×

An algorithm for the construction of exceptional modules over tubular canonical algebras. (English) Zbl 1231.16011

Let \(\Lambda=kQ/I\) be a finite-dimensional algebra over a field \(k\), defined by a quiver \(Q\) and an admissible ideal \(I\). A basic problem of representation theory is to classify all indecomposable finite-dimensional \(\Lambda\)-modules, up to isomorphism, for a fixed algebra \(\Lambda\) provided \(\Lambda\) is not of wild representation type. The most satisfactory realization of this task would be the construction of a complete list of matrix representations representing all isomorphism classes of indecomposable \(\Lambda\)-modules.
By C. M. Ringel [Tame algebras and integral quadratic forms. Lect. Notes Math. 1099. Berlin: Springer-Verlag (1984; Zbl 0546.16013)] tubular canonical algebras were introduced and studied. For these algebras, the classification of indecomposables is well known, but not their explicit description by matrices. However for this case all so-called exceptional \(\Lambda\)-modules admit matrix representatives consisting only of \(0\), \(1\) and \(-1\) as coefficients when \(\Lambda\) has only three arms.
The main goal of the paper under review is to present a precise recursive algorithm for computing a matrix representative for any isoclass of modules from exceptional tubes over a tubular canonical algebra of quiver type. The input of the algorithm is a quadruple consisting of the slope, the number of the tube, the quasi-socle and the quasi-length, and it computes all regular exceptional \(\Lambda\)-modules, or more generally all indecomposable modules in exceptional tubes.
The algorithm as well as the collection of ingredient routines are implemented as system of procedures in Maple 9.5 programming language, constituting the main part of the package EXCEPTIONAL.

MSC:

16G20 Representations of quivers and partially ordered sets
16Z05 Computational aspects of associative rings (general theory)
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0546.16013

Software:

TUBULAR; Maple
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Dowbor, P.; Hübner, T., A computer algebra approach to sheaves over weighted projective lines, (Computational Methods for Representations of Groups and Algebras. Computational Methods for Representations of Groups and Algebras, Progr. Math., vol. 173 (1999), Birkhäuser), 187-200 · Zbl 0951.16019
[2] Dowbor, P.; Mróz, A., The multiplicity problem for indecomposable decompositions of modules over domestic canonical algebras, Colloq. Math., 111, 2, 221-282 (2008) · Zbl 1185.16020
[3] Geigle, W.; Lenzing, H., A class of weighted projective curves arising in representation theory of finite dimensional algebras, (Singularities, Representations of Algebras, and Vector Bundles. Singularities, Representations of Algebras, and Vector Bundles, Springer Lecture Notes in Math., vol. 1273 (1987)), 265-297
[4] Kostrikin, A. I., Introduction to Algebra (1982), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0482.00001
[5] Kussin, D.; Meltzer, H., The braid group action for exceptional curves, Arch. Math., 79, 335-344 (2002) · Zbl 1062.16016
[6] Kussin, D.; Meltzer, H., Indecomposable modules for domestic canonical algebras, J. Pure Appl. Algebra, 211, 471-483 (2007) · Zbl 1185.16016
[7] Lenzing, H.; Meltzer, H., Sheaves on a weighted projective line of genus one and representations of a tubular algebra, (Representations of Algebras, Sixth International Conference. Representations of Algebras, Sixth International Conference, Ottawa, 1992. Representations of Algebras, Sixth International Conference. Representations of Algebras, Sixth International Conference, Ottawa, 1992, CMS Conf. Proc., vol. 14 (1993)), 317-337 · Zbl 0809.16012
[8] Meltzer, H., Exceptional sequences for canonical algebras, Arch. Math., 64, 304-312 (1995) · Zbl 0818.16016
[9] Meltzer, H., Tubular mutations, Colloq. Math., 74, 2, 267-274 (1997) · Zbl 0886.16013
[10] Meltzer, H., Exceptional modules for tubular canonical algebras, Algebr. Represent. Theory, 10, 481-496 (2007) · Zbl 1150.16014
[11] Ringel, C. M., Tame Algebras and Integral Quadratic Forms, Springer Lecture Notes in Math., vol. 1099 (1984), Springer · Zbl 0546.16013
[12] Ringel, C. M., Exceptional modules are tree modules, Linear Algebra Appl., 275-276, 471-493 (1998) · Zbl 0964.16014
[13] (Rudakov, A. N., Helices and vector bundles: Seminaire Rudakov. Helices and vector bundles: Seminaire Rudakov, London Math. Soc. Lecture Notes, vol. 148 (1990)) · Zbl 0727.00022
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.