This paper, which is the first of five series of papers, is devoted to studying the representation theory of a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with the symmetrizable Cartan matrix. To each symmetrizable Cartan matric \(C\) and an orientation \(\Omega\) of \(C\), the authors attach an infinite series of 1-Iwanaga-Gorenstein algebras \(H=H(C,D,\Omega)\) indexed by the different symmetrizers \(D\) of \(C\). These algebras are defined by quivers with relations over arbitrary field \(K\). The algebras \(H\) can be identified with tensor algebras of modulations of the orientation valued graph \(\Gamma\) corresponding to \((C,\Omega)\). However, in contrast to the classical notion of modulation, the rings attached to the vertices of \(\Gamma\) are truncated rings instead of division rings. They also introduce a series of algebras \(\prod=\prod(C,D)\), defined by quivers and relations. These algebras \(\prod\) can be regarded as preprojective algebras of quivers (or more generally of modulated graph) over truncated polynomial rings.
The main aim of the paper under review is to show that analogues of the five following results about the finite connected acyclic quiver \(Q\) hold for the algebras \(H\) and \(\prod\):
1. Gabriel theorem: The quiver \(Q\) is representation-finite if and only if \(Q\) is a Dynkin quiver of type \(A_n,D_n,E_k,k=6,7,8\). In this case, there is a bijection between the isomorphism classes of indecomposable representations of \(Q\) and the set of positive roots of the corresponding simple complex Lie algebra.
2. The discovery of I. N. Bernstein et al. [Usp. Mat. Nauk 28, No. 2(170), 19–33 (1973; Zbl 0269.08001)] of Coxeter functors \(C^\pm(-)=F_{i_n}^\pm\circ\cdots\circ F_{i_1}^\pm:\mathrm{rep}(KQ)\to\mathrm{rep}(KQ)\), which are defined as compositions of reflection functors. They lead to a more conceptual proof of Gabriel’s theorem. Applied to the indecomposable projective (resp., injective) representations they yield a family of indecomposable representations, called preprojective (resp., preinjective) representations.
3. Gabriel’s theorem saying that there are functorial isomorphisms \(TC^\pm(-)\cong\tau^\pm(-)\), where \(T\) is a twist functor and \(\tau(-)\) is the Auslander-Reiten translation.
4. Auslander, Platzeck and Reiten theorem saying that the functors \(F_k^+\) and Hom\(_{KQ} (T,.-)\), where \(F_k^+\) is a BGP-reflection functor and \(T\) is the associated APR-tilting module, are equivalent.
5.Gelfand and Ponomarevs discovery of the preprojective algebra \(\prod(Q)\) of the quiver \(Q\), and their result that \(\prod(Q)\), seen as a module over \(KQ\), is isomorphic to the direct sum of all preprojective \(KQ\)-modules. The algebra \(\prod(Q)\) is isomorphic to the tensor algebra \(T_{KQ}(\mathrm{Ext}^1_{KQ}(D(KQ),KQ))\), where \(D\) denotes the duality with respect to the base field \(K\).
For Part II, see [the authors, Int. Math. Res. Not. 2018, No. 9, 2866–2898 (2018; Zbl 1408.16012)].