Benjamin Dequêne
Published 2022-05-17Version 1
Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowro\'{n}ski 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. If $\mathcal{Q} = (Q, I)$ is a finite connected gentle quiver, we show a characterization of the vertices $m$ such that representations consisting of direct sums of indecomposable representations all including $m$ in their support, the subcategory of which we denote by $\mathscr{C}_{\mathcal{Q},m}$, are determined up to isomorphism by this invariant.
On $τ$-tilting subcategories
Triangulated subcategories of extensions, stable t-structures, and triangles of recollements
On the indecomposability in the category of representations of a quiver and in its subcategory of orthoscalar representations
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