Homotopy Lie algebras and fundamental groups via deformation theory

Martin Markl; Stefan Papadima

doi:10.5802/aif.1315

Schottky groups acting on homogeneous rational manifolds

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2016-0065 ◽

2019 ◽

Vol 2019 (753) ◽

pp. 23-56 ◽

Cited By ~ 1

Author(s):

Christian Miebach ◽

Karl Oeljeklaus

Keyword(s):

Deformation Theory ◽

Group Actions ◽

Picard Group ◽

Schottky Group ◽

Fundamental Groups ◽

Complex Manifolds ◽

Schottky Groups ◽

Algebraic Dimension ◽

Geometric Invariants ◽

Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

Download Full-text

On the role of semisimple subalgebras in the deformation theory of lie algebras and their homomorphisms general theory and applications

Reports on Mathematical Physics ◽

10.1016/0034-4877(73)90001-3 ◽

1973 ◽

Vol 4 (4) ◽

pp. 255-274 ◽

Cited By ~ 1

Author(s):

U. Cattaneo

Keyword(s):

General Theory ◽

Lie Algebras ◽

Deformation Theory

Download Full-text

Holonomy Lie algebras, logarithmic connections and the lower central series of fundamental groups

Singularities - Contemporary Mathematics ◽

10.1090/conm/090/1000601 ◽

1989 ◽

pp. 171-182 ◽

Cited By ~ 5

Author(s):

Toshitake Kohno

Keyword(s):

Lie Algebras ◽

Lower Central Series ◽

Central Series ◽

Fundamental Groups

Download Full-text

The tangent complex and Hochschild cohomology of -rings

Compositio Mathematica ◽

10.1112/s0010437x12000140 ◽

2012 ◽

Vol 149 (3) ◽

pp. 430-480 ◽

Cited By ~ 26

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.

Download Full-text

VERSAL DEFORMATIONS OF THREE-DIMENSIONAL LIE ALGEBRAS AS L∞ ALGEBRAS

Communications in Contemporary Mathematics ◽

10.1142/s0219199705001702 ◽

2005 ◽

Vol 07 (02) ◽

pp. 145-165 ◽

Cited By ~ 15

Author(s):

ALICE FIALOWSKI ◽

MICHAEL PENKAVA

Keyword(s):

Vector Space ◽

Lie Algebras ◽

Moduli Space ◽

Deformation Theory ◽

Three Dimensional ◽

3 Dimensional ◽

Exterior Degree ◽

Versal Deformations

We consider versal deformations of 0|3-dimensional L∞ algebras, also called strongly homotopy Lie algebras, which correspond precisely to ordinary (non-graded) three-dimensional Lie algebras. The classification of such algebras is well-known, although we shall give a derivation of this classification using an approach of treating them as L∞ algebras. Because the symmetric algebra of a three-dimensional odd vector space contains terms only of exterior degree less than or equal to three, the construction of versal deformations can be carried out completely. We give a characterization of the moduli space of Lie algebras using deformation theory as a guide to understanding the picture.

Download Full-text