We study the Yangian Yn associated to the general linear Lie algebra gl n over a field of positive characteristic, as well as its shifted analog Ynpσq. Our main result gives a description of the centre of Ynpσq: it is a polynomial algebra generated by its Harish-Chandra centre (which lifts the centre in characteristic zero) together with a large p-centre. Moreover, Ynpσq is free as a module over its center. In future work, it will be seen that every reduced enveloping algebra Uχpgl n q is Morita equivalent to a quotient of an appropriate choice of shifted Yangian, and so our results will have applications in classical representation theory.

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 ge=Lie(Ge) where Ge stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [ge,ge] in ge and use it to determine those e∈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∈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 [ge,ge] in ge 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 Ge 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

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 (g, e) associated to a nilpotent element e ∈ g = 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 (g, e), which allows us to define reduced finite W -algebras U η (g, e) and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin's equivalence of categories, generalizing recent work of the second author.

Let g = gl N ( ), where is an algebraically closed field of characteristic p > 0, and N ∈ Z ≥1 . Let χ ∈ g * and denote by U χ (g) the corresponding reduced enveloping algebra. The Kac-Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional U χ (g)-module has dimension divisible by p dχ , where d χ is half the dimension of the coadjoint orbit of χ. Our main theorem gives a classification of U χ (g)-modules of dimension p dχ . As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for U 0 (h) for a certain Levi subalgebra h of g; we view this as a modular analogue of Moeglin's theorem on completely primitive ideals in U (gl N (C)). To obtain these results, we reduce to the case χ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted W -algebra.2010 Mathematics Subject Classification: 17B10, 17B37.

We study Frobenius extensions which are free-filtered by a totally ordered, finitely generated abelian group, and their free-graded counterparts. First we show that the Frobenius property passes up from a free-graded extension to a free-filtered extension, then also from a free-filtered extension to the extension of their Rees algebras. Our main theorem states that, under some natural hypotheses, a free-filtered extension of algebras is Frobenius if and only if the associated graded extension is Frobenius. In the final section we apply this theorem to provide new examples and non-examples of Frobenius extensions.