zbMATH — the first resource for mathematics

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

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
Full Text: DOI EuDML
[1] [B1] Bautista, R.: Irreducible morphisms and the radical of a category. An. Inst. Mat. Univ. Nac. Aut?noma M?xico22, 83-135 (1982) · Zbl 0594.16020
[2] [B2] Bautista, R.: On algebras of strongly unbounded representation type. Comment. Math. Helv. (To appear)
[3] [BLS1] Bautista, R., Larri?n, F., Salmer?n, L.: On simply connected algebras. J. Lond. Math. Soc. (2),27, 212-220 (1983) · Zbl 0511.16022
[4] [BLS2] Bautista, R., Larri?n, F., Salmer?n, L.: Cleaving diagrams in representation theory. (To appear)
[5] [Bo1] Bongartz, K.: Zykellose Algebren sind nicht z?gellos. Proc. Ottawa 1979. Springer Lect. Notes 832, pp. 97-102
[6] [Bo2] Bongartz, K.: Treue einfach zusammenh?ngender Algebren I. Comment. Math. Helv.57, 282-330 (1982) · Zbl 0502.16022
[7] [Bo3] Bongartz, K.: Algebras and quadratic forms. J. Lond. Math. Soc.28, 461-469 (1983) · Zbl 0532.16020
[8] [Bo4] Bongartz, K.: A criterion for finite representation type. Math. Ann.269, 1-12 (1984) · Zbl 0552.16012
[9] [Bo5] Bongartz, K.: Critical simply connected algebras. Manuscr. Math.46, 117-136 (1984) · Zbl 0537.16024
[10] [Bo6] Bongartz, K.: Indecomposable modules are standard. Comment. Math. Hellv. (To appear)
[11] [BoG] Bongartz, K., Gabriel, P.: Covering spaces in representation theory. Invent. Math.65, 331-378 (1982) · Zbl 0482.16026
[12] [BrG] Bretscher, O., Gabriel, P.: The standard form of a representation-finite algebra. Bull. Soc. Math. Fr.111, 21-40 (1983) · Zbl 0527.16021
[13] [EM] Eilenberg, S., Mac Lane, S.: Acyclic models. Am. J. Math.75, 189-199 (1953) · Zbl 0050.17205
[14] [EZ] Eilenberg, S., Zilber, J.A.: Semi-simplicial complexes and singular homology. Ann. Math.51, 499-513 (1950) · Zbl 0036.12601
[15] [F1] Fischbacher, U.: The representation-finite algebras with three simple modules. Proceedings of the Fourth Intern. Confer. on Representations of Algebras. Ottawa, Carleton Univ., August 1984
[16] [F2] Fischbacher, U.: Une nouvelle preuve d’un th?or?me de nazarova et Roiter. Comptes-Rendus Ac. Sc. Paris300, 1-9, 259-263 (1984)
[17] [G1] Gabriel, P.: Unzerlegbare Darstellungen I. Manuscr. Math.6, 71-103 (1972) · Zbl 0232.08001
[18] [G2] Gabriel, P.: Finite representation type is open. Proc. Ottawa 1974. Springer Lect. Notes 488, pp. 132-155
[19] [G3] Gabriel, P.: Auslander-Reiten sequences and representation finite algebras. Proc. Ottawa 1979. Springer Lect. Notes 831, pp. 1-71
[20] [G4] Gabriel, P.: The universal cover of a representation-finite algebra. Proc. Puebla 1980. Springer Lect. Notes 903, pp. 68-105
[21] [GZ] Gabriel, P., Zisman, M.: Calculus of fractions and Homotopy Theory. In: Ergebnisse der Mathematik, vol. 35. Berlin-Heidelberg-New York: Springer 1967 · Zbl 0186.56802
[22] [HV] Happel, D., Vossieck, D.: Minimal algebras of infinite representation type with preprojective component. Manuscr. Math.42, 221-243 (1983) · Zbl 0516.16023
[23] [J1] Jans, J.: On the indecomposable representations of algebras. Ann. Math.66, 418-429 (1957) · Zbl 0079.05203
[24] [J2] Jans, J.: The representation type of algebras and subalgebras. Can. J. Math.10, 39-44 (1958) · Zbl 0101.02504
[25] [Kf] Kraft, H.: Geometric methods in representation theory. Proc. Puebla 1980. Springer Lect. Notes 944, pp. 180-257
[26] [K1] Kupisch, H.: Symmetrische Algebren mit endlich vielen unzerlegbaren Darstellungen, I, II. J. Reine Angew. Math.219, 1-25 (1965);245, 1-14 (1970) · Zbl 0132.28002
[27] [K2] Kupisch, H.: Basis-algebren symmetrischer Algebren und eine Vermutung von Gabriel. J. Algebra55, 58-73 (1978) · Zbl 0406.16020
[28] [KS] Kupisch, H., Scherzler, E.: Symmetric algebras of finite representation type. Proc. Ottawa 1979. Springer Lect. Notes 832, pp. 328-368 · Zbl 0471.16021
[29] [Kg] Krugliak, C.A.: Representations of algebras with zero radical square. Zap. Nau?n. Sem. LOMI28, 60-68 (1972)
[30] [ML1] Mac Lane, S.: Homology, 422p. Berlin-Heidelberg-New York: Springer 1970
[31] [ML2] Mac Lane, S.: Categories for the working mathematician, 262p. Berlin-Heidelberg-New York: Springer 1971 · Zbl 0232.18001
[32] [MP1] Mart?nez-Villa, R., De La Pe?a, J.A.: Automorphisms of representation-finite algebras. Invent. Math.72, 359-362 (1983) · Zbl 0505.16013
[33] [MP2] Mart?nez-Villa, R., De La Pe?a, J.A.: Multiplicative basis for algebras whose universal cover has no oriented cycle. J. Algebra87, 389-395 (1984) · Zbl 0537.16026
[34] [Mz1] Mazzola, G.: The algebraic and geometric classification of associative algebras of dimension five. Manuscr. Math.27, 81-101 (1979) · Zbl 0446.16033
[35] [Mz2] Mazzola, G.: Generic finite schemes and Hochschild cocycles. Comment. Math. Helv.55, 267-293 (1980) · Zbl 0463.14004
[36] [NR] Nazarova, L.A., Roiter, A.V.: Categorical matrix problems and the Brauer-Thrall conjecture. Preprint Kiev (1973). German version in: Mitt. Math. Semin. Giessen115, 1-153 (1975)
[37] [O] Ovsienko, S.A.: Comments on Roiter’s existence proof of multiplicative bases. Private communication (June 1982)
[38] [Re] Reiten, I.: Introduction to the representation theory of Artin algebras. Preprint 67p. University of Trondheim (1983)
[39] [Rd] Riedtmann, Ch.: Many algebras with the same Auslander-Reiten quiver. Bull. London Math. Soc.15, 43-47 (1983) · Zbl 0499.16020
[40] [Ri] Ringel, C.M.: Indecomposable representations of finite dimensional algebras. Report ICM 82 Warszawa, 14p. (1983)
[41] [R] Roiter, A.V.: Generalization of Bongartz’ theorem. Preprint Math. Inst. Ukranian Acad. of Sciences, pp. 1-32, Kiev (1981)
[42] [SW] Skowronski, A., Waschb?sch, J.: Representation finite biserial algebras. J. Reine Angew. Math.345, 172-181 (1983) · Zbl 0511.16021
[43] [St] Stallings, J.R.: On torsion-free groups with infinitely many ends. Ann. Math.88, 312-334 (1968) · Zbl 0238.20036
[44] [Sw] Swan, R.G.: Groups of cohomological dimension one. J. Algebra12, 585-601 (1969) · Zbl 0188.07001
[45] [Vo] Voghera, G.: Zusammenstellung der irreduziblen complexen Zahlensysteme in sechs Einheiten. Denkschr. der math.-naturwiss. Klasse der K. Akademie der Wiss. zu Wien84, 269-328 (1908) · JFM 39.0499.09
[46] [VW] Von H?hne, H., Waschb?sch, J.: Die StrukturN-reihiger Algebren. Manuskript, Freie Universitat Berlin, 16 S. (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.