Hall algebras,hereditary algebras and quantum groups

4-VIII-1994 & 4-X-1994     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Stable equivalences of graded algebras

We extend the notion of stable equivalence to the class of locally finite graded algebras. For such an algebra Λ, we focus on the Krull–Schmidt category gr_Λ of finitely generated -graded Λ-modules with degree 0 maps, and the stable category obtained by factoring out those maps that factor through a graded projective module. We say that Λ and Γ are graded stably equivalent if there is an equivalence that commutes with the grading shift. Adapting arguments of Auslander and Reiten involving functor categories, we show that a graded stable equivalence α commutes with the syzygy operator (where defined) and preserves finitely presented modules. As a result, we see that if Λ is right noetherian (resp. right graded coherent), then so is any graded stably equivalent algebra. Furthermore, if Λ is right noetherian or k is artinian, we use almost split sequences to show that a graded stable equivalence preserves finite length modules. Of particular interest in the nonartinian case, we prove that any graded stable equivalence involving an algebra Λ with socΛ=0 must be a graded Morita equivalence.     

The representation dimension of Hecke algebras and symmetric groups

We establish a lower bound for the representation dimension of all the classical Hecke algebras of types A, B and D. For all the type A algebras, and “most” of the algebras of types B and D, we also establish upper bounds. Moreover, we establish bounds for the representation dimension of group algebras of some symmetric groups.     

On pointed Hopf algebras associated with the symmetric groups

Let G be the symmetric group . It is an important open problem whether the dimension of the Nichols algebra is finite when is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs might give rise to finite-dimensional Nichols algebras. This work was partially supported by CONICET, ANPCyT and Secyt (UNC).     

Topological equivalences for differential graded algebras

We investigate the relationship between differential graded algebras (dgas) and topological ring spectra. Every dga C gives rise to an Eilenberg-Mac Lane ring spectrum denoted HC. If HC and HD are weakly equivalent, then we say C and D are topologically equivalent. Quasi-isomorphic dgas are topologically equivalent, but we produce explicit counterexamples of the converse. We also develop an associated notion of topological Morita equivalence using a homotopical version of tilting.     

Derived equivalences and Cohen-Macaulay Auslander algebras

Let A and B be Artin R-algebras of finite Cohen-Macaulay type. Then we prove that, if A and B are standard derived equivalent, then their Cohen-Macaulay Auslander algebras are also derived equivalent. And we show that Gorenstein projective conjecture is an invariant under standard derived equivalence between Artin R-algebras.     

Affine cellular algebras and Morita equivalences

In this paper we prove that Morita equivalences under particular conditions are compatible with affine cellular structures.     

Tilting simply connected algebras

Let A be a representation-finite simply connected algebra, and T_Abe an arbitrary tilting module (in the sense of Happei and Ringel). We show that B = End T_A satisfies the separation condition of Bautista, Larrion and Salmeron. This implies that B is also simply connected.     

Derived equivalences between triangular matrix algebras

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.     

Socle equivalences of weighted surface algebras

We describe the structure and properties of the finite-dimensional symmetric algebras over an algebraically closed field K which are socle equivalent to the general weighted surface algebras of triangulated surfaces, investigated in [11]. In particular, we prove that all these algebras are tame periodic algebras of period 4. The main results of this paper form an essential step towards a classification of all symmetric tame periodic algebras of period 4.     

Double Hall algebras and derived equivalences

We show that the reduced Drinfeld double of the Ringel-Hall algebra of a hereditary category is invariant under derived equivalences. By associating an explicit isomorphism to a given derived equivalence, we also extend the results of Burban and Schiffmann [5] and [006], Sevenhant and Van den Bergh (1999) [25], and Xiao and Yang (2001) [31].     

On graded Cartan invariants of symmetric groups and Hecke algebras

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A at roots of unity. These matrices are \({\mathbb {Z}}[v,v^{-1}]\)-valued and may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring \({\mathbb {Z}}[v,v^{-1}]\). This proves and generalizes a conjecture of Ando-Suzuki-Yamada on the invariants of these matrices over \({\mathbb {Q}}[v,v^{-1}]\) and also generalizes the first author’s recent proof of the Külshammer-Olsson-Robinson conjecture over \({\mathbb {Z}}\).     

Tilting up iterated tilted algebras

We show that, if is a representation-finite iterated tilted algebra of euclidean type , then there exist a sequence of algebras , and a sequence of modules , where , such that each is an APR-tilting -module, or an APR-cotilting -module, and is tilted representation-finite.

     

Derived equivalences of selfinjective algebras preserve singularities

We prove that the derived equivalences (more generally the stable equivalences of Morita type) of finite dimensional selfinjective algebras over algebraically closed fields preserve the types of singularities in the orbit closures of module varieties. As an application, we obtain that the orbit closures in the module varieties of the Brauer tree algebras are normal and Cohen-Macaulay. Mathematics Subject Classification (2000):14B05, 14L30, 16D50, 16G20     

Morita equivalences of Ariki–Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle. Received: 22 March 2000; in final form: 19 September 2001 / Published online: 29 April 2002