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Let $mathfrak g$ be a simple Lie algebra over $mathbb C$ and let $e in mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(mathfrak g,e)$ associated to $e$ and the problem of determining the variety $mathcal E(mathfrak g,e)$ of 1-dimensional representations of $U(mathfrak g,e)$. For $mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $mathcal E(mathfrak g,e)$, and the action of the component group $Gamma_e$ of the centralizer of $e$ on $mathcal E(mathfrak g,e)$. As a consequence, we provide two examples where the nilpotent orbit of $e$ is induced, but there is a 1-dimensional $Gamma_e$-stable $U(mathfrak g,e)$-module which is not induced via Losevs parabolic induction functor. In turn this gives examples where there is a non-induced multiplicity free primitive ideal of $U(mathfrak g)$.
Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(mathfrak g,e)$ associated
Let $mathfrak g = mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N in mathbb Z_{ge 1}$. Let $chi in mathfrak g^*$ and denote by $U_chi(mathfrak g)$ the corresponding reduced enveloping algebra. The Kac-
Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main result shows t
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.
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 d