scholarly journals Higher-dimensional kinematical Lie algebras via deformation theory

2018 ◽  
Vol 59 (6) ◽  
pp. 061702 ◽  
Author(s):  
José M. Figueroa-O’Farrill
Keyword(s):  
Lie Algebras ◽  
Deformation Theory ◽  
Higher Dimensional

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.

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

2005 ◽  
Vol 07 (02) ◽  
pp. 145-165 ◽  
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.

scholarly journals Kinematical Lie algebras via deformation theory

2018 ◽  
Vol 59 (6) ◽  
pp. 061701 ◽  
Author(s):  
José M. Figueroa-O’Farrill
Keyword(s):  
Lie Algebras ◽  
Deformation Theory

scholarly journals Special tensors in the deformation theory of quadratic algebras for the classical Lie algebras

2007 ◽  
Vol 57 (12) ◽  
pp. 2539-2546 ◽  
Author(s):  
Michael Eastwood ◽  
Petr Somberg ◽  
Vladimír Souček
Keyword(s):  
Lie Algebras ◽  
Deformation Theory ◽  
Quadratic Algebras

scholarly journals Homotopy Lie algebras and fundamental groups via deformation theory

1992 ◽  
Vol 42 (4) ◽  
pp. 905-935 ◽  
Author(s):  
Martin Markl ◽  
Stefan Papadima
Keyword(s):  
Lie Algebras ◽  
Deformation Theory ◽  
Fundamental Groups

scholarly journals Conformal Lie algebras via deformation theory

2019 ◽  
Vol 60 (2) ◽  
pp. 021702 ◽  
Author(s):  
José M. Figueroa-O’Farrill
Keyword(s):  
Lie Algebras ◽  
Deformation Theory

scholarly journals Geometric Models for Lie–Hamilton Systems on ℝ2

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1053
Author(s):  
Julia Lange ◽  
Javier de Lucas
Keyword(s):  
Lie Algebras ◽  
Poisson Structure ◽  
Quotient Space ◽  
Geometric Description ◽  
Natural Manner ◽  
Geometric Models ◽  
Hamilton System ◽  
Higher Dimensional ◽  
Symplectic Leaves ◽  
Lie Systems

This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

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