$A_\infty$-Algebras from Lie Pairs
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915820630704128 |
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| author | Stiénon, Mathieu Vitagliano, Luca Xu, Ping |
| author_facet | Stiénon, Mathieu Vitagliano, Luca Xu, Ping |
| contents | Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $Γ(Λ^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdotΓ(A)}$, where $R=C^\infty(M)$, admits an $A_\infty$-algebra structure, unique up to $A_\infty$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H^\bullet_{CE} \big( A, \frac{U(L)}{U(L)\cdotΓ(A)} \big)$ admits a canonical associative algebra structure. This $A_\infty$-algebra can be considered as the universal enveloping algebra of the $L_\infty$-algebroid $A[1]\times_M L/A$. Our construction is based on the homotopy equivalence of the $L_\infty$-algebroid $A[1]\times_M L/A$ and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz-Mackenzie. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16769 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | $A_\infty$-Algebras from Lie Pairs Stiénon, Mathieu Vitagliano, Luca Xu, Ping Differential Geometry Algebraic Geometry Quantum Algebra Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $Γ(Λ^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdotΓ(A)}$, where $R=C^\infty(M)$, admits an $A_\infty$-algebra structure, unique up to $A_\infty$-isomorphisms. As a consequence, the Chevalley-Eilenberg cohomology $H^\bullet_{CE} \big( A, \frac{U(L)}{U(L)\cdotΓ(A)} \big)$ admits a canonical associative algebra structure. This $A_\infty$-algebra can be considered as the universal enveloping algebra of the $L_\infty$-algebroid $A[1]\times_M L/A$. Our construction is based on the homotopy equivalence of the $L_\infty$-algebroid $A[1]\times_M L/A$ and the dg Lie algebroid corresponding to the comma double Lie algebroid of Jotz-Mackenzie. |
| title | $A_\infty$-Algebras from Lie Pairs |
| topic | Differential Geometry Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2210.16769 |