Cohomology and deformations of twisted Rota–Baxter operators and NS-algebras

Abstract

The aim of this paper is twofold. In the first part, we consider twisted Rota–Baxter operators on associative algebras that were introduced by Uchino as a noncommutative analogue of twisted Poisson structures. We construct an $L_\infty$-algebra whose Maurer–Cartan elements are given by twisted Rota–Baxter operators. This leads to cohomology associated to a twisted Rota–Baxter operator. This cohomology can be seen as the Hochschild cohomology of a certain associative algebra with coefficients in a suitable bimodule. We study deformations of twisted Rota–Baxter operators by means of the above-defined cohomology. Application is given to Reynolds operators. In the second part, we consider NS-algebras of Leroux that are related to twisted Rota–Baxter operators in the same way dendriform algebras are related to Rota–Baxter operators. We define cohomology of NS-algebras using non-symmetric operads and study their deformations in terms of the cohomology.

Appendix

In this appendix, we recall $L_\infty$-algebras and their Maurer–Cartan theory [27, 31, 32]. We will follow the sign conventions of [28].

Let $L = \bigoplus _i L_i$ be a graded vector space. A multilinear map $l : L^{\otimes k} \rightarrow L$ is said to be skew-symmetric if $l ( x_{\sigma (1)}, \ldots , x_{\sigma (k)} ) = (-1)^\sigma \epsilon (\sigma ) l (x_1, \ldots , x_k)$, for $\sigma \in S_k$. Here $\epsilon (\sigma )$ is the Koszul sign.

Definition 6.1

A $L_\infty$-algebra consists of a graded vector space $L = \bigoplus _i L_i$ together with a collection $\{ l_k : L^{\otimes k} \rightarrow L |~ \mathrm {deg}(l_k) = 2-k \}_{k \ge 1}$ of skew-symmetric multilinear maps satisfying the following higher Jacobi identities: for each $n \ge 1$, we have

$$\begin{aligned} \sum _{i+j = n+1}^{} \sum _{\sigma }^{} (-1)^\sigma \epsilon (\sigma ) ~ (-1)^{i (j-1)} ~ l_j \big ( l_i (x_{\sigma (1)}, \ldots , x_{\sigma (i)}), x_{\sigma (i+1)}, \ldots , x_{\sigma (n)} \big ) = 0, \end{aligned}$$

where $\sigma$ runs over all $(i, n-i)$-unshuffles with $i \ge 1$.

An element $\alpha \in L_1$ is called a Maurer–Cartan element in the $L_\infty$-algebra $(L, l_1, l_2, \ldots )$ if $\alpha$ satisfies

$$\begin{aligned} l_1 (\alpha ) + \frac{1}{2!} l_2 (\alpha , \alpha ) - \frac{1}{3!} l_3 (\alpha , \alpha , \alpha ) - \cdots = 0. \end{aligned}$$

If $\alpha$ is a Maurer–Cartan element, then one can construct a new $L_\infty$-algebra twisted by $\alpha$ on the graded vector space L. In particular, if the given $L_\infty$-algebra $(L, l_1, l_2, \ldots )$ has $l_i =0$, for $i=1$ and $i > 3$, then the structure maps for the twisted $L_\infty$-algebra are given by

$$\begin{aligned}&l_1' (x) = l_2 (\alpha , x) - \frac{1}{2}~ l_3 (\alpha , \alpha , x) ,~\quad l_2' (x,y) = l_2 (x, y) - l_3 (\alpha , x, y) \\&\text { and } ~~ l_3' (x, y, z)= l_3 (x, y, z). \end{aligned}$$