Alexander Premet

Nilpotent orbits in good characteristic and the Kempf–Rousseau theory

Method and apparatus are provided for controlling the advancement of a strand by directing a fluid along the strand in a first direction opposite to the direction of advancement of the strand to retard the advancement of the strand sufficiently to create slack in the strand, and directing a fluid in a second direction transverse to the first direction to control slack in the strand.

Nilpotent commuting varieties of reductive Lie algebras

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p≥0, and 𝔤=Lie G. In positive characteristic, suppose in addition that p is good for G and the derived subgroup of G is simply connected. Let 𝒩=𝒩(𝔤) denote the nilpotent variety of 𝔤, and ℭnil(𝔤):={(x,y)∈𝒩×𝒩 |  [x,y]=0}, the nilpotent commuting variety of 𝔤. Our main goal in this paper is to show that the variety ℭnil(𝔤) is equidimensional. In characteristic 0, this confirms a conjecture of Vladimir Baranovsky; see [2]. When applied to GL(n), our result in conjunction with an observation in [2] shows that the punctual (local) Hilbert scheme ℋn⊂Hilbn(ℙ2) is irreducible over any algebraically closed field.

Representations of restricted Lie algebras and families of associative ℒ-algebras

Abstract Let ℒ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K, we introduce an associative algebra Uξ,λ (ℒ) of dimension pn over K. The algebra Uξ,1 (ℒ) is isomorphic to the reduced enveloping algebra Uξ (ℒ), while the algebra Uξ,0 (ℒ) is nothing but the reduced symmetric algebra Sξ (ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function.

An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field of characteristic p > 0. Suppose that G(1) is simply connected and p is good for the root system of G. Given a one-dimensional torus A C G let g(A, i) denote the weight component of Ad(i) corresponding to weight i E X(i) Z. It is proved in the paper that, for any nonzero nilpotent element e e g, there is a one-dimentional torus Ae C G such that e E g(Ae, 2) and Kerade C fi>og(Ae, i).

The Recognition Theorem for Graded Lie Algebras in Prime Characteristic

The Recognition Theorem for graded Lie algebras is an essential ingredient in the classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic p > 3. The main goal of this monograph is to present the first complete proof of this fundamental result.

Block representation type of reduced enveloping algebras

Let K be an algebraically closed field of characteristic p, G a connected, reductive K-group, g = Lie(G), X E g * and U X (g) the reduced enveloping algebra of g associated with X. Assume that G (1) is simply-connected, p is good for G and g has a non-degenerate G-invariant bilinear form. All blocks of U X (g) having finite and tame representation type are determined.

Derived subalgebras of centralisers and finite

Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [g_e, g_e] in g_e and use it to determine those e in g for which the largest commutative quotient U(g,e)^{ab} of the finite W-algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g. The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov we prove that a nilpotent element e in g is non-singular if and only if the maximal dimension of the geometric quotients S/G, where S is a sheet of g containing e, coincides with the codimension of [g_e,g_e] in g_e and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element e in a classical Lie algebra g the closed subset of Specm U(g,e)^{ab} consisting of all points fixed by the natural action of the component group of G_e is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.

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Abstract Let k be a global field and let kυ be the completion of k with respect to υ a non-archimedean place of k. Let G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of kυ -rank 1. Let G = G(kυ ). Let Г be an arithmetic lattice in G and let C = C(Г) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Г by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is , the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example , where is the ring of S-integers in k, with S = {υ}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Г on the Bruhat-Tits tree associated with G.

-algebras

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan–Reichstein–Vistoli (Annals of Math., 2010) and Chernousov–Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2. We also complete the determination of generic stabilizers in spin and half-spin groups of low rank.

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let K be an algebraic number field and O be the ring of integers of K. Let G be a finite group and M be a finitely generated torsion free OG-module. We say that M is a globally irreducible OG-module if, for every maximal ideal p of O, the kp G-module M⊗Okp is irreducible, where kp stands for the residue field O/p. Answering a question of Pham Huu Tiep, we prove that the symmetric group Σn does not have non-trivial globally irreducible modules. More precisely we establish that if M is a globally irreducible OΣn-module, then M is an O-module of rank 1 with the trivial or sign action of Σn.