Wild Algebras Research Articles

Dense Orbits of a Two Point Algebra

Finite dimensional algebras over a field can be classified into three classes by representation type. These are finite, tame, and wild. Wild algebras contain copies of the representations of all finite dimensional algebras so classifying the indecomposable representations is not feasible. We then look at representations whose orbit is dense in their associated irreducible component of their representation variety. With this geometric perspective, we can approximate our representations. We give an example of a two point dense orbit algebra of wild representation type and show that it has a dense orbit for dimension vectors of certain types.

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.

Representation embeddings, interpretation functors and controlled wild algebras

We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: that a representation embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse interpretation functor from its image and hence that, in this case, ${\rm Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance, that every wild category of modules interprets the (undecidable) word problem for (semi)groups. We show that the conjecture holds for finitely controlled representation embeddings. Finally we prove that if $R,S$ are finite dimensional algebras over an algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$ then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$ also is domestic.

Quiver Grassmannians and Auslander varieties for wild algebras

Let k be an algebraically closed field and Λ a finite-dimensional k-algebra. Given a Λ-module M, the set Ge(M) of all submodules of M with dimension vector e is called a quiver Grassmannian. If D,Y are Λ-modules, then we consider Hom(D,Y) as a Γ(D)-module, where Γ(D)=End(D)op, and the Auslander varieties for Λ are the quiver Grassmannians of the form GeHom(D,Y). Quiver Grassmannians, thus also Auslander varieties are projective varieties and it is known that every projective variety occurs in this way. There is a tendency to relate this fact to the wildness of quiver representations and the aim of this note is to clarify these thoughts: We show that for an algebra Λ which is (controlled) wild, any projective variety can be realized as an Auslander variety, but not necessarily as a quiver Grassmannian.

Wild algebras have one-point extensions of representation dimension at least four

We show that any wild algebra has a one-point extension of representation dimension at least four, and more generally that it has an n -point extension of representation dimension at least n + 3 . We give two explicit constructions, and obtain new examples of small algebras of representation dimension four.

Tame tensor products of algebras

With the help of Galois coverings, we describe the tame tensor products A K B of basic, connected, nonsimple, nite-dimensional algebras A and B over an algebraically closed eld K. In particular, the description of all tame group algebras AG of nite groups G over nite-dimensional algebras A is completed. Introduction. Throughout the paperK will denote a xed algebraically closed eld. By an algebra we mean a nite-dimensional K-algebra (associa- tive, with an identity) which we moreover assume to be basic and connected. An algebra A can be written as a bound quiver algebra A= KQ=I, where Q = QA is the Gabriel quiver of A and I is an admissible ideal in the path algebra KQ of Q. By Drozd's Tame and Wild Theorem (9) the class of algebras may be divided into two disjoint classes. One class consists of the tame algebras for which the indecomposable modules occur, in each dimension d, in a - nite number of discrete and a nite number of one-parameter families. The second class is formed by the wild algebras whose representation theory com- prises the representation theories of all nite-dimensional algebras over K. Accordingly, we may realistically hope to classify the indecomposable nite- dimensional modules only for the tame algebras. The representation theory of arbitrary tame algebras is still only emerging. We are concerned with the problem of describing when the tensor prod- uct A KB of two nonsimple algebras A and B is tame. The class of tensor product algebras contains several important classes of algebras, including: (1) the group algebrasAG= A KKG of nite groupsG with coecien ts in algebras A;

Cleaving functors and controlled wild algebras

We show that the cleaving functors introduced in [Bautista et al., Invent. Math. 81 (1985) 217] as a tool for proving infinite representation type of finite-dimensional algebras can also be used to establish controlled wildness. The main application is that an algebra is controlled wild if there is an indecomposable projective module with a Loewy factor having a homogeneous direct summand which is of length at least 3. As a second application we derive Han's covering criterion.

Controlled Wild Algebras

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras. 2000 Mathematical Subject Classification: 16G20, 16G60.

TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

We determine the representation type of the algebras whose quiver has precisely two vertices and admits no loops by listing all minimal wild algebras of this form. It turns out that such an algebra A is tame if and only if A/rad3 A is tame, and in this case A degenerates to a special biserial algebra. Moreover, A is wild if and only if it is controlled wild.

Strictly wild algebras with radical square zero

It is proved that a wild algebra with radical square zero is strictly wild if and only if it has a wild hereditary algebra as its factor algebra, which, on the one hand supports one conjecture in [3], on the other hand can be used to construct many non-strictly wild algebras.

The Tits and Euler forms of a tame algebra

Since the introduction by Gabriel in 1972 of the Tits quadratic form associated to finite dimensional hereditary algebras, quadratic forms have played an important role in the representation theory of algebras and related categories. There have been quadratic forms associated to finite dimensional algebras, partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding Grothendieck group (see for example [3,9,14,21]). The class of finite dimensional algebras (associative, with in identity) over an algebraically closed field k is divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and one-parameter families. The second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation. In this paper we establish properties of the Tits and Euler forms associated to tame simply connected algebras. Let A be a basic connected finite dimensional algebra over k. Then A has a presentation A = kQA/I , where QA = (Q0, Q1) is the ordinary quiver of A with set of vertices Q0 and set of arrows Q1 and I is an admissible ideal of the path algebra kQA (see [9]). We shall assume that QA has no oriented cycles (and hence g`dim A < ∞). Then the Euler form χA : ZQ0 → Z is defined by

Geometry of Directing Modules over Tame Algebras

Throughout the paper K denotes a fixed algebraically closed field. By an Ž algebra we mean a finite dimensional K-algebra associative, with an . identity and by a module a finite dimensional left A-module. The class of algebras may be divided into two disjoint subclasses. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory is as complicated as the study of finite dimensional vector spaces together with two noncommuting endomorphisms, for which the classification up to isomorphism is a well-known unsolved problem. Hence, we can realistically hope to describe modules only for tame algebras. Given an algebra A and a nonnegative vector d in Ž . the Grothendieck group K A of A, it is an interesting task to study the 0 Ž . affine variety mod d of A-modules of dimension-vector d and the action A Ž . of the corresponding product G d of general linear groups. For example, Ž . we may ask when the variety mod d is irreducible, smooth, complete A intersection, Gorenstein, Cohen]Macaulay, normal, . . . . The main aim of this paper is to describe the geometry of module Ž . varieties mod d for the dimension-vectors d of arbitrary directing modA w x ules over tame algebras. Recall that following 20 an indecomposable A-module M is said to be directing if it does not belong to a cycle M a M a ??? a M a M of nonzero nonisomorphisms between inde1 r composable A-modules. Directing modules have played an important role

Coxeter polynomials and domination of wild quiver algebras

Coxeter polynomials and domination of wild quiver algebras

On perpendicular categories of stones over quiver algebras

Let H = kzt be the path algebra of a quiver A, where k denotes an arbitrary field. J will always be assumed to be finite without oriented cycles. We denote by mod H the category of finite dimensional left modules over H. An indecomposable H-module X is called a stone provided Ext}~(X, X) = 0. Note that for a stone X the endomorphism ring End X = k (see for example part 1). For a stone X we denote by X l the full subcategory of mod H of the modules Y with Homn(X, Y) = 0 = Ext,(X, Y). Dually, • consists of those H-modules Y satisfying Homn(Y, X) = 0 = Ext,(Y, X). The categories X • and Xare called the right respectively left perpendicular category of X. They are equivalent to mod Ar respectively mod At, where A r and At are again finite dimensional, hereditary k-algebras, and both have one simple module less than H (compare [GL] and [S1]). We denote by Ar respectively zt t the quivers with kAr = Ar and kJz = A,. Since their introduction to Geigle, Lenzing and Schofield, perpendicular categories have become a powerful tool in the representation theory of finite dimensional algebras. They inherit essential properties from mod H, furnish a reduction procedure and open the possibility for proofs by induction. In general, it is difficult problem to determine the (right or left) perpendicular category of a given stone X. We briefly recall what is known. It is quite easy to see (Proposition 2.1) that the quiver Ar differs from a full subquiver of zl by an admissible change of orientation if and only if X is preprojective or preinjective over H. Moreover, if H is tame, then there are only finitely many regular stones X over H, and the structure of its perpendicular categories is contained in [Rill. Hence we may restrict to the context where H is wild and X is a regular stone. Baer and Straul3 proved [B2], [St] that in this situation the perpendicular categories are wild again, but combining results of Xi IX] and Kerner [K3] it follows that certain wild module categories, namely of those algebras classified in [X], do not arise as perpendicular categories of regular stones over wild algebras. On the other hand, investigations of growth numbers in [K3] prove, that for a fixed hereditary, connected, wild algebra with at least three simple modules, there are infinitely many pairwise non isomorphic algebras Ai such that mod Ai is equivalent to X • for a regular stone X over H.

Semicontinuity for representations of one-dimensional Cohen-Macaulay rings

We show that the number of parameters for CM-modules of prescribed rank is semi-continuous in families of CM rings of Krull dimension 1. This transfers a result of Knoerrer from the commutative to the not necessarily commutative case. For this purpose we introduce the notion of ``dense subrings'' which seems rather technical but, nevertheless, useful. It enables the construction of ``almost versal'' families of modules for a given algebra and the definition of the ``number of parameters''. The semi--continuity implies, in particular, that the set of so-called ``wild algebras'' in any family is a countable union of closed subsets. A very exciting problem is whether it is actually closed, hence whether the set of tame algebras is open. However, together with the results of a former paper of the authors the semi-continuity implies that tame is indeed an open property for curve singularities (commutative CM rings). An analogous procedure leads to the semicontinuity of the number of parameters in other cases, like representations of finite dimensional algebras or finite dimensional bimodules.

Epimorphisms of rings, interpretations of modules and strictly wild algebras

Abstract It is shown that an epimorphism from a ring R to a ring S induces an embedding of the Ziegler spectrum of S as a closed subset of the Ziegler spectrum of R.

On degenerations of tame and wild algebras

On degenerations of tame and wild algebras

Stable components of wild tilted algebras

Stable components of wild tilted algebras

On wild tilted algebras which are squids

On wild tilted algebras which are squids

Strongly indefinite quadratic forms and wild algebras

Strongly indefinite quadratic forms and wild algebras