Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.     

Cohomology of Algebras of Semidihedral Type. III: The Family SD(3 $$\mathcal{K}$$ )

The present paper continues a series of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebras are described (in terms of quivers with relations) for the algebras of semidihedral type that form the family SD(3 ). Bibliography: 10 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 84–100.     

On Schur algebras and related algebras V: some quasi-hereditary algebras of finite type

In a recent paper, Erdmann determined which of the Schur algebras S(n, r) have finite representation type and described the finite type Schur algebras up to Morita equivalence. The present paper grew out of a desire to see Erdmann's results in the more general context of algebras which are quasi-hereditary in the sense of Cline et al. (1988). Weconsider here the class of quasi-hereditary algebras which have a duality fixing simples. This class includes the “generalized Schur algebras” defined and studied by the first author, and the Schur algebras themselves in particular. In the first part we describe the possible Morita types of the quasi-hereditary algebras of finite representation type over an algebraically closed field with duality fixing simples. This is then applied, in the second part, to give the block theoretic refinement of Erdmann's results.     

Cohomology of algebras of semidihedral type. VI

The present paper continues a cycle of papers of the author (certain of them in collaboration) in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the family . Bibliography: 19 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 183–198.     

Computation of the Yoneda Algebras for Algebras of Dihedral Type

We continue a series of papers in which the Yoneda algebra is computed for algebras of dihedral and semidihedral types. In this paper, the Yoneda algebra is computed for one more family of algebras, namely, for the family D(3 ) (in the classification of K. Erdmann). In order to find the minimal resolutions of simple modules, we used for the first time a C⁺⁺ program implemented by the second author. Bibliography: 10 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 101–120.     

Cohomology of algebras of semidihedral type. V

The present paper continues a cycle of papers of the author (some of them are written in collaboration), in which the Yoneda algebras are calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the paper, the Yoneda algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the family SD(3 )₁. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 131–154.     

Cartan determinants for gentle algebras

The determinant of the Cartan matrix of a finite dimensional algebra is an invariant of the derived category and can be very helpful for derived equivalence classifications. In this paper we determine the determinants of the Cartan matrices for all gentle algebras. This is a class of algebras of tame representation type which occurs naturally in various places in representation theory. The definition of these algebras is of a purely combinatorial nature, and so are our formulae for the Cartan determinants.Received: 29 October 2004     

Cohomology of algebras of semidihedral type. IV

The present paper continues a series of papers by the author (some of them are written in collaboration) in which the Yoneda algebra is calculated for several families of algebras of dihedral and semidihedral type (in K. Erdmann’s classification). In the present paper, the Yoneda algebra is described (in terms of quivers with relations) for algebras of semidihedral type, namely, of the families SD(2A)₁, SD(2A)₂, and SD(3A)₂. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 71–116.     

Higher-dimensional orders,graph-orders,and derived equivalences

Green-orders (tree-orders) in the classical one-dimensional case are the setting, to understand p-adic blocks with cyclic defect of finite groups. Blocks with “cyclic defect” of Hecke orders however, are Green-orders over two-dimensional rings. Hecke orders of dihedral groups of order divisible by 4 are even defined over a three-dimensional ring. We extend the notion of Green-orders to orders associated to a locally embedded graph instead of a tree, and to general complete regular local noetherian ground rings of finite dimension. We extend the result, that classical tree-orders are derived equivalent to star-orders. We then use these results to clarify the derived equivalence classes of tame algebras of Dihedral type.     

Representation Type of Borel-Schur Algebras

Algebras and Representation Theory - In our previous work (Erdmann et al. J. Algebra Appl. 17(2),1850028, 2018), we found all Borel-Schur algebras of finite representation type. In the present...     

Higher spherical algebras

We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in Erdmann and Skowroński (Algebras Represent Theory 22:387–406, 2019), and hence that it is a tame symmetric periodic algebra of period 4.     

Representation type of Schur algebras

We give a complete classification of the classical Schur algebras and the infinitesimal Schur algebras which have tame representation type. In combination with earlier work of some of the authors on semisimplicity and finiteness, this completes the classification of representation type of all classical and infinitesimal Schur algebras in all characteristics. Received October 17, 1997; in final form March 5, 1998     

Stable equivalence preserves representation type

Given two finite dimensional algebras and , it is shown that is of wild representation type if and only if is of wild representation type provided that the stable categories of finite dimensional modules over and $\Gamma$ are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over and , and the result follows from certain additional properties of this bijection. In the second part of this paper the Auslander-Reiten translation is extended to an operation on the category of all modules. It is shown that various finiteness conditions are preserved by this operation. Moreover, the Auslander-Reiten translation induces a homeomorphism between the set of non-projective and the set of non-injective points in the Ziegler spectrum. As a consequence one obtains that for an algebra of tame representation type every generic module remains fixed under the Auslander-Reiten translation. Received: July 24, 1996     

Completed cohomology of Shimura curves and a p-adic Jacquet–Langlands correspondence

We study indefinite quaternion algebras over totally real fields F, and give an example of a cohomological construction of p-adic Jacquet–Langlands functoriality using completed cohomology. We also study the (tame) levels of p-adic automorphic forms on these quaternion algebras and give an analogue of Mazur’s ‘level lowering’ principle.     

Morava -theory rings for the dihedral, semidihedral and generalized quaternion groups in Chern classes

Morava -theory rings of classifying spaces of the dihedral, semidihedral and generalized quaternion groups are presented in terms of Chern classes.

     

Erdős–Gyárfás conjecture for some families of Cayley graphs

The Paul Erd?s and András Gyárfás conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order \(p^3\).     

Representation Theory of some Infinite-dimensional Algebras Arising in Continuously Controlled Algebra and Topology

In this paper we determine the representation type of some algebras of infinite matrices continuously controlled at infinity by a compact metrizable space. We explicitly classify their finitely presented modules in the finite and tame cases. The algebra of row-column-finite (or locally finite) matrices over an arbitrary field is one of the algebras considered in this paper, its representation type is shown to be finite.Received October 2003     

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.