The $L_\infty$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
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| Format: | Preprint |
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2020
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| _version_ | 1866913395062603776 |
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| author | Das, Apurba Mishra, Satyendra Kumar |
| author_facet | Das, Apurba Mishra, Satyendra Kumar |
| contents | A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$. Using Voronov's derived bracket and a recent work of Lazarev et al., we construct an $L_\infty [1]$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter algebras. By a standard twisting, we define a new $L_\infty [1]$-algebra that controls Maurer-Cartan deformations of a relative Rota-Baxter algebra $(A,M,T)$. We introduce the cohomology of a relative Rota-Baxter algebra $(A, M, T)$ and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of coboundary skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota-Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2008_11076 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The $L_\infty$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators Das, Apurba Mishra, Satyendra Kumar Rings and Algebras Representation Theory 16E40, 16S80, 16W99, 18G55 A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$. Using Voronov's derived bracket and a recent work of Lazarev et al., we construct an $L_\infty [1]$-algebra whose Maurer-Cartan elements are precisely relative Rota-Baxter algebras. By a standard twisting, we define a new $L_\infty [1]$-algebra that controls Maurer-Cartan deformations of a relative Rota-Baxter algebra $(A,M,T)$. We introduce the cohomology of a relative Rota-Baxter algebra $(A, M, T)$ and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of coboundary skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota-Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras. |
| title | The $L_\infty$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators |
| topic | Rings and Algebras Representation Theory 16E40, 16S80, 16W99, 18G55 |
| url | https://arxiv.org/abs/2008.11076 |