In this talk we are going to discuss various aspects of gradings on finite-dimensional algebras and their representations. Among topics covered are the description of gradings by finite groups on simple finite-dimensional algebras, the structure of not necessarily simple finite-dimansional algebras, graded modules over graded algebras, the connection between the properties of the identity component of the grading and those of the whole of algebra. The talk will be based on joint results with V.  Drensky, S. Montgomery, S. Sehgal and M. Zaicev.

Let A be an artin algebra over a local commutative artinian ring k. We consider the category p(A) whose objects are morphisms f: P® Q with P and Q finitely generated left A-modules. We introduce an exact structure on p(A). We prove a relation between the Hom-functor and the Ext-functor of the exact structure. From here we can prove the existence of almost split sequences in p(A) and its relation with the almost split sequences for finitely generated A-modules.

Let k be a field of positive characteristic p, let G be a finite group and let P be a Sylow p-subgroup of G. Suppose X is a non-singular projective variety over k with a faithful linear action of G. We discuss the question of whether X has an embedding into projective N-space P^N for some N such that the corresponding coordinate ring S(X) of X can be written as a direct sum of indecomposable kG-modules which lie in finitely many kG-isomorphism classes. This was shown to be the case by Symonds and Karagueuzian in case k is finite and X=P^N (embedding into itself). We show that this is also true if either (i)  X is a projective curve, or (ii)  X is a projective surface, H¹(X,O_X)=0 = H²(X,O_X) and X^P ¹ Æ.

It is well-known that the positive part g⁺ of a Kac-Moody algebra g is isomorphic to Ringel's composition algebra c(Q), defined by a quiver Q with the same root system as g. Using vertex operator algebras, one can construct a Lie algebra g(L) for every generalized root lattice L. If L is elliptic and simply laced, Saito and Yoshii gave a description of the Lie algebras g(L) by generators and relations. Our aim is to compare in some cases g(L) with the composition algebra c(A) of certain tubular algebra A=kQ/I. Here the quiver Q encodes the Serre relations of g(L) and the ideal I of the path algebra kQ encodes the additional relations described by Saito and Yoshii. These additional relations (which Saito and Yoshii consider to be strange, since they involve more than two simples) are very natural from the point of view of finite-dimensional algebras.

Surely, one of the deepest results in homological algebra is Gerstenhaber's theorem that Hochschild cohomology of any associative algebra is commutative with respect to the Yoneda product.

It has been observed by several authors recently that for flat algebras the result follows very elegantly from the Eckmann-Hilton argument originally used to prove commutativity of higher homotopy groups.

Here, we investigate the scope of this method and show that various incarnations of Hochschild cohomology, such as selfextensions of the identity functor on the category of endofunctors, are amenable to the same treatment.

We prove further that flatness can be replaced by the weaker hypothesis that the algebra K® A satisfies Tor₊^K(A,A) = 0.

According to the canonical isomorphism between the positive part U⁺_q(g) of the Drinfeld-Jimbo quantum group U _q (g) and the generic composition algebra C (D) of L, where the Kac-Moody Lie algebra g and the finite dimensional hereditary algebra L have the same diagram, in specially, we get a realization of quantum root vectors of the generic composition algebra of the Kronecker algebra by using the Ringel-Hall approach. The commutation relations among all root vectors are given and an integral PBW-basis of this algebra is also obtained.

This is a report on joint work with J. Schröer and B. Leclerc.

Let g be a simple Lie algebra of type A, D, E and n a maximal nilpotent subalgebra of g. Moreover, let N be a maximal unipotent subgroup of a simple Lie group with Lie algebra g. Finally, let P denote the corresponding preprojective algebra.

Lusztig's semicanonical basis B of U(n) is parametrized by irreducible components of the corresponding preprojective varieties mod (P,d). The dual B^* is a basis of C[N]. We can show:

$\bullet$ For two elements of B^* holds b_C·b_D Î B^* if for the corresponding irreducible components holds ext_P¹(C,D)=0.
$\bullet$ The dual canonical basis B_q^* of U_q(n) specializes for q=1 to B if and only if P is representation finite.

This explains the multiplicative properties of the dual canonical basis observed previously in the cases A_2,3,4. On the other hand it gives us a good control over the dual semicanonical basis in the cases A₅ and D₄, i.e. when P is tame, since we have in this case a precise combinatorial description of the irreducible components of mod(P,d) in terms of indecomposable components.

In this talk I describe some joint work with Eduardo N. Marcos of University of São Paulo, Brazil. We give a definition of a Koszul-like algebra in terms of graded projective resolutions and finite generation of the Ext-algebra. This class includes Koszul algebras and D-Koszul algebras (introduced by R. Berger and studied by Green-Marcos-Martínez-Zhang). For such algebras, we find necessary and sufficient condition on the degrees of generators of the projective modules in the resolution for the Ext-algebra to be finitely generated. We investigate the finite generation of a restricted class of bigraded algebras.

Another consequence of our work is studying certain subalgebras of Ext-algebras. In particular, we prove the following proposition. Let A be a Koszul algebra and E(A) be its Ext-algebra å_{k ³ 0}Ext_A^k(A/J,A/J) where J is the graded Jacobson radical of A. Then for n ³ 1, the subalgebra sum_{k ³ 0}Ext_A^nk(A/J,A/J) is always a Koszul algebra.

Let A be a graded algebra having the Auslander-Gorenstein property, and let G be a group of graded automorphisms of A. P. Jørgensen and J. Zhang and N. Jing and J. Zhang have given conditions when the fixed subring A^G also satisfies the Auslander-Gorenstein property; these conditions involve the "homological determinant" of the automorphisms in G. We study groups of graded automomorphisms of down-up algebras and generalized Weyl algebras, and compute the homological determinants of these automorphisms. We use these results to produce examples of fixed subrings having the Auslander-Gorenstein property.

(Joint work with Jean-Philippe Morin)

Let A be a finite dimensional algebra over a field given by a quiver with relations. Let S be a simple A-module with a non-split self-extension, that is, the quiver has a loop at the corresponding vertex. The strong no loop conjecture claims that S is of infinite projective dimension. First we show that Extⁱ_A(S,S), for all i ³ 1, does not vanish if the self-extension is almost split. As a by-product of the proof, we get a new characterization of Nakayama algebras, strengthening the one given by Auslander-Reiten-Smalø. Further, using a result of Green-Solberg-Zacharia, we deduce that Extⁱ_A(S,S) does not vanish for all i ³ 1 if no power of the loop is a component of a polynomial relation. Finally, we show that Extⁱ_A(S,S) does not vanish for all i ³ 1 if the convex support of S is a special biserial algebra.

(Joint work with Alexandr Zubkov)

Classical Schur algebras are known to be quasi-hereditary and cellular and the Schur superalgebras in "large" characteristics are semisimple. In case of "small" characteristics we determine that the Schur superalgebras are no longer quasi-hereditary or cellular. In case when the characteristic divides the degree r we determine all Schur superalgebras S(m|n,r) of finite representation type.

As was shown by R. Martinez Villa and A. Martsinkovsky, noncommutative Serre duality for generalized Artin-Schelter regular algebras can be deduced from the Auslander-Reiten formula for quasi-Frobenius (QF) algebras. This is based on a functorial isomorphism, provided by Koszul duality, between cohomology of tails and stable cohomology and on the interpretation of the the Auslander-Reiten formula for QF algebras as a duality in stable cohomology. This result opens a possibility for establishing Serre-type duality formulas for much more general classes of algebras. Their Koszul duals would not be QF and, therefore, the original Auslander-Reiten formula would have to be replaced by a hypothetical formula expressing a duality in stable cohomology. We establish such a formula by subjecting the Auslander-Reiten formula to the process of stabilization, which will be explained in the lecture. This is joint work with Idun Reiten.

Let L be an artin algebra. We denote by S(L) the category of pairs (A,B) where A is a finitely generated L-module and B is a submodule of A; a map f:(A,B)®(A¢,B¢) is just a L-linear map f: B® B¢ such that f(A) Ì A¢. The case of L = Z/(pⁿ) with p a prime number and n a positive integer has attracted a lot of interest since the categories S(Z/(pⁿ)) describe the possible embeddings of a subgroup in a finite abelian pⁿ-bounded group. The category S(Z/(pⁿ)) has finitely many indecomposables for n £ 5, is tame for n=6 and wild for n ³ 7.

For L a commutative local uniserial ring of Loewy length n, and (A,B) an object in S(L), the radical and socle series of A and B give rise to a filtration of B; the subsequent factors form the vector spaces for a subspace representation of a poset. In case n=6, this poset is tame of tubular type E₈; from its indecomposable representations we can reconstruct parts of the category S(L).

This is a report about joint work with Claus Michael Ringel.

Let A be the category of projective L modules for L an Artin algebra. In this talk we show the existence of almost split sequence in the category of projective finite complexes C(A).