1. Introduction and main result 1.1. Let k be an algebraically closed eld. For basic notation and results we refer to GR] and Ri]. All algebras which we will consider are nite-dimensional basic algebras over k. Sometimes we will consider such an algebra as a nite spectroid. For an algebra A we denote by mod?A the category of nite-dimensional right A-modules and by ind?A a spectroid of this aggr...

0. Introduction 1 1. Loop algebras and quantum groups. 3 2. Coherent sheaves and Quot schemes 6 3. Transfer functors 12 4. Induction and restriction functors 15 5. The algebra UA 22 6. Proof of Theorem 5.1 i) 28 7. Harder-Narasimhan filtration 32 8. Some Lemmas. 34 9. Proof of Theorem 5.1.ii) (finite type) 38 10. Proof of Theorem 5.1ii) (tame type) 41 11. Corollaries 48 12. Canonical basis of U...

A nite algebra C is called minimal with respect to a pair < of its congruences if every unary polynomial f of C is either a permutation, or f( ) . It is the basic idea of tame congruence theory developed by Ralph McKenzie and David Hobby [7] to describe nite algebras via minimal algebras that sit inside them. As shown in [7], minimal algebras have a very restricted structure. This paper present...

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. prove that every such algebra is derived equivalent to a higher spherical studied by Erdmann Skowroński (Arch. Math. 114, 25–39, 2020), hence it tame symmetric periodic period 4. This together with the results (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. 2020) shows trivial exten...

By Drozd's Tame and Wild Theorem, ®nite-dimensional algebras over an algebraically closed ®eld k may be divided into two disjoint classes: one class consists of tame algebras whose indecomposable modules can be parametrized using only one continuous parameter, the other class consists of wild algebras which have families of indecomposable modules depending on arbitrarily many continuous paramet...

In [CK] and [SZ], the authors constructed the bases of cluster algebras of finite types and of type e A1,1, respectively. In this paper, we will deduce Z-bases for cluster algebras of affine types and compare Z-bases with generic variables in [Du2]. Moreover, we give inductive formulas for computing the multiplication between two generalized cluster variables associated to objects in a tube.

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (in...

The Auslander-Reiten quiver of a finite-dimensional associative algebra A encodes information about the indecomposable finite-dimensional representations of A and their homomorphisms. A component of the AuslanderReiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the AuslanderReiten translation. Preproj...