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Register a new userLeibniz algebras associated with representations of the Diamond Lie algebra
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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 cite{Cas} 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 $BL^2$ and $ZL^2$ are constructed, as well.
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In the present paper we describe Leibniz algebras with three-dimensional Euclidean Lie algebra $mathfrak{e}(2)$ as its liezation. Moreover, it is assumed that the ideal generated by the squares of elements of an algebra (denoted by $I$) as a right $mathfrak{e}(2)$-module is associated to representations of $mathfrak{e}(2)$ in $mathfrak{sl}_2({mathbb{C}})oplus mathfrak{sl}_2({mathbb{C}}), mathfrak{sl}_3({mathbb{C}})$ and $mathfrak{sp}_4(mathbb{C})$. Furthermore, we present the classification of Leibniz algebras with general Euclidean Lie algebra ${mathfrak{e(n)}}$ as its liezation $I$ being an $(n+1)$-dimensional right ${mathfrak{e(n)}}$-module defined by transformations of matrix realization of $mathfrak{e(n)}.$ Finally, we extend the notion of a Fock module over Heisenberg Lie algebra to the case of Diamond Lie algebra $mathfrak{D}_k$ and describe the structure of Leibniz algebras with corresponding Lie algebra $mathfrak{D}_k$ and with the ideal $I$ considered as a Fock $mathfrak{D}_k$-module.
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 $L$.
The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-graded Lie algebra L which are integrable as representations of g and whose weights are bounded by some dominant weight of g. We link this category to the module category of an associative algebra, whose structure we determine for map algebras and sl_n(A). Our results unify previous work of Chari and her collaborators on map algebras and of Seligman on isotropic Lie algebras.
In this paper we construct a minimal faithful representation of the $(2m+2)$-dimensional complex general Diamond Lie algebra, $mathfrak{D}_m(mathbb{C})$, which is isomorphic to a subalgebra of the special linear Lie algebra $mathfrak{sl}(m+2,mathbb{C})$. We also construct a faithful representation of the general Diamond Lie algebra $mathfrak{D}_m$ which is isomorphic to a subalgebra of the special symplectic Lie algebra $mathfrak{sp}(2m+2,mathbb{R})$. Furthermore, we describe Leibniz algebras with corresponding $(2m+2)$-dimensional general Diamond Lie algebra $mathfrak{D}_m$ and ideal generated by the squares of elements giving rise to a faithful representation of $mathfrak{D}_m$.
Let $L$ be a Lie algebra of Block type over $C$ with basis ${L_{alpha,i},|,alpha,iinZ}$ and brackets $[L_{alpha,i},L_{beta,j}]=(beta(i+1)-alpha(j+1))L_{alpha+beta,i+j}$. In this paper, we shall construct a formal distribution Lie algebra of $L$. Then we decide its conformal algebra $B$ with $C[partial]$-basis ${L_alpha(w),|,alphainZ}$ and $lambda$-brackets $[L_alpha(w)_lambda L_beta(w)]=(alphapartial+(alpha+beta)lambda)L_{alpha+beta}(w)$. Finally, we give a classification of free intermediate series $B$-modules.
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