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We show that if $B$ is a block of a finite group algebra $kG$ over an algebraically closed field $k$ of prime characteristic $p$ such that $HH^1(B)$ is a simple Lie algebra and such that $B$ has a unique isomorphism class of simple modules, then $B$ is nilpotent with an elementary abelian defect group $P$ of order at least $3$, and $HH^1(B)$ is in that case isomorphic to the Jacobson-Witt algebra $HH^1(kP)$. In particular, no other simple modular Lie algebras arise as $HH^1(B)$ of a block $B$ with a single isomorphism class of simple modules.
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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.
Let $min N$, $P(t)in C[t]$. Then we have the Riemann surfaces (commutative algebras) $R_m(P)=C[t^{pm1},u | u^m=P(t)]$ and $S_m(P)=C[t , u| u^m=P(t)].$ The Lie algebras $mathcal{R}_m(P)=Der(R_m(P))$ and $mathcal{S}_m(P)=Der(S_m(P))$ are called the $m$-th superelliptic Lie algebras associated to $P(t)$. In this paper we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras.
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.
We prove that the tensor product of a simple and a finite dimensional $mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $mathfrak{q}(n)$-supermodules to that of simple $mathfrak{sl}_n$-modules. Rough structure of simple $mathfrak{q}(n)$-supermodules, considered as $mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $mathcal{O}$.
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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