On the connection between group cohomology and Lie algebras

1982 ◽  
Vol 37 (4) ◽  
pp. 123-124 ◽  
Author(s):  
Yu V Kuz'min
Keyword(s):  
Lie Algebras ◽  
Group Cohomology

scholarly journals EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS

2013 ◽  
Vol 05 (01) ◽  
pp. 57-85 ◽  
Author(s):  
MATTHEW B. DAY
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Lie Algebras ◽  
Mapping Class Group ◽  
Finite Rank ◽  
Class Group ◽  
Homogeneous Spaces ◽  
Group Cohomology ◽  
Mapping Class ◽  
Nilpotent Lie Algebras

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [5]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.

scholarly journals Transporting cohomology in Lazard correspondence

2017 ◽  
Vol 16 (06) ◽  
pp. 1750119
Author(s):  
Oihana Garaialde Ocaña ◽  
Jon González-Sánchez
Keyword(s):  
Lie Algebras ◽  
Group Cohomology ◽  
Nilpotency Class ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Relative Cohomology ◽  
Category Of Modules ◽  
Lazard Correspondence

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-[Formula: see text] groups of nilpotency class smaller than [Formula: see text] and finitely generated nilpotent [Formula: see text]-Lie algebras of nilpotency class smaller than [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] the group cohomology functors and the Lie cohomology functors respectively. The aim of this paper is to show that for [Formula: see text], [Formula: see text] and [Formula: see text], and for a given category of modules the cohomology functors [Formula: see text] and [Formula: see text] are naturally equivalent. A similar result is proved for [Formula: see text] with the relative cohomology groups.

Spectrum of group cohomology and support varieties

2013 ◽  
Vol 11 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Eric M. Friedlander
Keyword(s):  
Lie Algebras ◽  
Finite Groups ◽  
Equivariant Cohomology ◽  
Group Cohomology ◽  
Qualitative Description ◽  
Spectrum Of Group ◽  
Representations Of Finite Groups ◽  
Restricted Lie Algebras ◽  
Arbitrary Finite Group ◽  
Support Varieties

AbstractWe give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

Infinitely divisible projective representations of the Lie algebras

1972 ◽  
Vol 72 (3) ◽  
pp. 357-368 ◽  
Author(s):  
D. Mathon
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Algebraic Method ◽  
Group Cohomology ◽  
Continuity Condition ◽  
Group Representations ◽  
Infinitely Divisible ◽  
Projective Representations ◽  
One To One ◽  
Representations Of Lie Groups

Infinitely divisible group representations were first defined by Streater(1) as an important concept closely related to continuous tensor product. Araki(2) analysed the factorizable representations of Lie groups and obtained a generalization of the Levy–Khinchine formula. A similar concept for Lie algebras was defined and studied by Streater in (3). Although the definition is not strictly an infinitesimal analogue of infinitely divisible representations of Lie groups, the results of (3) in the cohomological formulation are very similar to Araki's main theorem. Parthasarathy and Schmidt(4) generalized the concept of infinite divisibifity to the projective representations of locally compact groups and obtained a one-to-one correspondence between infinitely divisible projective representations and 1-co-cycles in the group cohomology with coefficients in a Hubert space. A similar generalization for Lie algebras is studied in the present paper. Infinitely divisible projective representations of Lie algebras are studied by a purely algebraic method, independently of (4) (since not all our projective representations are necessarily integrable). As expected, a one-to-one relation is obtained between the infinitely divisible projective representations and 1-co-cycles in the cohomology on the corresponding enveloping algebra with coefficients in a Hilbert space. The present problem is simpler than the group case since there is no continuity condition on the multiplier in a Lie algebra. A similar algebraic method was used in a discussion of infinitely divisible representations of canonical anticommutation relations (9).

Lie Groups, Lie Algebras, Cohomology and some Applications in Physics

1995 ◽  
Author(s):  
Josi A. de Azcárraga ◽  
Josi M. Izquierdo
Keyword(s):  
Lie Groups ◽  
Lie Algebras

The rigidity of some solvable Lie algebras

2018 ◽  
Vol 2018 (2) ◽  
pp. 43-49
Author(s):  
R.K. Gaybullaev ◽  
Kh.A. Khalkulova ◽  
J.Q. Adashev
Keyword(s):  
Lie Algebras ◽  
Solvable Lie Algebras

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

scholarly journals Abelian extensions and crossed modules of Hom-Lie algebras

2020 ◽  
Vol 224 (3) ◽  
pp. 987-1008
Author(s):  
José Manuel Casas ◽  
Xabier García-Martínez
Keyword(s):  
Lie Algebras ◽  
Abelian Extensions ◽  
Crossed Modules
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