A remark on finitely presented infinite-dimensional algebras (original) (raw)

Correction and Addendum to "On Algebras of Finite Representation Type

Transactions of The American Mathematical Society, 1969

In this note we retain the notation and terminology of our original paper [2]. Our purpose is to repair the statement and proof of Theorem 2.3 of [2], and using its generalization (Theorem A below) as a prototype, we can by use of an extension of Corollary 3.3 of [2] (Theorem B below) remove the weighty hypothesis of "large kernels" from Theorems 3.4, 3.5, 3.6 and Corollary 3.7 of [2]. This latter task derives some urgency from the conjecture of J. P. Jans [3] that an algebra with large kernels has finite module type. This conjecture, if true, would of course annihilate any importance of the above-mentioned results of [2] in its original form as far as the Brauer-Thrall conjecture is concerned^).

Infinite-dimensional algebras without simple bases

Linear and Multilinear Algebra, 2019

The study of the recently introduced notions of amenability, congeniality and simplicity of bases for infinite dimensional algebras is furthered. A basis B over an infinite dimensional F-algebra A is called amenable if F B , the direct product indexed by B of copies of the field F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. If B is congenial to C but C is not congenial to B, then we say that B is properly congenial to C. An amenable basis B is called simple if it is not properly congenial to any other amenable basis and it is called projective if there does not exist any amenable basis which is properly congenial to B. We introduce a family of algebras and study these notions in that context; in particular, we show that the family includes examples of algebras without simple or projective bases. This same family also serves to illustrate the one-sided nature of amenability and simplicity as we produce examples of bases which are amenable only on one side and, likewise, bases which are only one one-sided simple.

Filtrations and distortion in infinite-dimensional algebras

Journal of Algebra, 2011

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also reflects the degree of complexity of the membership problem for the elements of this algebra in this subalgebra. One of our goals here is to investigate, mostly in the case of associative or Lie algebras, if a tame filtration of an algebra can be induced from the degree filtration of a larger algebra.

On the noetherianity of some associative finitely presented algebras

Journal of Algebra, 1991

We consider fmitely generated aasocnttive algebras over a tixed held K of arbitrary characteristic. For such an algebra A we impose some structural restrictions (WC call A strictly ordcrcd). We arc interested in the implication of strict order on A for its noetherian properties. In particular, we prove that if A is a graded standard hmtely presented strictly ordered algebra, then .4 is left noetherian If and only if it is almost commutative. In this case A has polynomial growth.

On classification of finite dimensional algebras

Classification and invariants, with respect to basis changes, of finite dimensional algebras are considered. An invariant open, dense (in the Zariscki topology) subset of the space of structural constants is defined. The algebras with structural constants from this set are classified and a basis to the field of invariant rational functions of structural constants is provided.

On algebras of finite representation type

Transactions of The American Mathematical Society, 1969

Introduction. Since D. G. Higman proved that bounded representation type and finite representation type are equivalent for group algebras at prime characteristic, there has been a renewed interest in the Brauer-Thrall conjecture that bounded representation type implies finite representation type for arbitrary algebras. The main purpose of this paper is to present a new approach to this conjecture by showing the relevance (when the base field is algebraically closed) of questions concerning the structure of indecomposable modules of certain special types, namely, the stable (every maximal submodule is indecomposable), the costable (having the dual property), and the stable-costable (having both properties) indecomposable modules. The main tools are the Sandwich Lemma (1.2) which is proved using an old observation of É. Goursat, an observation of A. Heller, C. W. Curtis, and D. Zelinsky concerning quasifrobenius (QF) rings (Proposition 2.1), and a general interlacing technique similar to methods used by Jans, Tachikawa, and Colby for building up large indecomposable modules of finite length which has validity in any abelian category (Theorem 3.1).