On transversal infinitesimal automorphisms for harmonic foliations

1987 ◽  
Vol 24 (2) ◽  
Author(s):  
Philippe Tondeur ◽  
Gabor Toth
Keyword(s):  
Harmonic Foliations ◽  
Infinitesimal Automorphisms

scholarly journals On Lie algebras of vector fields with invariant submanifolds

1974 ◽  
Vol 55 ◽  
pp. 91-110 ◽  
Author(s):  
Akira Koriyama
Keyword(s):  
Lie Algebras ◽  
Compact Support ◽  
Vector Fields ◽  
Finite Dimensional ◽  
Invariant Submanifolds ◽  
Infinitesimal Automorphisms

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

scholarly journals The tangent complex and Hochschild cohomology of -rings

2012 ◽  
Vol 149 (3) ◽  
pp. 430-480 ◽  
Author(s):  
John Francis
Keyword(s):  
Lie Algebras ◽  
Deformation Theory ◽  
Hochschild Cohomology ◽  
Homology Theory ◽  
Configuration Spaces ◽  
Algebra Structure ◽  
Koszul Duality ◽  
Derived Algebraic Geometry ◽  
Factorization Homology ◽  
Infinitesimal Automorphisms

AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

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