Maurer-Cartan equation in the DGLA of graded derivations
Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d* and i* components of D. Under the hypothesis that...
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2021
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oai:doaj.org-article:3d3765efb72548cd93f043031b2428822021-12-05T14:10:45ZMaurer-Cartan equation in the DGLA of graded derivations2300-744310.1515/coma-2020-0113https://doaj.org/article/3d3765efb72548cd93f043031b2428822021-06-01T00:00:00Zhttps://doi.org/10.1515/coma-2020-0113https://doaj.org/toc/2300-7443Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d* and i* components of D. Under the hypothesis that IdT(M) + Ψ is invertible we prove that Ψ=b(Ψ)=-12_(IdTM+Ψ)-1∘[Ψ,Ψ]ℱ𝒩{\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}}, where [·, ·]𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions eΨ= ℒΨ+𝒥b(Ψ) of the Maurer-Cartan equation according to their type: eΨ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒩 = 0 and r is minimal with this property, where [·, ·]𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution eΨ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1.de Bartolomeis PaoloIordan AndreiDe Gruyterarticledifferential graded lie algebrasmaurer-cartan equationfoliationsgraded derivationsprimary 32g1016w2553c12MathematicsQA1-939ENComplex Manifolds, Vol 8, Iss 1, Pp 183-195 (2021) |
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DOAJ |
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differential graded lie algebras maurer-cartan equation foliations graded derivations primary 32g10 16w25 53c12 Mathematics QA1-939 |
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differential graded lie algebras maurer-cartan equation foliations graded derivations primary 32g10 16w25 53c12 Mathematics QA1-939 de Bartolomeis Paolo Iordan Andrei Maurer-Cartan equation in the DGLA of graded derivations |
| description |
Let M be a smooth manifold and D = ℒΨ+𝒥Ψ a solution of the Maurer-Cartan equation in the DGLA of graded derivations D* (M) of differential forms on M, where Ψ, Ψ are differential 1-form on M with values in the tangent bundle TM and ℒΨ, 𝒥Ψ are the d* and i* components of D. Under the hypothesis that IdT(M) + Ψ is invertible we prove that Ψ=b(Ψ)=-12_(IdTM+Ψ)-1∘[Ψ,Ψ]ℱ𝒩{\rm{\Psi = }}b\left( {\rm{\Psi }} \right) = - {1 \over {}}{\left( {I{d_{TM}} + {\rm{\Psi }}} \right)^{ - 1}} \circ {\left[ {{\rm{\Psi }},{\rm{\Psi }}} \right]_{\mathcal{F}\mathcal{N}}}, where [·, ·]𝒩 is the Frölicher-Nijenhuis bracket. This yields to a classification of the canonical solutions eΨ= ℒΨ+𝒥b(Ψ) of the Maurer-Cartan equation according to their type: eΨ is of finite type r if there exists r∈ 𝒩 such that Ψr∘ [Ψ, Ψ]𝒩 = 0 and r is minimal with this property, where [·, ·]𝒩 is the Frölicher-Nijenhuis bracket. A distribution ξ ⊂TM of codimension k ⩾ 1 is integrable if and only if the canonical solution eΨ associated to the endomorphism Ψ of TM which is trivial on ξ and equal to the identity on a complement of ξ in TM is of finite type ⩽ 1, respectively of finite type 0 if k = 1. |
| format |
article |
| author |
de Bartolomeis Paolo Iordan Andrei |
| author_facet |
de Bartolomeis Paolo Iordan Andrei |
| author_sort |
de Bartolomeis Paolo |
| title |
Maurer-Cartan equation in the DGLA of graded derivations |
| title_short |
Maurer-Cartan equation in the DGLA of graded derivations |
| title_full |
Maurer-Cartan equation in the DGLA of graded derivations |
| title_fullStr |
Maurer-Cartan equation in the DGLA of graded derivations |
| title_full_unstemmed |
Maurer-Cartan equation in the DGLA of graded derivations |
| title_sort |
maurer-cartan equation in the dgla of graded derivations |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/3d3765efb72548cd93f043031b242882 |
| work_keys_str_mv |
AT debartolomeispaolo maurercartanequationinthedglaofgradedderivations AT iordanandrei maurercartanequationinthedglaofgradedderivations |
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1718371760282271744 |