Dmitriy Rumynin

Singular localization and intertwining functors for reductive Lie algebras in prime characteristic

In [BMR] we observed that,, on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of finite characteristic with a given (generalized) regular central character can be identified with coherent sheaves on the formal neighborhood of the corresponding (generalized) Springer fiber. In the present paper we treat singular central characters. The basic step is the Beilinson-Bernstein localization of modules with a fixed (generalized) central character a as sheaves on the partial flag variety corresponding to the singularity of A. These sheaves are modules over a sheaf of algebras which is a version of twisted crystalline, differential operators. We discuss translation functors and intertwining functors. The latter generate an action of the affine braid group on the derived category of modules with a regular (generalized) central character, which intertwines different localization functors. We also describe the standard duality on Lie algebra modules in terms of D-modules and coherent sheaves.

Geometric representation theory of restricted lie algebras

We modify the Hochschild φ-map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a goup scheme that leads to a geometric construction of unrestricted representations. For a classical semisimple Lie algebra, we construct equivariant line bundles whose global sections afford representations with a nilpotentp-character.

Lie algebras in symmetric monoidal categories

We study the algebras that are defined by identities in the symmetric monoidal categories; in particular, the Lie algebras. Some examples of these algebras appear in studying the knot invariants and the Rozansky-Witten invariants. The main result is the proof of the Westbury conjecture for a K3-surface: there exists a homomorphism from a universal simple Vogel algebra into a Lie algebra that describes the Rozansky-Witten invariants of a K3-surface. We construct a language that is necessary for discussing and solving this problem, and we formulate nine new problems.

Subregular representations of _{} and simple singularities of type _{-1}. II

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.

Subregular Representations of

Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.Alexander Premet has stated the following problem: what is a relation between subregular nilpotent representations of a classical semisimple restricted Lie algebra and non-commutative deformations of the corresponding singularities? We solve this problem for type A.

{\frak s} {\frak l}_{n}

Throughout Chapters 3-11, k∨ is an injective cogenerator of the category of k-modules. One can always take k∨ = HomZ(k,Q/Z).

and Simple Singularities of Type A n −1

Throughout Chapters 1-11, k is a commutative ring. All our rings, bimodules, abelian groups, … will be k-modules; all additive categories will be k-linear.

Semicontramodules and Semihomomorphisms

This chapter contains some known results and some results deemed to be new, but no proofs. Its goal is to prepare the reader for the more technically involved constructions of the main body of the monograph (where the proofs are given). In particular, we do not have to worry about nonassociativity of the cotensor product and partial definition of the semitensor product here, distinguish between the myriad of notions of absolute/relative coflatness/coprojectivity/injectivity of comodules and analogously for contramodules, etc., because we only consider coalgebras over fields.

Semialgebras and Semitensor Product

Let ϱ → D be a map of corings compatible with a k-algebra map A → B. Let S be a semialgebra over the coring ϱ and T be a semialgebra over the coring D.

Preliminaries and Summary

Let A be an exact category in which all infinite products exist and the functors of infinite product are exact. A complex C• over A is called Italic if it belongs to the minimal triangulated subcategory Acycl ctr (A) of the homotopy category Hot(A) containing all the total complexes of exact triples ′K• → K• → ″K• of complexes over A and closed under infinite products. Any contraacyclic complex is acyclic. It follows from the next lemma that any acyclic complex bounded from above is contraacyclic.