Classification Of Finite-dimensional Algebras Research Articles

The Tits and Euler forms of a tame algebra

Since the introduction by Gabriel in 1972 of the Tits quadratic form associated to finite dimensional hereditary algebras, quadratic forms have played an important role in the representation theory of algebras and related categories. There have been quadratic forms associated to finite dimensional algebras, partially ordered sets and other objects in order to determine their representation type and the classes of indecomposable representations in the corresponding Grothendieck group (see for example [3,9,14,21]). The class of finite dimensional algebras (associative, with in identity) over an algebraically closed field k is divided into two disjoint classes. One class consists of tame algebras for which the indecomposable modules occur, in each dimension, in a finite number of discrete and one-parameter families. The second class consist of wild algebras whose representation theory is as complicated as the classification of pairs of matrices up to simultaneous conjugation. In this paper we establish properties of the Tits and Euler forms associated to tame simply connected algebras. Let A be a basic connected finite dimensional algebra over k. Then A has a presentation A = kQA/I , where QA = (Q0, Q1) is the ordinary quiver of A with set of vertices Q0 and set of arrows Q1 and I is an admissible ideal of the path algebra kQA (see [9]). We shall assume that QA has no oriented cycles (and hence g`dim A < ∞). Then the Euler form χA : ZQ0 → Z is defined by

Left monomial rings---a generalization of monomial algebras

A particularly tractable class of finite dimensional algebras defined by quivers and relations is that of monomial algebras, i.e., those for which the ideal of relations is generated by a collection of paths. The homological structure of these algebras is very well understood and some constructions for them are even algorithmic. There is, for example, an algorithm due to Green, Happel and Zacharia (see [11], where the algebras are called 0-relations algebras) for constructing the projective resloutions of the simple modules which determines their projective dimensions in a predictable number of steps. The Cartan determinant conjecture is known to be true for these algebras since they are positively graded ([18]) and the finitistic dimensions are finite ([12] and [13]) and are thoroughly understood due to the recent work of B.Z. Huisgen ([13] and [14]). Other properties of monomial algebras will be cited below. Here we introduce a class of left artinian rings which includes that of monomial algebras and we show that many of the above results remain valid within it. The proposed rings, called left monomial rings (see Definition 2.2) will include monomial algebras and the more general 0-relations algebras given by species and 0-relations, as well as left (almost) serial rings, right serial rings, hereditary artinian rings and more. To each such ring R is associated a monomial algebra A so that, in many ways, R and A have the same homological properties (see Theorem 2.3); enough so that, for example, the projective dimensions of the corresponding simple modules are the same. (See Theorem 2.3 and its corollary.)

Algebras with a finite number of classes of nonequivalent indecomposable modules

It is proved that the class of finite-dimensional algebras over fields with the property stated in the title is closed with respect to extensions by means of semi-simple artinian algebras.

Commutative QF-1 Artinian Rings are QF

In a recent paper, D. R. Floyd proved several results on algebras, each of whose faithful representations is its own bicommutant (=R. M. Thrall's QF-1 algebras, a generaliza- tion of QF-algebras) among which was the theorem in the title for algebras. We obtain our extension of Floyd's result by use of inter- lacing modules, replacing his arguments involving the representa- tions themselves. In (10), Thrall observed that the class of finite-dimensional algebras over which every faithful representation has the double centralizer property (i.e., is its own bicommutant) properly contains the class of quasi-Frobenius ( = QF) algebras. He called the members of the former class QF-1 algebras and posed the intriguing problem of characterizing these algebras in terms of ideal structure. Solutions for this problem have been given for generalized uniserial algebras (5) and for commutative algebras (4). Every faithful module M over a QF ring R has the double centralizer property in the sense that the natural homomorphism X: R--Homc(M, M) (where C= HomR(M, M)) is onto (see (2, ?59)). Thus Thrall's definition and his problem extend naturally to QF-1 artinian rings. In view of recent results on the dominant dimension of an artinian ring (see (1, Theorem 2) or (8, Lemma 9)), the characterization of QF-1 generalized uniserial algebras (and its proof) given in (5) re- mains valid for generalized uniserial rings. In this note we prove the theorem of the title, thus extending a theorem that Floyd proved for finite-dimensional algebras by means of matrix representations (4). It is not difficult to show that a direct sum of rings is QF or QF-1 if and only if so is each of the direct summands. Thus for our purposes we may assume that R is a commutative local artinian ring. Accord- ing to Nakayama's original definition (9, p. 8), such a ring is QF if and only if its R-socle (i.e., its largest semisimple R-submodule) S(R) = S(RR) is simple. We shall prove the theorem by constructing, in the event that S(R) is not simple, a faithful module whose double cen- tralizer has an R-socle larger than that of R. The methods used in this