2015/07/06 by Selman Uǧuz, S. Uguz, I. A. Karimjanov +4 · 1 citation
Mathematics · #17A32 #17B10 #17B30 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:17A32 #msc:17B10 #msc:17B30
paper · pdf · doi:10.48550/arxiv.1507.01349
18 pages
openalex publication_date 2015/07/06 · arxiv created 2016/05/01 · arxiv updated 2016/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra \mathfrakD and the ideal generated by the squares of elements (further denoted by I) is a right \mathfrakD-module. Using description \citeCas of representations of algebra \mathfrakD in \mathfraksl(3,ℂ) and \mathfraksp(4,\mathbbF) where \mathbbF=ℝ or ℂ we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra \mathfrakD. Classification of Leibniz algebras with corresponding Lie algebra \mathfrakD and with the ideal I as a Fock right \mathfrakD-module is presented. The linear integrable deformations in terms of the second cohomology groups of obtained finite-dimensional Leibniz algebras are described. Two computer programs in Mathematica 10 which help to calculate for a given Leibniz algebra the general form of elements of spaces BL2 and ZL2 are constructed, as well.