Representation and central extension of hom-Lie algebroids

2018 ◽  
Vol 17 (11) ◽  
pp. 1850219 ◽  
Author(s):  
S. Merati ◽  
M. R. Farhangdoost
Keyword(s):  
Central Extension ◽  
Lie Algebroids ◽  
Central Extensions ◽  
Fundamental Properties ◽  
Direct Sum ◽  
One To One

The aim of this paper is to develop the theory of representation of hom-Lie algebroids. After introducing some key constructions and examples of hom-Lie algebroids involving sub-Lie hom-algebroids and direct sum hom-Lie algebroids, we describe the notion and some properties of infinitesimal action of hom-Lie algebroids. We introduce concept of representation of hom-Lie algebroids and prove some fundamental properties to show a one to one correspondence between representations and exterior differentials. Finally, we review trivial representations and its associated cohomology to introduce the central extensions. Also, we show that the central extensions induced by two trivial representations are isomorphic if their [Formula: see text]-forms are cohomologous.

scholarly journals Universal central extensions of Lie–Rinehart algebras

2018 ◽  
Vol 17 (07) ◽  
pp. 1850134 ◽  
Author(s):  
J. L. Castiglioni ◽  
X. García-Martínez ◽  
M. Ladra
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Central Extension ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Definition Of ◽  
Universal Central Extensions

In this paper, we study the universal central extension of a Lie–Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie–Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.

scholarly journals Invariant metrics on central extensions of quadratic Lie algebras

2019 ◽  
Vol 19 (12) ◽  
pp. 2050224
Author(s):  
R. García-Delgado ◽  
G. Salgado ◽  
O. A. Sánchez-Valenzuela
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bilinear Form ◽  
General Structure ◽  
Invariant Metrics ◽  
Invariant Metric ◽  
Central Extensions ◽  
Direct Sum ◽  
Quadratic Lie Algebra

A quadratic Lie algebra is a Lie algebra endowed with a symmetric, invariant and nondegenerate bilinear form; such a bilinear form is called an invariant metric. The aim of this work is to describe the general structure of those central extensions of quadratic Lie algebras which in turn have invariant metrics. The structure is such that the central extensions can be described algebraically in terms of the original quadratic Lie algebra, and geometrically in terms of the direct sum decompositions that the invariant metrics involved give rise to.

scholarly journals Central extensions and rational quadratic forms

1993 ◽  
Vol 130 ◽  
pp. 177-182 ◽  
Author(s):  
Yoshiomi Furuta ◽  
Tomio Kubota
Keyword(s):  
Central Extension ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Narrow Sense ◽  
Maximal Extension ◽  
Central Extensions ◽  
Class Field ◽  
Ray Class Field

The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as followswhere a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.Moreover is decomposed completely in if and only if .

scholarly journals A note on induced central extensions

1979 ◽  
Vol 20 (3) ◽  
pp. 411-420 ◽  
Author(s):  
L.R. Vermani
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Central Extension ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Central Extensions ◽  
Baer Sum ◽  
Necessary And Sufficient

A characterization of induced central extensions which gives an explicit relationship between induced central extensions and n-stem extensions is obtained. Using the characterization, necessary and sufficient conditions for a central extension of an abelian group by a nilpotent group of class n to be a Baer sum of an induced central extension and an extension of class n are obtained.

Lie triple systems and Leibniz algebras

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Revaz Kurdiani
Keyword(s):  
Lie Algebra ◽  
Central Extension ◽  
Triple System ◽  
Leibniz Algebra ◽  
Universal Central Extension ◽  
Central Extensions ◽  
Lie Triple System ◽  
Triple Systems ◽  
Leibniz Algebras ◽  
Universal Central Extensions

AbstractThe present paper deals with the Lie triple systems via Leibniz algebras. A perfect Lie algebra as a perfect Leibniz algebra and as a perfect Lie triple system is considered and the appropriate universal central extensions are studied. Using properties of Leibniz algebras, it is shown that the Lie triple system universal central extension is either the universal central extension of the Leibniz algebra or the universal central extension of the Lie algebra.

scholarly journals On central extensions of a Galois extension of algebraic number fields

1984 ◽  
Vol 93 ◽  
pp. 133-148 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Central Extension ◽  
Algebraic Closure ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Central Extensions

Let k be an algebraic number field of finite degree, and K a finite Galois extension of k. A central extension L of K/k is an algebraic number field which contains K and is normal over k, and whose Galois group over K is contained in the center of the Galois group Gal(L/k). We denote the maximal abelian extensions of k and K in the algebraic closure of k by kab and Kab respectively, and the maximal central extension of K/k by MCK/k. Then we have Kab⊃MCK/k⊃kab·K.

Universal smooth k-tame central extension

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Finite Type ◽  
Central Extension ◽  
Group Scheme ◽  
Reductive Groups ◽  
Central Extensions ◽  
Weil Restriction ◽  
Group A ◽  
Subgroup Scheme ◽  
Minimal Type ◽  
Simply Connected

This chapter describes the construction of canonical central extensions that are analogues for perfect smooth connected affine k-groups of the simply connected central cover of a connected semisimple k-group. A commutative affine k-group scheme of finite type is k-tame if it does not contain a nontrivial unipotent k-subgroup scheme. The chapter establishes good properties of the universal smooth k-tame central extension, noting that the property “locally of minimal type” is inherited by pseudo-reductive central quotients of pseudo-reductive groups. Although inseparable Weil restriction does not generally preserve perfectness, the chapter shows that the formation of the universal smooth k-tame central extension interacts with derived groups of Weil restrictions.

Projective Representations of Minimum Degree of Group Extensions

1978 ◽  
Vol 30 (5) ◽  
pp. 1092-1102 ◽  
Author(s):  
Walter Feit ◽  
Jacques Tits
Keyword(s):  
Simple Group ◽  
Central Extension ◽  
Finite Simple Group ◽  
Minimum Degree ◽  
Projective Representation ◽  
Closed Field ◽  
Natural Question ◽  
Central Extensions ◽  
Ordinary Representation ◽  
Projective Representations

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

Ganea Term for Homology of Leibniz n-Algebras

2005 ◽  
Vol 12 (04) ◽  
pp. 629-634 ◽  
Author(s):  
J. M. Casas
Keyword(s):  
Exact Sequence ◽  
Central Extension ◽  
Central Extensions ◽  
Leibniz Algebras

We extend the five-term exact sequence of homology with trivial coefficients of Leibniz n-algebras [Formula: see text] associated to a central extension of Leibniz n-algebras [Formula: see text] by means of a sixth term which is a generalization of the Ganea term for homology of Leibniz algebras. We use this sequence in order to analyze several questions related with the centre and central extensions of a Leibniz n-algebra.

On elements of order four in certain free central extensions of groups

1989 ◽  
Vol 106 (1) ◽  
pp. 13-28 ◽  
Author(s):  
Ralph Stöhr
Keyword(s):  
Finite Order ◽  
Central Extension ◽  
Commutator Subgroup ◽  
Metabelian Group ◽  
The Other ◽  
Central Extensions ◽  
Image Position ◽  
Order Four ◽  
Torsion Free ◽  
Number Of Authors

Let F be a non-cyclic free group, R a normal subgroup of F and G = F/R, i.e.where π is the natural projection of F onto G, is a free presentation of G. Let R′ denote the commutator subgroup of R. The quotient F/[R′,F] is a free central extensionof the group F/R′, the latter being a free abelianized extension of G. While F/R′ is torsion-free (see, e.g. [2], p. 23), elements of finite order may occur in R′/[R′,F], the kernel of the free central extension (l·2). Since C. K. Gupta [1] discovered elements of order 2 in the free centre-by-metabelian group F/[F″,F] (i.e. (1·2) in the case R = F′), torsion in F/[R′,F] has been studied by a number of authors (see, e.g. [4–13]). Clearly the elements of finite order in F/[R′,F] form a subgroup T of the abelian group R′/[R′,F]. It will be convenient to write T additively. By a result of Kuz'min [5], any element of T has order 2 or 4. Moreover, it was pointed out in [5] that elements of order 4 may really occur. On the other hand, it has been shown in [11] that, if G has no 2-torsion, then T is an elementary abelian 2-group isomorphic to H4(G, ℤ2). So if T contains an element of order 4, then G must have 2-torsion. We also mention a result of Zerck [13], who proved that 2T is an invariant of G, i.e. it does not depend on the particular choice of the free presentation (1·1).

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