L. M. Camacho

Naturally graded quasi-filiform Leibniz algebras

The classification of naturally graded quasi-filiform Lie algebras is known; they have the characteristic sequence (n-2,1,1) where n is the dimension of the algebra. In the present paper we deal with naturally graded quasi-filiform non-Lie-Leibniz algebras which are described by the characteristic sequence or . The first case has been studied in [Camacho, L.M., Gomez, J.R., Gonzalez, A.J., Omirov, B.A., 2006. Naturally graded 2-filiform Leibniz Algebra and its applications, preprint, MA1-04-XI06] and now, we complete the classification of naturally graded quasi-filiform Leibniz algebras. For this purpose we use the software Mathematica (the program used is explained in the last section).

The Classification of Naturally Graded p-Filiform Leibniz Algebras

In the present article the classification of n-dimensional naturally graded p-filiform (1 ≤ p ≤ n − 4) Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.

SOME PROPERTIES OF EVOLUTION ALGEBRAS

Abstract. The paper is devoted to the study of finite dimensional com-plex evolution algebras. The class of evolution algebras isomorphic toevolution algebras with Jordan form matrices is described. For finitedimensional complex evolution algebras the criterium of nilpotency is es-tablished in terms of the properties of corresponding matrices. Moreover,it is proved that for nilpotent n-dimensional complex evolution algebrasthe possible maximal nilpotency index is 1 + 2 n−1 . 1. IntroductionIn 20s and 30s of the last century a new object was introduced to math-ematics, which was the product of interactions between Mendelian geneticsand mathematics. Mendel established the basic laws for inheritance, which aresummarized as Mendel’s Law of Segregation and Mendel’s Law of IndependentAssortment. This laws were mathematically formulated by Serebrowsky [10],who was also the first to give an algebraic interpretation of the “ × ” sign,which indicated sexual reproduction. Later Glivenkov [6] used the notion ofMendelian algebras in his work. Also Kostitzin [7] independently introduceda “symbolic multiplication” to express Mendel’s laws. In his several papersEtherington [3]-[5] introduced the formal language of abstract algebra to thestudy of genetics. These algebras, in general, are non-associative.However, in the beginning of the XX century in genetics there were discov-ered several examples of inheritances, where traits do not segregate in accor-dance with Mendel’s laws. In the present day, non-Mendelian genetics is a basiclanguage of molecular genetics. Non-Mendelian inheritance plays an importantrole in several disease processes. Naturally, the question arises: What non-Mendelian genetics offers to mathematics? The evolution algebras, introducedin [12] serve as the answer to this question.The concept of evolution algebras lies between algebras and dynamical sys-tems. Algebraically, evolution algebras are non-associative Banach algebra;dynamically, they represent discrete dynamical systems. Evolution algebras

ON NILPOTENT LEIBNIZ n-ALGEBRAS

We study the nilpotency of Leibniz n-algebras related with the adapted version of Engels theorem to Leibniz n-algebras. We also deal with the characterization of finite-dimensional nilpotent complex Leibniz n-algebras.

3-Filiform Leibniz algebras of maximum length, whose naturally graded algebras are Lie algebras

In this article we present the classification of the 3-filiform Leibniz algebras of maximum length, whose associated naturally graded algebras are Lie algebras. Our main tools are a previous existence result by Cabezas and Pastor [J.M. Cabezas and E. Pastor, Naturally graded p-filiform Lie algebras in arbitrary finite dimension, J. Lie Theory 15 (2005), pp. 379–391] and the construction of appropriate homogeneous bases in the connected gradation considered. This is a continuation of the work done in Ref. [J.M. Cabezas, L.M. Camacho, and I.M. Rodríguez, On filiform and 2-filiform Leibniz algebras of maximum length, J. Lie Theory 18 (2008), pp. 335–350].

NATURALLY GRADED 2-FILIFORM LEIBNIZ ALGEBRAS.

In the present article the classification of naturally graded 2-filiform Leibniz algebras is obtained. A splitting of the set of naturally graded Leibniz algebras into the families of Lie and non-Lie Leibniz algebras by means of characteristic sequences (isomorphism invariants) is proved.

Leibniz algebras associated with representations of filiform Lie algebras

Abstract In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra n n , 1 . We introduce a Fock module for the algebra n n , 1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra n n , 1 with condition that the ideal I is a Fock n n , 1 -module, where I is the ideal generated by squares of elements from L . We also consider Leibniz algebras with corresponding Lie algebra n n , 1 and such that the action I × n n , 1 → I gives rise to a minimal faithful representation of n n , 1 . The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4 .

THE DERIVATIONS OF SOME EVOLUTION ALGEBRAS

In this work, we investigate the derivations of n-dimensional complex evolution algebras, depending on the rank of the appropriate matrices. For evolution algebra with non-singular matrices we prove that the space of derivations is zero. The spaces of derivations for evolution algebras with matrices of rank n − 1 are described.

Quasi-filiform Leibniz algebras of maximum length

The n-dimensional p-filiform Leibniz algebras of maximum length have already been studied with 0 ≤ p ≤ 2. For Lie algebras whose nilindex is equal to n−2 there is only one characteristic sequence, (n − 2, 1, 1), while in Leibniz theory we obtain the two possibilities: (n − 2, 1, 1) and (n − 2, 2). The first case (the 2-filiform case) is already known. The present paper deals with the second case, i.e., quasi-filiform non-Lie-Leibniz algebras of maximum length. Therefore this work completes the study of the maximum length of the Leibniz algebras with nilindex n − p with 0 ≤ p ≤ 2.

Leibniz Algebras Associated to Extensions of sl 2

In this work, we investigate the structure of Leibniz algebras whose associated Lie algebra is a direct sum of sl 2 and the solvable radical. In particular, we obtain the description of such algebras when: the ideal generated by the squares of elements of a Leibniz algebra is irreducible over sl 2 and when the dimension of the radical is equal to two.