Leibniz Algebras Associated with Representations of the Diamond Lie Algebra

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• 作者：Selman Uguz ; Iqbol A. Karimjanov ; Bakhrom A. Omirov
• 关键词：Diamond lie algebra ; Leibniz algebra ; Representation of diamond lie algebra ; Fock representation ; Heisenberg lie algebra ; Linear deformation ; The second group of cohomology
• 刊名：Algebras and Representation Theory
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：20
• 期：1
• 页码：175-195
• 全文大小：
• 刊物主题：Commutative Rings and Algebras; Associative Rings and Algebras; Non-associative Rings and Algebras;
• 出版者：Springer Netherlands
• ISSN：1572-9079
• 卷排序：20

文摘

In this paper we describe some Leibniz algebras whose corresponding Lie algebra is four-dimensional Diamond Lie algebra \(\mathfrak {D}\) and the ideal generated by the squares of elements (further denoted by I) is a right \(\mathfrak {D}\)-module. Using description (Casati et al J. Math. Phys. 51: 033515 (2010)) of representations of algebra \(\mathfrak {D}\) in \(\mathfrak {sl}(3,{\mathbb {C}})\) and \(\mathfrak {sp}(4,{\mathbb {F}})\) where \({\mathbb {F}}={\mathbb {R}}\) or \({\mathbb {C}}\) we obtain the classification of above mentioned Leibniz algebras. Moreover, Fock representation of Heisenberg Lie algebra was extended to the case of the algebra \(\mathfrak {D}.\) Classification of Leibniz algebras with corresponding Lie algebra \(\mathfrak {D}\) and with the ideal I as a Fock right \(\mathfrak {D}\)-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.