scholarly journals From simplicial homotopy to crossed module homotopy in modified categories of interest

2020 ◽  
Vol 27 (4) ◽  
pp. 541-556
Author(s):  
Kadir Emir ◽  
Selim Çetin
Keyword(s):  
Lie Algebras ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Algebraic Structures ◽  
Homotopy Equivalence ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules ◽  
Simplicial Objects ◽  
Simplicial Homotopy

AbstractWe address the (pointed) homotopy of crossed module morphisms in modified categories of interest that unify the notions of groups and various algebraic structures. We prove that the homotopy relation gives rise to an equivalence relation as well as to a groupoid structure with no restriction on either domain or co-domain of the corresponding crossed module morphisms. Furthermore, we also consider particular cases such as crossed modules in the categories of associative algebras, Leibniz algebras, Lie algebras and dialgebras of the unified homotopy definition. Finally, as one of the major objectives of this paper, we prove that the functor from simplicial objects to crossed modules in modified categories of interest preserves the homotopy as well as the homotopy equivalence.

SEMI-COMPLETE CROSSED MODULES OF LIE ALGEBRAS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250096
Author(s):  
A. AYTEKIN ◽  
J. M. CASAS ◽  
E. Ö. USLU
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Crossed Module ◽  
Crossed Modules ◽  
Sufficient And Necessary Conditions

We investigate some sufficient and necessary conditions for (semi)-completeness of crossed modules in Lie algebras and we establish its relationships with the holomorphy of a crossed module. When we consider Lie algebras as crossed modules, then we recover the corresponding classical results for complete Lie algebras.

Some new results on Gröbner–Shirshov bases for Lie algebras and around

2018 ◽  
Vol 28 (08) ◽  
pp. 1403-1423
Author(s):  
L. A. Bokut ◽  
Yuqun Chen ◽  
Abdukadir Obul
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Leibniz Algebras ◽  
Novikov Algebras

We review Gröbner–Shirshov bases for Lie algebras and survey some new results on Gröbner–Shirshov bases for [Formula: see text]-Lie algebras, Gelfand–Dorfman–Novikov algebras, Leibniz algebras, etc. Some applications are given, in particular, some characterizations of extensions of groups, associative algebras and Lie algebras are given.

scholarly journals Braiding for categorical algebras and crossed modules of algebras II: Leibniz algebras

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1443-1469
Author(s):  
Alejandro Fernández-Fariña ◽  
Manuel Ladra
Keyword(s):  
Lie Algebras ◽  
Leibniz Algebras ◽  
Crossed Modules

In this paper, we study the category of braided categorical Leibniz algebras and braided crossed modules of Leibniz algebras, and we relate these structures with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras using the Loday-Pirashvili category.

CHARACTERIZING n-ISOCLINIC CLASSES OF CROSSED MODULES

2018 ◽  
Vol 61 (03) ◽  
pp. 637-656
Author(s):  
HAJAR RAVANBOD ◽  
ALI REZA SALEMKAR ◽  
SAJEDEH TALEBTASH
Keyword(s):  
Equivalence Relation ◽  
Crossed Module ◽  
Basic Properties ◽  
Crossed Modules

AbstractIn this paper, we introduce the notion of the equivalence relation, called n-isoclinism, between crossed modules of groups, and give some basic properties of this notion. In particular, we obtain some criteria under which crossed modules are n-isoclinic. Also, we present the notion of n-stem crossed module and, under some conditions, determine them within an n-isoclinism class.

scholarly journals Hamiltonian operators and related differential-algebraic Balinsky-Novikov, Riemann and Leibniz type structures on nonassociative noncommutative algebras

2019 ◽  
Vol 12 (4) ◽  
Author(s):  
Orest Artemovych ◽  
Alexandr Balinsky ◽  
Anatolij Prykarpatski
Keyword(s):  
Lie Algebras ◽  
Structural Properties ◽  
Poisson Manifold ◽  
Structural Constraints ◽  
Algebraic Structures ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Noncommutative Algebras ◽  
The Right ◽  
Hamiltonian Operators

We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative  noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson  structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. \ As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed investigation both of their structural properties and finite-dimensional representations of the right Leibniz algebras defined by the corresponding structural constraints. \ Subject to these important  aspects we stop in the work mostly on the structural properties of the right Leibniz algebras, especially on their derivation algebras and their generalizations. We have also added a short Supplement within which we \ revisited \ the classical Poisson manifold approach, closely related to our construction of \ Hamiltonian operators, generated by nonassociative and noncommutative algebras. In particular, \ we presented its natural and simple generalization allowing effectively to describe  a wide class\ of Lax type integrable nonlinear Kontsevich type Hamiltonian systems on associative noncommutative algebras.

scholarly journals More on crossed modules in Lie, Leibniz, associative and diassociative algebras

2017 ◽  
Vol 16 (06) ◽  
pp. 1750107 ◽  
Author(s):  
J. M. Casas ◽  
R. F. Casado ◽  
E. Khmaladze ◽  
M. Ladra
Keyword(s):  
Associative Algebras ◽  
Adjoint Functors ◽  
Leibniz Algebras ◽  
Crossed Modules

Adjoint functors between the categories of crossed modules in dialgebras and Leibniz algebras are constructed. The well known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.

Isoclinism of crossed modules

2016 ◽  
Vol 09 (03) ◽  
pp. 1650091 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Hamid Mohammadzadeh ◽  
Saeed Shahrokhi
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Crossed Module ◽  
Direct Products ◽  
Group Case ◽  
Crossed Modules ◽  
Equivalent Conditions

In this paper, we introduce the notion of the equivalence relation, isoclinism, on crossed modules of groups, and give some equivalent conditions for crossed modules to be isoclinic. In particular, it is shown that if two crossed modules [Formula: see text] and [Formula: see text] are isoclinic then [Formula: see text] can be constructed from [Formula: see text] and vice versa using the operations of forming direct products, taking crossed submodules, and factoring crossed modules, which generalizes the work of Weichsel. Also, similar to a result of Hall in the group case, we show that any equivalence class of crossed modules contains at least one stem crossed module.

scholarly journals Topics on n-ary Algebraic Structures

10.14311/1179 ◽  
2010 ◽  
Vol 50 (3) ◽  
Author(s):  
J. A. de Azcárraga ◽  
J. M. Izquierdo
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Structures ◽  
Central Extensions ◽  
Generalized Poisson ◽  
Leibniz Algebras ◽  
Infinitesimal Deformations ◽  
Generalized Lie Algebras

We review the basic definitions and properties of two types of n-ary structures, the Generalized Lie Algebras (GLA) and the Filippov (≡ n-Lie) algebras (FA), as well as those of their Poisson counterparts, the Generalized Poisson (GPS) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology complexes relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends Whitehead’s lemma to all n ≥ 2, n = 2 being the original Lie algebra case. Some comments onn-Leibniz algebras are also made.

scholarly journals Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras

2019 ◽  
Vol 19 (09) ◽  
pp. 2050176
Author(s):  
A. Fernández-Fariña ◽  
M. Ladra
Keyword(s):  
Lie Algebras ◽  
Associative Algebras ◽  
Crossed Modules

In this paper, the categories of braided categorical associative algebras and braided crossed modules of associative algebras are studied. These structures are also correlated with the categories of braided categorical Lie algebras and braided crossed modules of Lie algebras.

Extensions of Lie Algebras and the Third Cohomology Group

1953 ◽  
Vol 5 ◽  
pp. 470-476 ◽  
Author(s):  
S. I. Goldberg
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Cohomology Group ◽  
Algebraic Structures ◽  
Cohomology Groups ◽  
Associative Algebras ◽  
The Third

Cohomology theories of various algebraic structures have been investigated by several authors. The most noteworthy are due to Hochschild, MacLane and Eckmann, Chevalley and Eilenberg, who developed the theory of cohomology groups of associative algebras, abstract groups, and Lie algebras respectively. In this paper we are concerned primarily with a characterization of the third cohomology group of a Lie algebra by its extension properties.

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