B. Huisgen‐Zimmermann

activity

19972024

most citedThe geometry of finite dimensional algebras with vanishing radical square

8 citations · 20 across the 32 of their papers we have counts for

collaborators

Showing 2014Show all

25 papers · 1 filter

math.RT2014

Irreducible components of varieties of representations I. The local case

Birge Huisgen-Zimmermann

Let $Λ$ be a local truncated path algebra over an algebraically closed field $K$ , i.e., $Λ$ is a quotient of a path algebra $K Q$ by the paths of length $L + 1$ , where $Q$ is the quiv…

math.RT2014★ 8 cited

The geometry of finite dimensional algebras with vanishing radical square

Frauke M. Bleher, Ted Chinburg, Birge Huisgen-Zimmermann

Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducib…

math.RT2014★ 2 cited

Fine and coarse moduli spaces in the representation theory of finite dimensional algebras

B. Huisgen-Zimmermann

We discuss the concepts of fine and coarse moduli spaces in the context of finite dimensional algebras over algebraically closed fields. In particular, our formulation of a moduli…

math.RT2014

Top-stable degenerations of finite dimensional representations II

H. Derksen, B. Huisgen-Zimmermann, J. Weyman

Let $Λ$ be a finite dimensional algebra over an algebraically closed field. We exhibit slices of the representation theory of $Λ$ that are always classifiable in stringent geometri…

math.RT2014★ 1 cited

A hierarchy of parametrizing varieties for representations

B. Huisgen-Zimmermann

The primary purpose is to introduce and explore projective varieties, $GRASS_{d} (Λ)$ , parametrizing the full collection of those modules over a finite dimensional algebr…

math.RT2014

Generic representation theory of quivers with relations

E. Babson, B. Huisgen-Zimmermann, R. Thomas

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $Λ$ are explored, with regard to both their geometry an…

math.RT2014★ 1 cited

Top-stable degenerations of finite dimensional representations I

Birge Huisgen-Zimmermann

Given a finite dimensional representation $M$ of a finite dimensional algebra, two hierarchies of degenerations of $M$ are analyzed in the context of their natural orders: the pose…

math.RT2014

Classifying representations by way of Grassmannians

Birge Huisgen-Zimmermann

Let $Λ$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isom…

math.RA2014

Stacks of algebras and their homology

Nancy Heinschel, Birge Huisgen-Zimmermann

For any increasing function $f : N \to N_{\geq 2}$ which takes only finitely many distinct values, a connected finite dimensional algebra $Λ$ is constructed, wi…

math.RT2014

Analyzing the Structure of Representations via Approximations

Birge Huisgen-Zimmermann

Primarily this paper presents an expository report on alternatives to the traditional methods of classifying representations of finite dimensional algebras. Some new results illust…

math.RT2014

The geometry of uniserial representations of finite dimensional algebras III: Finite uniserial type

Birge Huisgen-Zimmermann

A description is given of those sequences $S = (S (0), S (1), \dots, S (l))$ of simple modules over a finite dimensional algebra for which there are only finitely many uniserial mo…

math.RT2014

The geometry of uniserial representations of finite dimensional algebras I

Birge Huisgen-Zimmermann

It is shown that, given any finite dimensional, split basic algebra $Λ = K Γ/ I$ (where $Γ$ is a quiver and $I$ an admissible ideal in the path algebra $K Γ$ ), there is a finite list…

math.RA2014

The finitistic dimension conjectures -- a tale of 3.5 decades

Birge Huisgen-Zimmermann

We review the history of several homological conjectures, both from a chronological and a methodological point of view.

math.RT2014

Direct products of modules and the pure semisimplicity conjecture. Part II

Birge Huisgen-Zimmermann, Manuel Saorín

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following p…

math.RT2014

Direct sums of representations as modules over their endomorphism rings

Birge Huisgen-Zimmermann, Manuel Saorín

This paper is devoted to the study of the endo-structure of infinite direct sums $⨁_{i \in I} M_{i}$ of indecomposable modules $M_{i}$ over a ring $R$ . It is centered on the fo…

math.RA2014

Direct products of modules and the pure semisimplicity conjecture

Birge Huisgen-Zimmermann, Frank Okoh

It is shown that, if $R$ is either an Artin algebra or a commutative noetherian domain of Krull dimension $1$ , then infinite direct products of $R$ -modules resist direct sum decomp…

math.RA2014

Purity, algebraic compactness, direct sum decompositions, and representation type

Birge Huisgen-Zimmermann

This survey article is devoted to the notions of purity, algebraic and $Σ$ -algebraic compactness, direct sum decompositions, and representation type in the category of modules over…

math.RT2014

The phantom menace in representation theory

Birge Huisgen-Zimmermann

Our principal goal in this overview is to explain and motivate the concept of a phantom in the representation theory of a finite dimensional algebra $Λ$ . In particular, we exhibit…

math.RT2014

Viewing finite dimensional representations through infinite dimensional ones

Dieter Happel, Birge Huisgen-Zimmermann

We develop criteria for deciding the contravariant finiteness status of a subcategory $A \subseteq Λ -mod$ , where $Λ$ is a finite dimensional algebra. In particular, given a…

math.RT2014★ 2 cited

A Homological Bridge Between Finite and Infinite Dimensional Representations of Algebras

B. Huisgen-Zimmermann, S. O. Smalø

Given a finite dimensional algebra $Λ$ , we show that a frequently satisfied finiteness condition for the category $P^{\infty} (Λ - mod)$ of all finitely generated (left) $…

math.RT2014★ 6 cited

Repetitive resolutions over classical orders and finite dimensional algebras

K. R. Goodearl, B. Huisgen-Zimmermann

Repetitiveness in projective and injective resolutions and its influence on homological dimensions are studied. Some variations on the theme of repetitiveness are introduced, and i…

math.RT2014

Co- Versus Contravariant Finiteness of Categories of Representations

B. Huisgen-Zimmermann, S. O. Smalø

This article supplements recent work of the authors. (1) A criterion for failure of covariant finiteness of a full subcategory of $Λ -mod$ is given, where $Λ$ is a finite dim…

math.RT2014

Varieties of uniserial representations IV. Kinship to geometric quotients

Klaus Bongartz, Birge Huisgen-Zimmermann

Let $Λ$ be a finite dimensional algebra over an algebraically closed field, and $S$ a finite sequence of simple left $Λ$ -modules. In [6, 9], quasiprojective algebraic variet…

math.RT2014

The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness

Klaus Bongartz, Birge Huisgen-Zimmermann

We provide two alternate settings for a family of varieties modeling the uniserial representations with fixed sequence of composition factors over a finite dimensional algebra. The…