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Register a new userOn the variety of 1-dimensional representations of finite $W$-algebras in low rank
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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 $U(mathfrak g)$.
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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 to a nilpotent element $e in mathfrak g = operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$-centre of $U(mathfrak g,e)$, which allows us to define reduced finite $W$-algebras $U_eta(mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabins equivalence of categories, generalizing recent work of the second author.
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--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_chi(mathfrak g)$-module has dimension divisible by $p^{d_chi}$, where $d_chi$ is half the dimension of the coadjoint orbit of $chi$. Our main theorem gives a classification of $U_chi(mathfrak g)$-modules of dimension $p^{d_chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(mathfrak h)$ for a certain Levi subalgebra $mathfrak h$ of $mathfrak g$; we view this as a modular analogue of M{oe}glins theorem on completely primitive ideals in $U(mathfrak{gl}_N(mathbb C))$. To obtain these results, we reduce to the case $chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra.
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 that if the Ext-quiver of $A$ has no loops and at most two parallel arrows in any direction, and if $HH^1(A)$ is a simple Lie algebra, then char(k) is not equal to $2$ and $HH^1(A)cong$ $sl_2(k)$. The third result investigates symmetric algebras with a quiver which has a vertex with a single loop.
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 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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