Emzar Khmaladze - Academia.edu (original) (raw)

Papers by Emzar Khmaladze

Centro de Matemática da Universidade de Coimbra eBooks, 2009

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebra... more From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

Journal of Pure and Applied Algebra, Sep 1, 2005

The tripleability of the category of crossed n-cubes is studied. The leading cotriple homology of... more The tripleability of the category of crossed n-cubes is studied. The leading cotriple homology of these homotopy (n + 1)-types is investigated, describing it as Hopf type formulas.

arXiv (Cornell University), Dec 23, 2016

We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investi... more We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.

Journal of Pure and Applied Algebra, Jun 1, 2010

We fit the homology with trivial coefficients of Leibniz n-algebras into the context of Quillen h... more We fit the homology with trivial coefficients of Leibniz n-algebras into the context of Quillen homology and provide the Hopf type formula for the higher homology.

Homology, Homotopy and Applications, 2014

We construct a pair of adjoint functors between the categories of crossed modules of Lie and asso... more We construct a pair of adjoint functors between the categories of crossed modules of Lie and associative algebras, which extends the classical one between the categories of Lie and associative algebras. This result is used to establish an equivalence of categories of modules over a Lie crossed module and its universal enveloping crossed module.

arXiv (Cornell University), Apr 21, 2021

arXiv (Cornell University), Jan 25, 2019

We study the capability property of Leibniz algebras via the non-abelian exterior product.

Journal of Geometry and Physics

Communications in Algebra, Aug 1, 2006

ABSTRACT The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical r... more ABSTRACT The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical results of solvable and nilpotent Lie algebras theory are extended to Leibniz n-algebras category. A homological criterion similar to Stallings Theorem for Lie algebras is obtained in Leibniz n-algebras category by means of the homology with trivial coefficients of Leibniz n-algebras.

Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie... more Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie algebra homology with Λ/qΛ coefficients is obtained, where Λ is a ground ring and q is a nonnegative integer. Hopf formulas for the second and third homology of a Lie algebra are proved. The condition for the existence and the description of the universal q-central relative extension of a Lie epimorphism in terms of relative homologies are given.

Extracta mathematicae, 2002

Georgian Mathematical Journal, Jul 16, 2020

We study the capability property of Leibniz algebras via the non-abelian exterior product.

Proceedings of the American Mathematical Society, Sep 1, 2010

By using purely algebraic methods of n-foldČech derived functors, the higher equivariant integral... more By using purely algebraic methods of n-foldČech derived functors, the higher equivariant integral group homology is investigated from the Hopf formulas point of view.

Homology, Homotopy and Applications, 1999

The notions of tensor end exterior products modulo q of two crossed P-modules, where q is a posit... more The notions of tensor end exterior products modulo q of two crossed P-modules, where q is a positive integer and P is a Lie algebra, are introduced and some properties are established. The condition for the existence of a universal q-central relative extension of a Lie epimorphism is given and this extension is described as an exterior product modulo q.

Journal of Algebra and Its Applications, Apr 12, 2017

Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are... more Adjoint functors between the categories of crossed modules of 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.

Journal of Algebra, Oct 1, 2015

We introduce the non-abelian tensor product of Lie superalgebras and study some of its properties... more We introduce the non-abelian tensor product of Lie superalgebras and study some of its properties. We use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

Homology, Homotopy and Applications, 2011

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebra... more From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

Forum Mathematicum, 2008

We introduce crossed modules for the category of Leibniz n-algebras and prove that they are equiv... more We introduce crossed modules for the category of Leibniz n-algebras and prove that they are equivalent to cat 1-Leibniz n-algebras and internal categories in Leibniz n-algebras. We interpret the set of equivalence classes of crossed extensions as the second cohomology of Leibniz n-algebras developed in [5].

arXiv (Cornell University), Jun 2, 2023

Given a non-negative integer q, we study two different notions of the qcapability of Lie algebras... more Given a non-negative integer q, we study two different notions of the qcapability of Lie algebras via the non-abelian q-exterior product of Lie algebras. The first is related to the q-crossed modules and inner q-derivations, and the second is the Lie algebra version of the q-capability of groups proposed by Ellis in 1995. 2010 Mathematics Subject Classification. 18G10, 18G50. Key words and phrases. Lie algebra, capable Lie algebra, non-abelian q-tensor and q-exterior products, tensor and exterior centers.

Homology, Homotopy and Applications, 2000

Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonab... more Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed P-module which shows that for q = 0 our definition is a natural extension of Conduché and Ellis' definition [CE]. Some other properties of homologies of precrossed P-modules are investigated.

Centro de Matemática da Universidade de Coimbra eBooks, 2009

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebra... more From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

Journal of Pure and Applied Algebra, Sep 1, 2005

The tripleability of the category of crossed n-cubes is studied. The leading cotriple homology of... more The tripleability of the category of crossed n-cubes is studied. The leading cotriple homology of these homotopy (n + 1)-types is investigated, describing it as Hopf type formulas.

arXiv (Cornell University), Dec 23, 2016

We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investi... more We introduce a non-abelian exterior product of two crossed modules of Leibniz algebra and investigate its relation to the low dimensional Leibniz homology. Later this non-abelian exterior product is applied to the construction of eight term exact sequence in Leibniz homology. Also its relationship to the universal quadratic functor is established, which is applied to the comparison of the second Lie and Leibniz homologies of a Lie algebra.

Journal of Pure and Applied Algebra, Jun 1, 2010

We fit the homology with trivial coefficients of Leibniz n-algebras into the context of Quillen h... more We fit the homology with trivial coefficients of Leibniz n-algebras into the context of Quillen homology and provide the Hopf type formula for the higher homology.

Homology, Homotopy and Applications, 2014

We construct a pair of adjoint functors between the categories of crossed modules of Lie and asso... more We construct a pair of adjoint functors between the categories of crossed modules of Lie and associative algebras, which extends the classical one between the categories of Lie and associative algebras. This result is used to establish an equivalence of categories of modules over a Lie crossed module and its universal enveloping crossed module.

arXiv (Cornell University), Apr 21, 2021

arXiv (Cornell University), Jan 25, 2019

We study the capability property of Leibniz algebras via the non-abelian exterior product.

Journal of Geometry and Physics

Communications in Algebra, Aug 1, 2006

ABSTRACT The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical r... more ABSTRACT The concepts of solvable and nilpotent Leibniz n-algebra are introduced, and classical results of solvable and nilpotent Lie algebras theory are extended to Leibniz n-algebras category. A homological criterion similar to Stallings Theorem for Lie algebras is obtained in Leibniz n-algebras category by means of the homology with trivial coefficients of Leibniz n-algebras.

Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie... more Using the long exact sequence of nonabelian derived functors, an eight term exact sequence of Lie algebra homology with Λ/qΛ coefficients is obtained, where Λ is a ground ring and q is a nonnegative integer. Hopf formulas for the second and third homology of a Lie algebra are proved. The condition for the existence and the description of the universal q-central relative extension of a Lie epimorphism in terms of relative homologies are given.

Extracta mathematicae, 2002

Georgian Mathematical Journal, Jul 16, 2020

We study the capability property of Leibniz algebras via the non-abelian exterior product.

Proceedings of the American Mathematical Society, Sep 1, 2010

By using purely algebraic methods of n-foldČech derived functors, the higher equivariant integral... more By using purely algebraic methods of n-foldČech derived functors, the higher equivariant integral group homology is investigated from the Hopf formulas point of view.

Homology, Homotopy and Applications, 1999

The notions of tensor end exterior products modulo q of two crossed P-modules, where q is a posit... more The notions of tensor end exterior products modulo q of two crossed P-modules, where q is a positive integer and P is a Lie algebra, are introduced and some properties are established. The condition for the existence of a universal q-central relative extension of a Lie epimorphism is given and this extension is described as an exterior product modulo q.

Journal of Algebra and Its Applications, Apr 12, 2017

Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are... more Adjoint functors between the categories of crossed modules of 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.

Journal of Algebra, Oct 1, 2015

We introduce the non-abelian tensor product of Lie superalgebras and study some of its properties... more We introduce the non-abelian tensor product of Lie superalgebras and study some of its properties. We use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.

Homology, Homotopy and Applications, 2011

From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebra... more From the viewpoint of semi-abelian homology, some recent results on homology of Leibniz n-algebras can be explained categorically. In parallel with these results, we develop an analogous theory for Lie n-algebras. We also consider the relative case: homology of Leibniz n-algebras relative to the subvariety of Lie n-algebras.

Forum Mathematicum, 2008

We introduce crossed modules for the category of Leibniz n-algebras and prove that they are equiv... more We introduce crossed modules for the category of Leibniz n-algebras and prove that they are equivalent to cat 1-Leibniz n-algebras and internal categories in Leibniz n-algebras. We interpret the set of equivalence classes of crossed extensions as the second cohomology of Leibniz n-algebras developed in [5].

arXiv (Cornell University), Jun 2, 2023

Given a non-negative integer q, we study two different notions of the qcapability of Lie algebras... more Given a non-negative integer q, we study two different notions of the qcapability of Lie algebras via the non-abelian q-exterior product of Lie algebras. The first is related to the q-crossed modules and inner q-derivations, and the second is the Lie algebra version of the q-capability of groups proposed by Ellis in 1995. 2010 Mathematics Subject Classification. 18G10, 18G50. Key words and phrases. Lie algebra, capable Lie algebra, non-abelian q-tensor and q-exterior products, tensor and exterior centers.

Homology, Homotopy and Applications, 2000

Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonab... more Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonabelian derived functors, where q is a nonnegative integer. The Hopf formula is proved for the second homology group modulo q of a precrossed P-module which shows that for q = 0 our definition is a natural extension of Conduché and Ellis' definition [CE]. Some other properties of homologies of precrossed P-modules are investigated.