Regular derivations of truncated polynomial rings

Let $\Bbbk$ be an algebraically closed field of characteristic $p>2$. Let $\mathcal{O}_n=\Bbbk[X_1,\ldots,X_n]/(X_1^p,\ldots, X_n^p)$, a truncated polynomial ring in $n$ variables, and denote by $\mathcal{L}$ the derivation algebra of $\mathcal{O}_n$. It is known that the ring of all polynomial functions on $\mathcal{L}$ invariant under the action of the group of $\mathrm{Aut}(\mathcal{L})$ is freely generated by $n$ elements. Furthermore, the related quotient morphism is faithfully flat and all its fibres are irreducible complete intersections. An element $x\in\mathcal{L}$ is called ${\it regular}$ if the centraliser of $x$ in $\mathcal{L}$ has the smallest possible dimension. In this preprint we give an explicit description of regular elements of $\mathcal{L}$ and show that a precise analogue of Kostant's differential criterion for regularity holds in $\mathcal{L}$. We also show that a fibre of the above mentioned quotient morphism is normal if and only if it consists of regular semisimple elements of $\mathcal{L}$.Comment: Many typos and a serious error in the proof of Theorem 3 are correcte

Primitive ideals, non-restricted representations and finite W-algebras

We prove that all finite W-algebras associated with nilpotent elements e in a complex semisimple Lie algebra g have finite-dimensional representations. In order to obtain this result we establish a connection between primitive ideals of U(g) attached to the nilpotent orbit containing e and finite-dimensional representations of the reduced enveloping algebra assiciated with e over an algebraically closed field of finite characteristic.Comment: 19 pages, minor corrections and up-dates, the new version is accepted for publicatio

Enveloping algebras of Slodowy slices and Goldie rank

It is known that any primitive ideal I of U(g) whose associated variety contains a nilpotent element e in its open G-orbit admits a finite generalised Gelfand-Graev model which is a finite dimensional irreducible module over the finite W-algebra U(g,e). We prove that if V is such a model for I, then the Goldie rank of the primitive quotient U(g)/I always divides the dimension of V. For g=sl(n), we use a result of Joseph to show that the Goldie rank of U(g)/I equals the dimension of V and we show that the equality conntinues to hold outside type A provided that the Goldie field of U(g)/I is isomorphic to a Weyl skew-field. As an application of this result, we disprove Joseph's conjecture on the structure of the Goldie fields of primitive quotients of U(g) formulated in the mid-70s.Comment: This version is accepted for publication (many typos and minor errors are corrected, several new remarks, etc

Multiplicity-free primitive ideals associated with rigid nilpotent orbits

We prove that any finite W-algebra U(g,e) admits a one-dimensional representation fixed by the action of the component group of the centraliser of e. As a consequence, for any nilpotent orbit O in g there exists a multiplicity-free (and hence completely prime) primitive ideal of the universal enveloping algebra U(g) whose associated variety coincides with the Zariski closure of O.Comment: minor changes and corrections; the present version has been accepted for publicatio

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$ and suppose that $p$ is a very good prime for $G$. We prove that any maximal Lie subalgebra $M$ of $\mathfrak{g} = {\rm Lie}(G)$ with ${\rm rad}(M) \ne 0$ has the form $M = {\rm Lie}(P)$ for some maximal parabolic subgroup $P$ of $G$. We show that the assumption on $p$ is necessary by providing a counterexample for groups type ${\rm E}_8$ over fields of characteristic $5$. Our arguments rely on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields prime characteristic.Comment: 42 pages; this version of the preprint is accepted for publication in "Advances in Mathematics

Classification of finite dimensional simple Lie algebras in prime characteristics

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.Comment: Revised version: a list of open problems has been added as suggested by the refere

Classification of the maximal subalgebras of exceptional Lie algebras over fields of good characteristic

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal \it{\mbox{exotic semidirect product}}. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\rm E}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\rm E}_7$.Comment: This version is accepted for publication in Journal of the American Mathematical Society; 40 page