Representation-finite algebras and multiplicative bases.

*(English)*Zbl 0575.16012Let 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.

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 |

##### Keywords:

multiplicative basis; indecomposable finite dimensional modules; representation-finite algebras; local primitive orthogonal idempotents; indecomposable projective right ideals; locally representation finite; normalization theorem; normed presentation; Jacobson radical; generators; mild algebra; lattice of ideals; base category; ray-category; algebra cohomology; fundamental group; universal cover##### References:

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