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

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.

Download Full-text

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

Georgian Mathematical Journal ◽

10.1515/gmj-2018-0069 ◽

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.

Download Full-text

Rota-Baxter Leibniz Algebras and Their Constructions

Advances in Mathematical Physics ◽

10.1155/2018/8540674 ◽

2018 ◽

Vol 2018 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Liangyun Zhang ◽

Linhan Li ◽

Huihui Zheng

Keyword(s):

Hopf Algebra ◽

Weak Hopf Algebra ◽

Associative Algebras ◽

Leibniz Algebras

In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

Download Full-text

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

International Journal of Algebra and Computation ◽

10.1142/s0218196718400027 ◽

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.

Download Full-text

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

Filomat ◽

10.2298/fil2005443f ◽

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.

Download Full-text

The Tensor Category of Linear Maps and Leibniz Algebras

Georgian Mathematical Journal ◽

10.1515/gmj.1998.263 ◽

1998 ◽

Vol 5 (3) ◽

pp. 263-276

Author(s):

J. L. Loday ◽

T. Pirashvili

Keyword(s):

Tensor Product ◽

Hopf Algebras ◽

Type Theorem ◽

Tensor Category ◽

Leibniz Algebra ◽

Vector Spaces ◽

Associative Algebras ◽

Enveloping Algebra ◽

Leibniz Algebras ◽

Linear Maps

Abstract We equip the category of linear maps of vector spaces with a tensor product which makes it suitable for various constructions related to Leibniz algebras. In particular, a Leibniz algebra becomes a Lie object in and the universal enveloping algebra functor UL from Leibniz algebras to associative algebras factors through the category of cocommutative Hopf algebras in . This enables us to prove a Milnor–Moore type theorem for Leibniz algebras.

Download Full-text