Iterated tubular algebras

Representation Type Of One Point Extensions Of Iterated Tubular Algebras

“…If M = M 0 , and M is a ray-module, by similar arguments to the one in 3.1, B[M ] is an algebra with acceptable projective modules and so, by theorem 3.4 of [18], the representation type is determined by the quadratic form. And by the other hand, if the support of M is contained in the branch, the result follows from [14].…”

“…The vector-space category Hom(M, −) is finite for those modules with the support contained in the branch, then the result follows from [14]. If Hom(M, −) is infinite there exists an injective module I, outside of the tube, such that Hom(M, I) = 0, but again this morphism factors through infinite families of tubes and the result follows as in theorem 3.1,or [18].…”

“…It was shown in [16] that if B is a tame algebra, then q B is weakly non negative, this important result was also obtained by Drozd for matrix problems in [12] . It is known that the converse is not true in general, see example in section 2.4 of [17], but it is true for some families of algebras, as tubular algebras [20], quasitilted algebras [21] and iterated tubular algebras [18]. The main motivation of our work was the result in [16] that if C is a tame concealed algebra, not of typẽ A n , and M an indecomposable C-module, then the one point extension C[M ] is tame if and only if q C[M] is weakly non negative.…”

“…It follows that M and I are separated by a separating tubular family, then this non zero morphism factor trough a orthogonal tubular family, in particular factors trough five orthogonal bricks, then by Nazarova theorem it follows that Hom B (M, modB) is wild and, see in prop. 3.3 of [18], the corresponding quadratic form is strongly indefinite. Now, consider the case that M belongs to Q ∞ .…”

“…We use strongly in this work the characterization of B as a semi-regular n-iterated tubular algebra, with n = 0 or n = 1 ( see [21], [18] and [4]). Proof: If B is a tilted algebra, then B is a tilted algebra of euclidian type and the result follows from [8].…”