The Representation Type of Algebras and Subalgebras

Bounded Representation

For A an associative algebra with identity over a field K, [A : K] < ∞, and d an integer, we define g Λ(d) to be the number of inequivalent indecomposable Λ-modules of degree d over K. Following (6), we define Λ to be of finite representation type if . Λis said to be of bounded representation type if there exists d 0 such that g Λ(d) = 0 for d ⩾ d 0; Λ is of unbounded representation type if not of bounded type.

Algebras of bounded finite dimensional representation type

Glasgow Mathematical Journal ◽

10.1017/s0017089500031578 ◽

1995 ◽

Vol 37 (3) ◽

pp. 289-302 ◽

Cited By ~ 1

Author(s):

Allen D. Bell ◽

K. R. Goodearl

Keyword(s):

Representation Type ◽

Dimensional Representation ◽

Dimensional Algebra ◽

Finite Dimensional Representation ◽

Finite Representation ◽

Finite Dimensional ◽

Left And Right ◽

Bounded Representation ◽

Algebra Over A Field

It is well known that for finite dimensional algebras, “bounded representation type” implies “finite representation type”; this is the assertion of the First Brauer-Thrall Conjecture (hereafter referred to as Brauer-Thrall I), proved by Roiter [26] (see also [23]). More precisely, it states that if R is a finite dimensional algebra over a field k, such that there is a finite upper bound on the k-dimensions of the finite dimensional indecomposable right R-modules, then up to isomorphism R has only finitely many (finite dimensional) indecomposable right modules. The hypothesis and conclusion are of course left-right symmetric in this situation, because of the duality between finite dimensional left and right R-modules, given by Homk(−, k). Furthermore, it follows from finite representation type that all indecomposable R modules are finite dimensional [25].

On algebras of finite representation type

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1969-0237558-9 ◽

1969 ◽

Vol 135 ◽

pp. 127-127 ◽

Cited By ~ 9

Author(s):

Spencer E. Dickson

Keyword(s):

Representation Type ◽

Finite Representation ◽

On the Auslander algebra of one commutative semigroup of finite representation type

Prykladni Problemy Mekhaniky i Matematyky ◽

10.15407/apmm2020.18.43-47 ◽

2020 ◽

Vol 18 (0) ◽

Author(s):

O. V. Zubaruk

Keyword(s):

Commutative Semigroup ◽

Representation Type ◽

Finite Representation ◽

Auslander Algebra

Auslander algebras Of finite representation type

Communications in Algebra ◽

10.1080/00927878708823424 ◽

1987 ◽

Vol 15 (1-2) ◽

pp. 377-424 ◽

Cited By ~ 8

Author(s):

Kiyoshi Igusa ◽

Maria-Ines Platzeck ◽

Gordana Todorov ◽

Dan Zachana

Keyword(s):

Representation Type ◽

Finite Representation ◽

On algebras whose trivial extensions are of finite representation type

Lecture Notes in Mathematics - Representations of Algebras ◽

10.1007/bfb0093006 ◽

1981 ◽

pp. 364-371 ◽

Cited By ~ 3

Author(s):

Kunio Yamagata

Keyword(s):

Representation Type ◽

Finite Representation ◽

A class of self-injective algebras of finite representation type

Representation Theory II - Lecture Notes in Mathematics ◽

10.1007/bfb0088481 ◽

1980 ◽

pp. 545-572 ◽

Cited By ~ 5

Author(s):

Eberhard Scherzler ◽

Josef Waschbüsch

Keyword(s):

Representation Type ◽

Finite Representation ◽

Integer sequences arising from Auslander–Reiten quivers of some hereditary artin algebras

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500511 ◽

2020 ◽

pp. 2150051

Author(s):

Agustín Moreno Cañadas ◽

Gabriel Bravo Rios ◽

Hernán Giraldo

Keyword(s):

Representation Type ◽

Integer Sequences ◽

Finite Representation ◽

Artin Algebras

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type.