I am going to talk about our joint results with Sudarshan Sehgal and Mikhail Zaicev on the structure of finite-dimensional graded simple algebras. These algebras have been described in term of twisted group rings and matrix algebras with fine and elementary gradings.

The general classification of finite dimensional Hopf algebras over a field of characteristic 0 is a difficult open problem. In the study of Hopf algebras of low dimension which are neither cosemisimple nor pointed, one approach is to study the simple coalgebras in the coradical H₀ of the Hopf algebra H and the H₀-bicomodules P_n. Note that this amounts to studying the irreducible representations of the dual Hopf algebra H^*. We outline some older and newer results for this problem.

Let k be an algebraically closed field of positive characteristic, and let G be a finite group. There are various classical results in the literature concerning the lifting of finitely generated kG-modules over complete discrete valuation rings, such as Green's liftability theorem. To understand and generalize these results, it is useful to reformulate them in terms of deformation rings.

Suppose B is a block of kG of tame representation type with defect group D. For certain B, we will show how to determine the universal deformation rings R(G,V) of finitely generated kG-modules V belonging to B which have stable endomorphism ring isomorphic to k. We will relate R(G,V) to the group ring WD where W is the ring of infinite Witt vectors over k.

Kirillov has described a geometric McKay correspondence for finite subgroups G Ì PSL₂ (C): for each `height function' on the affine Dynkin diagram associated to G, there is a derived equivalence from G-equivariant sheaves on P¹ to the path algebra of an orientation of the diagram. These equivalences for various height functions are related by reflection functors.

I develop an analogous McKay correspondence for the cotangent bundle T^* P¹ in which each height function gives a derived equivalence from equivariant sheaves on T^* P¹ to the preprojective algebra of the affine Dynkin diagram. These various equivalences are related by so-called spherical twists, which generate an action of the Artin group of the diagram on the derived category of equivariant sheaves.

For a pair (a,b) of relatively prime natural numbers, the Christoffel word C(a,b) is defined by the path with integral vertices which is closest to the line segment from (0,0) to (a,b). Viewing this line segment as an arc in the once-punctured torus, we define a J-module M(a,b) for each Christoffel word. Here J is the Jacobian algebra of the once-punctured torus. We show that one obtains the Markoff number associated with C(a,b) by counting submodules of M(a,b).

Recent work on noncommutative desingularisations of determinantal varieties led us to take a closer look at the object in the title. While its linear properties, such as local cohomology and projective resolution, are well within reach of a first course in homological algebra-and provide interesting examples-the multiplicative properties are more intriguing.

Based on results by T. Bridgeland, we know that the n-th Veronese subalgebra, if n is the number of variables, is Koszul, Calabi-Yau, of finite global dimension, and provides an algebraic model of the anti-canonical bundle of the underlying projective space, realising that bundle as a moduli space of representations of that algebra.

We will indicate how these results relate to the theory of quiver representations and helices.

Joint work with Thuy Pham.

The lecture will provide a complete description of the possibilities for global dimension of the endomorphism algebras described in the title in terms of the cardinalities of the Auslander-Reiten orbits of indecomposable modules.

A joint result with C. M. Ringel.

Let A be an artin algebra, and consider the category modA of finitely generated right A-modules. A module M in modA is called almost projective if Ext¹_A (M,X) ¹ 0 for at most finitely many non-isomorphic indecomposable modules in modA; and almost injective if Ext¹_A (X,M) ¹ 0 for at most finitely many non-isomorphic indecomposable modules in modA. We shall show that the almost projective or injective modules are distributed in finitely many DTr-orbits in the Auslander-Reiten quiver of A. In particular, A is of finite representation type if and only if every module in modA is almost projective or injective. As a consequence, if A is a finite-dimensional algebra over an algebraically closed field of infinite representation type, then there exists infinitely many non-isomorphic indecomposable modules in modA which are neither almost projective nor almost injective.

We describe recent results for Schur superalgebras in positive characteristics. We discuss similarities and differences of the structure of simple and costandard modules for Schur superalgebras and for classical Schur algebras. In particular, we consider the structure of these modules for Schur superalgebras S(2|1) and S(2|2).