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.

     

The Algebras of Semi-Invariants of Euclidean Quivers

We give a new short proof of Skowroński and Weyman's theorem about the structure of the algebras of semi-invariants of Euclidean quivers, in the case of quivers without oriented cycles and in characteristic zero. Our proof is based essentially on Derksen and Weyman's result about the generators of these algebras and properties of Schofield semi-invariants.     

COILS FOR VECTORSPACE CATEGORIES

Coils as components of Auslander–Reiten quivers of algebras and coil algebras are introduced by Assem and Skowroński. This concept is applied in the present paper to vectorspace categories. The four admissible operations on an Auslander–Reiten component of a vectorspace category, and the notions of v-coils and of vcoil vectorspace categories are introduced. A detailed study on the indecomposable objects of factorspace category of a vcoil vectorspace category is carried out.     

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.     

Pseudo equality algebras

A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabziński and Wroński: we provide an equational characterization for the equivalence operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety.     

Tame strongly simply connected algebras form an open scheme

Using van den Dries’s test and Brüstle, de la Peña and Skowroński’s characterization of tame strongly simply connected algebras we prove that such algebras of fixed dimension form an open Z-scheme. There is also an open Z-scheme of all strongly simply connected algebras.     

Stable Equivalence of Selfinjective Artin Algebras of Dynkin Type

We prove new results on the stable equivalences of selfinjective Artin algebras of tilted Dynkin type, extending the main results of Skowroński and Yamagata to arbitrary tilted type.Presented by I. Reiten     

Topological dynamics for multidimensional perturbations of maps with covering relations and Liapunov condition

In this paper, we study topological dynamics of high-dimensional systems which are perturbed from a continuous map on Rm×Rk of the form (f(x),g(x,y)). Assume that f has covering relations determined by a transition matrix A. If g is locally trapping, we show that any small C⁰ perturbed system has a compact positively invariant set restricted to which the system is topologically semi-conjugate to the one-sided subshift of finite type induced by A. In addition, if the covering relations satisfy a strong Liapunov condition and g is a contraction, we show that any small C¹ perturbed homeomorphism has a compact invariant set restricted to which the system is topologically conjugate to the two-sided subshift of finite type induced by A. Some other results about multidimensional perturbations of f are also obtained. The strong Liapunov condition for covering relations is adapted with modification from the cone condition in Zgliczyński (2009) [11]. Our results extend those in Juang et al. (2008) [1], Li et al. (2008) [2], Li and Malkin (2006) [3], Misiurewicz and Zgliczyński (2001) [4] by considering a larger class of maps f and their multidimensional perturbations, and by concluding conjugacy rather than entropy. Our results are applicable to both the logistic and Hénon families.     

The Derived Category of Surface Algebras: the Case of the Torus with One Boundary Component

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobiński and Malicki on gentle 2-cycle algebras.     

Socle Deformations of Selfinjective Algebras of Euclidean Type

We classify all selfinjective finite dimensional algebras over an algebraically closed field which are socle equivalent to the selfinjective algebras of Euclidean type. In particular, nonstandard domestic selfinjective algebras in any characteristic of the base field are exhibited.     

Corrigendum to “A Schneider type theorem for Hopf algebroids” [J. Algebra 318 (1) (2007) 225–269]

In our paper we heavily used the result that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on [T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3, Theorem 2.6], whose proof turned out to contain an unjustified step. Here we prove the main results in our paper without using [T. Brzeziński, A note on coring extensions, Ann. Univ. Ferrara Sez. VII Sci. Mat. LI (2005) 15–27. A corrected version is available at http://arxiv.org/abs/math/0410020v3, Theorem 2.6] and the derived isomorphism of comodule categories.     

Singularity Categories of some 2-CY-tilted Algebras

We define a class of finite-dimensional Jacobian algebras, which are called (simple) polygon-tree algebras, as a generalization of cluster-tilted algebras of type \(\mathbb {D}\). They are 2-CY-tilted algebras. Using a suitable process of mutations of quivers with potential (which are also BB-mutations) inducing derived equivalences, and one-pointed (co)extensions which preserve singularity equivalences, we find a connected selfinjective Nakayama algebra whose stable category is equivalent to the singularity category of a simple polygon-tree algebra. Furthermore, we also give a classification of algebras of this kind up to representation type.     

Hausdorff dimension of the limit sets of some planar geometric constructions

We determine the Hausdorff and box dimension of the limit sets for some class of planar non-Moran-like geometric constructions generalizing the Bedford-McMullen general Sierpiński carpets. The class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines, as well as some non-affine examples, e.g. the flexed Sierpiński gasket.     

Invariance of selfinjective algebras of quasitilted type under stable equivalences

We prove that the class of finite dimensional selfinjective algebras over a field which admit Galois coverings by the repetitive algebras of the quasitilted algebras, with Galois groups generated by compositions of the Nakayama automorphisms with strictly positive automorphisms, is invariant under stable and derived equivalences. Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday