W. W. Crawley-Boevey's research while affiliated with Bielefeld University and other places

... where the orientation of the arrows in (G o (x)) −1 (Q sp 1 ) does not depend on the relation > for words. Following Crawley-Boevey [CrB88], this simplified construction is sufficient to describe the indecomposable representations of the skewed-gentle algebras (a special case of clannish algebras) in our situation, where we consider kQ with char(k) = 2 and defining relations X 2 − 1 for all special loops. ...

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On homomorphism and generically $\tau$-reduced components for skewed-gentle algebras

... There is also an extensive literature for the construction of embeddings of groups H into the profinite completions of torsion-free, finitely generated nilpotent and solvable groups. For example, the work [18] shows that if G is a finitely-generated, torsion-free nilpotent group, then the profinite completion G is torsion free, so if D ⊂ G is a closed subgroup, then it must be a Cantor group. ...

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Molino theory for matchbox manifolds

... Our work also differs from theirs in the methods used. They used geometric models to study gentle algebras and their representations (developed in [ABCJP10], [OPS18], [PPP19], [BS21]), whereas we use combinatorial descriptions of representations and morphisms between them (developed in [GP68], [BR87], [CB89], [Kra91]). ...

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CHARACTERISATION OF BAND BRICKS OVER CERTAIN STRING ALGEBRAS AND A VARIANT OF PERFECTLY CLUSTERING WORDS

... (i) Q is the matrix algebra In each one of these cases, it is shown in the proof of [3, (6.8)], using previous work by V. Dlab and M. Ringel [15]& [14], and Crawley-Boevey [10], that there is a bounded principal ideal domain Γ and an exact full and faithful functor Ψ : Γ-Mod−−→Q-Mod mapping indecomposable Γ-modules of finite length onto regular Q-modules. Moreover, Q-reg ∼ = Ψ(Γ-mod) U, where U is a uniserial subcategory of Q-mod generated by a simple regular module. ...

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A representation embedding for algebras of infinite type

... The purpose of this section is to deduce Theorem B from the results of [6]. We fix an algebraically closed field of characteristic p>O. ...

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Torsion-free soluble groups, completions, and the zero divisor conjecture

... Skew-gentle algebras are a special case of clannish algebras [CB89]; however, when working over a field of characteristic different from 2, skew-gentle algebras are conveniently described (up to Morita-equivalence) as skew-group algebras of gentle algebras with an action of a group of order two on their Gabriel quiver. The group action lifts to an action on the graded surface with dissection giving rise to the gentle algebra, and thus to the partially wrapped Fukaya category. ...

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Skew-group $A_{\infty}$-categories as Fukaya categories of orbifolds

... The trichotomy theorem of Drozd [62] (see also Crawley-Boevey [55]) for finite dimensional algebras states that every finite dimensional algebra is is of finite, tame or wild representation type, and these types are mutually exclusive. Roughly speaking, finite representation type means that there are only finitely many isomorphism classes of finitely generated indecomposable modules. ...

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Classifying spaces of finite groups of tame representation type

... The proof of (1.4) relies on the theory of differential tensor algebras (ditalgebras) and reduction functors first developed by the Kiev School of representation theory, see [18] and [16] (see also [11]). For the general background on ditalgebras and their module categories, we refer the reader to [5]. ...

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A representation embedding for algebras of infinite type