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Deformed preprojective algebras of generalized Dynkin type. (English) Zbl 1117.16005

The authors introduce the class of deformed preprojective algebras of generalized Dynkin graphs \(\mathbb{A}_n\), \(\mathbb{D}_n\), \(\mathbb{E}_6\), \(\mathbb{E}_7\), \(\mathbb{E}_8\) and \(\mathbb{L}_n\). Let \(\Delta\) be any of the generalized Dynkin graphs and let \(R(\Delta)\) be a finite-dimensional self-injective algebra, which is isomorphic to the local algebra \(eP(\Delta)e\) of the preprojective algebra \(P(\Delta)\) at the exceptional primitive idempotent \(e\) of \(P(\Delta)\). For each element \(f\) which belongs to the radical square of \(R(\Delta)\), the authors define the deformed preprojective algebra \(P^f(\Delta)\) of type \(\Delta\).
The main result of this paper is the following Theorem 1.2. Let \(\Lambda\) be a basic, connected, finite-dimensional, self-injective algebra over an algebraically closed field \(K\). The following statements are equivalent. (i) \(\Lambda\) is isomorphic to a deformed preprojective algebra \(P^f(\Delta)\) of generalized Dynkin type \(\Delta\). (ii) \(\Omega^3S\cong\nu^{-1}S\), for any non-projective simple right \(\Lambda\)-module \(S\), where \(\nu^{-1}\) is the inverse Nakayama shift and \(\Omega^3S\) is the third syzygy of \(S\).
Moreover, in Theorem 1.1, the authors describe basic homological properties (such that Cartan matrix, Nakayama automorphisms, syzygy, …) of the algebras \(P^f(\Delta)\).
The final result (Theorem 1.3) of the paper shows that there exist deformed preprojective algebras \(P^f(\Delta)\), which are not isomorphic to the ordinary preprojective algebras \(P(\Delta)\).

MSC:

16G20 Representations of quivers and partially ordered sets
16G10 Representations of associative Artinian rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16D50 Injective modules, self-injective associative rings
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
16E05 Syzygies, resolutions, complexes in associative algebras
16S80 Deformations of associative rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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