Classification of three dimensional complex Leibniz algebras

Abstract

Application of abelian holonomy formalism to the elementary theory of numbers J. Math. Phys. 53, 052303 (2012) Asymptotic behavior of the Verblunsky coefficients for the OPUC with a varying weight J. Math. Phys. 53, 043510 (2012) Construction of time-dependent dynamical invariants: A new approach J. Math. Phys. 53, 042104 (2012) Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes Abstract. The aim of this paper is to complete the classification of three-dimensional complex Leibniz algebras. The description of isomorphism classes of three-dimensional complex Leibniz algebras has been given by Ayupov and Omirov in 1999. However, we found that this list has a little redundancy. In this paper we apply a method which is more elegant and it gives the precise list of isomorphism classes of these algebras. We compare our list with that of Ayupov-Omirov and show the corrections which should be made.

Classification of 5-dimensional complex nilpotent Leibniz algebras

“…The lack of antisymmetry property in Leibniz algebras makes the problem of classification of non-Lie nilpotent Leibniz algebras even harder. The complete classification of non-Lie nilpotent Leibniz algebras over C of dimension n ≤ 4 is known (see [1], [2], [6], [7], [9], [12], [15]), and some partial results on higher dimensions (see [8], [14]). In this paper we give the classification of 5−dimensional non-Lie nilpotent Leibniz algebras using congruence classes of bilinear forms as in ([7], [8], [9]) for the classification of 5−dimensional non-Lie nilpotent Leibniz algebras with dim(A 2 ) = 3 and dim(Leib(A)) = 1.…”

Alternative description of three dimensional complex diassociative algebras with some constraints

“…In [10] ( 1 ) In [10] can be found the classification of complex diassociative algebras in dimension two, where author excluded consideration of cases D=Ann(D) and Ann(D)=0, according to theorem 3 [10]. In this paper we concern to apply the method used in [4], [5], [10] for three dimension diassociative algebras case. Onwards all algebras are supposed to be over the field of complex numbers C and all omitted products of basis vectors are supposed to be zero.…”

“…Description related results for low dimensional complex Leibniz algebras can be found in [4], [2], and [5].…”

“…We would like to note that in our recent papers we considered classification of diassociative algebras in dimension two and three [5], [9]. Where, we obtained list of diassociative algebras using Associative algebras in respective dimensions.…”

“…For further details please refer to [10]. In current paper we refer to [4], [5], [10] where classification of algebras has been obtained applying notion of Annihilators. In [10] ( 1 ) In [10] can be found the classification of complex diassociative algebras in dimension two, where author excluded consideration of cases D=Ann(D) and Ann(D)=0, according to theorem 3 [10].…”