Deformed preprojective algebras of generalized Dynkin type

By J. Białkowski, K. Erdmann, Andrzej Skowronski. 2007

55 citations recorded by the source corpus.

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs An (n > 1), D n (n > 4), EG, E 7 , Eg and L n (n ≥ 1) and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift v -1 S of every non-projective simple module S is isomorphic to its third syzygy Ω 3 S.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2038c1d7982a0808366c72af73ce2831a90c9082

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Representations of Algebras and Related Topics (2011, background): “[1]). They have ﬁnitely many indecomposables. A 1-cluster-tilting object corresponds to a basic generator-cogenerator, therefore Π2(kQ) is the only 1-cluster-tilting object in C1(kQ). From results in [18] and [1], we deduce the following. Theorem 3.33. Let Q be a Dynkin quiver. (a) [1] The deformed ”
Calabi-Yau Frobenius algebras (2007, methodology): “from ΠA2n by passing to ﬁxed points under the Nakayama automorphism, ΠTn := (ΠA2n) η. In other words, this is associated to a graph of Tn type. In [Eu] (using the same resolution that appears in e.g. [BES07], which may be obtained from the Schoﬁeld resolution for type A2n), it is proved that ΠTn is ”
Periodic resolutions and self-injective algebras of finite type (2008, methodology): “If char(k) 6= 2 we find that the syzygy functor has order 6, even though Ω(3)(M) ∼= M for every nonprojective indecomposable A-module M [4].”
Calabi–Yau triangulated categories (2008, background): “Namely, as shown in [1], the k-linear categories underlying the 1-Calabi-Yau triangulated categories with finitely many isomorphism classes of indecomposables are precisely the categories of finite-dimensional projective modules over deformed preprojective algebras of generalized Dynkin type introdu”