohomology theory of a Rota-Baxter algebra. As applications of our cohomology, we study formal one-parameter deformations and abelian extensions of modified Rota-Baxter algebras.

Motivation & Objective

• To develop a cohomology theory for modified Rota-Baxter algebras with coefficients in bimodules.
• To relate the cohomology of modified Rota-Baxter algebras to the known cohomology of Rota-Baxter algebras.
• To apply the cohomology to classify formal one-parameter deformations and abelian extensions of modified Rota-Baxter algebras.
• To generalize deformation and extension theory from Rota-Baxter to modified Rota-Baxter algebras.
• To suggest future directions, including $L_∞$-algebra structures and homotopy theory for modified Rota-Baxter algebras.

Proposed method

• Define bimodules over modified Rota-Baxter algebras $(A,R)$ as pairs $(M,S)$ with compatibility conditions between $R$, $S$, and the $A$-bimodule structure on $M$.
• Construct a new associative algebra $A_R$ on the underlying space of $A$ using the product $a*_R b = R(a)\cdot b + a\cdot R(b)$.
• Endow the bimodule $M$ with an $A_R$-bimodule structure $\widetilde{M}$, enabling the use of Hochschild cochain complexes.
• Define the cohomology complex of $(A,R)$ with coefficients in $(M,S)$ as a combination of Hochschild complexes of $A$ and $A_R$, yielding $H^*_{\mathrm{mRBA}}((A,R),(M,S))$.
• Prove that the cohomology of a Rota-Baxter algebra $(A,P)$ of weight $\lambda$ is isomorphic to the cohomology of the modified Rota-Baxter algebra $(A,R=\lambda\mathrm{id}+2P)$ with weight $\kappa=-\lambda^2$.
• Use the cohomology to classify formal deformations and abelian extensions via 2-cocycles and equivalence classes.

Experimental results

Research questions

• RQ1 How can a cohomology theory be defined for modified Rota-Baxter algebras with coefficients in bimodules?
• RQ2 What is the relationship between the cohomology of modified Rota-Baxter algebras and the cohomology of Rota-Baxter algebras?
• RQ3 How can this cohomology be used to classify formal one-parameter deformations of modified Rota-Baxter algebras?
• RQ4 How can abelian extensions of modified Rota-Baxter algebras be classified using cohomological invariants?
• RQ5 What are the prospects for extending this theory to $L_\infty$-algebras, homotopy theory, and Lie or group-theoretic analogues?

Key findings

• The cohomology of a Rota-Baxter algebra $(A,P)$ of weight $\lambda$ is isomorphic to the cohomology of the modified Rota-Baxter algebra $(A,R=\lambda\mathrm{id}+2P)$ of weight $\kappa=-\lambda^2$.
• The underlying space $A$ of a modified Rota-Baxter algebra $(A,R)$ carries a new associative algebra structure $A_R$ defined by $a*_R b = R(a)\cdot b + a\cdot R(b)$.
• Any bimodule $(M,S)$ over a modified Rota-Baxter algebra $(A,R)$ inherits an $A_R$-bimodule structure $\widetilde{M}$, enabling the construction of a Hochschild cochain complex.
• Formal deformations of modified Rota-Baxter algebras are classified by the second cohomology group $H^2_{\mathrm{mRBA}}((A,R),(M,S))$.
• Abelian extensions of modified Rota-Baxter algebras are classified by the same second cohomology group, with a well-defined map from extension classes to cohomology classes.
• The classification is invariant under isomorphism of extensions, confirming the cohomological invariant is well-defined and canonical.