No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Biderivations of Low-Dimensional Leibniz Algebras
Abstract
In this paper we give a complete classification of the Leibniz algebras of biderivations of right Leibniz algebras of dimension up to three over a field F, with characteristic different than two. We describe the main properties of such class of Leibniz algebras and we also compute the biderivations of the four-dimensional Dieudonné Leibniz algebra d_1. Eventually we give an algorithm for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis.
... The first three equations state that the pair (l x , r x ) is a biderivation of the Leibniz algebra S, for any x ∈ L 0 . Biderivations of low-dimensional Leibniz algebras were classified in [20] and it turns out that ...
... , f n } of L n and we recall that the only non-trivial commutator is [e 2 , e 1 ] = e 1 . A straightforward application of the algorithm proposed in [20] for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis produces the following. ...
In this paper we study non-nilpotent non-Lie Leibniz F -algebras with one-dimensional derived subalgebra, where F is a field with char ( F ) ≠ 2 . We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by L n , where n = dim F L n . This generalizes the result found in Demir et al. (Algebras and Representation Theory 19:405-417, 2016), which is only valid when F = C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of L n . Eventually, we solve the coquecigrue problem for L n by integrating it into a Lie rack.
... The first three equations state that the pair (l x , r x ) is a biderivation of the Leibniz algebra S, for any x ∈ L 0 . Biderivations of low-dimensional Leibniz algebras were classified in [20] and it turns out that ...
... , f n } of L n and we recall that the only non-trivial commutator is [e 2 , e 1 ] = e 1 . A straightforward application of the algorithm proposed in [20] for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis produces the following. ...
In this paper we study non-nilpotent non-Lie Leibniz F-algebras with one-dimensional derived subalgebra, where F is a field with char(F) ≠ 2. We prove that such an algebra is isomorphic to the direct sum of the two dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by Ln, where n = dimF Ln. This generalizes the result found in [11], which is only valid when F = C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of Ln. Eventually, we solve the coquecigrue problem for Ln by integrating it into a Lie rack.
... The first three equations state that the pair (l x , r x ) is a biderivation of the Leibniz algebra S, for any x ∈ L 0 . Biderivations of low-dimensional Leibniz algebras were classified in [20] and it turns out that ...
... , f n } of L n and we recall that the only non-trivial commutator is [e 2 , e 1 ] = e 1 . A straightforward application of the algorithm proposed in [20] for finding derivations and anti-derivations of a Leibniz algebra as pair of matrices with respect to a fixed basis produces the following. ...
In this paper we study non-nilpotent non-Lie Leibniz F-algebras with one-dimensional derived subalgebra, where F is a field with char(F)≠2. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by L_n, where n=dimL_n. This generalizes the result found in [11], which is only valid when F=C . Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of L_n. Eventually, we solve the coquecigrue problem for L_n by integrating it into a Lie rack.
... The pair (− ad x , Ad x ) is called inner biderivation of g and the set of all inner biderivations forms a Leibniz subalgebra of Bider(g). We refer the reader to [14] for a complete classification of the Leibniz algebras of biderivations of low-dimensional Leibniz algebras over a general field F with char(F) = 2. ...
In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.
... The pair p´ad x , Ad x q is called inner biderivation of g and the set of all inner biderivations forms a Leibniz subalgebra of Biderpgq. We refer the reader to [14] for a complete classification of the Leibniz algebras of biderivations of low-dimensional Leibniz algebras over a general field F with charpFq ‰ 2. ...
In a recent paper, motivated by the study of central extensions of associative algebras, George Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly action representable and we characterize the class of acting morphisms. Moreover, we study the representability of actions of the category of Poisson algebras and we prove that the subvariety of commutative Poisson algebras is not weakly action representable.
We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field.
Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam.
Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object E(X) for the actions on (or split extensions of) an object X. We actually obtain a partial algebra E(X), which we call external weak actor of X, together with a monomorphism of functors SplExt(-,X) >--> Hom(-,E(X)), which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor USGA(X) and that of E(X) are described in detail. We end with some open questions.
This paper is devoted to the study of structure of Leibniz algebras. The classical methods to obtain classifications are essentially to solve a system of equations given by the identities of specified classes of algebras. In this work, some invariants were employed to obtain a complete list of three-dimensional Leibniz algebras over an arbitrary field and it is applied to classify three-dimensional Leibniz algebras over R, Z_3, Z_5, Z_7.
In this paper, we study non-abelian extensions of Leibniz algebras using two different approaches. First we construct two Leibniz 2-algebras using biderivations of Leibniz algebras, and show that under a condition on centers, a non-abelian extension of Leibniz algebras can be described by a Leibniz 2-algebra morphism. Furthermore, under this condition, non-abelian extensions are classified by homotopy classes of Leibniz 2-algebra morphisms. Then we give a description of non-abelian extensions of Leibniz algebras in terms of Maurer-Cartan elements in a differential graded Lie algebra.
We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimizing geodesics from identity to any given endpoint. (Several of these points have been covered, to varying degrees, by other authors.)
This paper deals with the low-dimensional cases of Leibniz algebras' derivations. We give an algorithm to find the derivation algebras. Then the algorithm is applied to find the basic derivations of Leibniz algebras. The characteristically nilpotency of Leibniz algebras has also been studied. The results are given in the form of tables.
For any category of interest ℂ we define a general category of groups with operations
\mathbbCG, \mathbbC\hookrightarrow\mathbbCG\mathbb{C_G}, \mathbb{C}\hookrightarrow\mathbb{C_G}, and a universal strict general actor USGA(A) of an object A in ℂ, which is an object of
\mathbbCG\mathbb{C_G}. The notion of actor is equivalent to the one of split extension classifier defined for an object in more general settings
of semi-abelian categories. It is proved that there exists an actor of A in ℂ if and only if the semidirect product
\textUSGA(A)\ltimes A{\text{USGA}}(A)\ltimes A is an object of ℂ and if it is the case, then USGA(A) is an actor of A. We give a construction of a universal strict general actor for any A ∈ ℂ, which helps to detect more properties of this object. The cases of groups, Lie, Leibniz, associative, commutative associative,
alternative algebras, crossed and precrossed modules are considered. The examples of algebras are given, for which always
exist actors.
In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation.
In this paper we give a complete classification of two-step nilpotent Leibniz algebras in terms of Kronecker modules associated with pairs of bilinear forms. Among these, we describe the complex and the real case of the indecomposable Heisenberg Leibniz algebras as a generalization of the classical (2n+1)-dimensional Heisenberg Lie algebra h2n+1. Then we use the Leibniz algebras - Lie local racks correspondence proposed by S. Covez to show that nilpotent real Leibniz algebras have always a global integration. As an application, we integrate the indecomposable nilpotent real Leibniz algebras with one-dimensional commutator ideal. Finally we show that every Lie quandle integrating a Leibniz algebra is induced by the conjugation of a Lie group and the Leibniz algebra is the Lie algebra of that Lie group.
Ideals and Homomorphisms.- Low-Dimensional Lie Algebras.- Solvable Lie Algebras and a Rough Classification.- Subalgebras of gl(V).- Engel's Theorem and Lie's Theorem.- Some Representation Theory.- Representations of sl(2, C).- Cartan's Criteria.- The Root Space Decomposition.- Root Systems.- The Classical Lie Algebras.- The Classification of Root Systems.- Simple Lie Algebras.- Further Directions.- Appendix A: Linear Algebra.- Appendix B: Weyl's Theorem.- Appendix C: Cartan Subalgebras.- Appendix D: Weyl Groups.- Appendix E: Answers to Selected Exercises.
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
We propose an algorithm using Gröbner bases that decides in terms of the existence of a non singular matrix P if two Leibniz algebra structures over a finite dimensional CC-vector space are representative of the same isomorphism class.We apply this algorithm in order to obtain a reviewed classification of the 3-dimensional Leibniz algebras given by Ayupov and Omirov. The algorithm has been implemented in a Mathematica notebook.
On 3-dimensional Leibniz algebras
- S Ayupov
- B Omirov
Ayupov S. and Omirov B., "On 3-dimensional Leibniz algebras", Uzbek Mathematical Journal 1 (1999), no. 7, pp. 9-14.
Categories for the working mathematician
- Mac Lane
Mac Lane S., "Categories for the working mathematician", Vol. 5 Graduate Texts
in Mathematics, Springer-Verlag, New York (2013), ISBN: 9781475747218.