## Representation-finite algebras and multiplicative bases.(English)Zbl 0575.16012

Let A be an algebra over an algebraically closed field K. A vector space K-basis of A is called multiplicative if the product of two basis-vectors is either a basis-vector or zero. It is proved in the paper that A has a multiplicative basis if A is representation-finite i.e. $$\dim_ KA$$ is finite and A admits only finitely many isoclasses of indecomposable finite dimensional modules. As a corollary the authors get that the number of isoclasses of representation-finite algebras of a given dimension is finite and that any representation-finite algebra is standard if char $$K\neq 2.$$
The main results of the paper are proved in a more general categorical context i.e. for algebras A having a complete set (in general infinite) of local primitive orthogonal idempotents $$e_ i$$ such that $$A\cong \oplus e_ iA\cong \oplus Ae_ i$$ (in general A has no identity). In this case A can be identified with the category consisting of indecomposable projective right ideals $$e_ iR$$. A is called locally bounded if A is basic and $$e_ iA$$, $$Ae_ i$$ are finite dimensional for all $$e_ i$$. If, in addition, the number of isoclasses of indecomposable right A-modules N with $$Ne_ i\neq 0$$ is finite for given $$e_ i$$ then A is called locally representation finite.
One of the main results of the paper is a ”normalization theorem” which asserts that every locally representation-finite algebra A admits a normed presentation. This means that there is a map $$\pi$$ carrying over points i of the quiver $$Q=Q_ A$$ of A on idempotents $$e_ i$$ and mapping arrows between i and j onto elements in $${\mathcal R}A(e_ i,e_ j)$$ whose classes modulo $${\mathcal R}^ 2A(e_ i,e_ j)$$ form a basis of $${\mathcal R}A/{\mathcal R}^ 2A(e_ i,e_ j)$$, where $${\mathcal R}A$$ is the Jacobson radical of A. Moreover, the kernel $$I^{\pi}$$ of the algebra homomorphism $$\Phi^{\pi}: KQ\to A$$ induced by $$\pi$$ admits a system of generators formed by some paths in Q and some differences of two parallel paths. The presentation $$\pi$$ is called semi-normed if $$I^{\pi}$$ admits a system of generators composed of differences v-tu where $$t\in K$$ and u,v are parallel paths in Q. The authors also prove a ”semi-normalization theorem” which asserts that every mild algebra A admits a semi-normed presentation. Here A is mild if the lattice of ideals of A is distributive and A/I is locally representation-finite for each ideal $$I\neq 0.$$
Proofs of the main results involve new useful concepts and notions. In particular notions of a base category, a ray-category, a standard form and a cleaving diagram are introduced in the paper and algebra cohomology groups are usefully involved in the proofs. It is proved that if P is a ray-category having no infinite chains of some special forms and such that at each point at most 3 arrows stop and at most 3 arrows start then $$H^ n(P,Z)=0$$ for $$n\geq 2$$ and any Abelian group Z. Moreover, the fundamental group of P is free (non-commutative). It is also proved that a distributive algebra is locally representation-finite (resp. mild) iff its standard form is locally representation-finite (resp. mild). Moreover, if char $$K\neq 2$$ then every mild algebra whose ray-category contains no infinite (special) chain is standard.
The results of the paper allow us to reduce the classification of representation-finite algebras to a combinatorial problem because of the following generalization of a result of Bongartz proved in the paper. The linearization K(P) of a ray-category P is locally representation-finite if the universal cover $$\tilde P$$ of P is interval-finite and contains no critical convex subcategory in the list of Bongartz-Happel-Vossieck presented in the paper. The reviewed paper contains many new ideas and among other things presents an interesting new approach to the study of representation-finite algebras as well as to the covering technique in the representation theory of finite dimensional algebras.
Reviewer: D.Simson

### MSC:

 16Gxx Representation theory of associative rings and algebras 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 16P10 Finite rings and finite-dimensional associative algebras 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16B50 Category-theoretic methods and results in associative algebras (except as in 16D90) 18A25 Functor categories, comma categories 55U10 Simplicial sets and complexes in algebraic topology
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