Computational Classification of Tubular Algebras

• Autorzy: Dowbor Piotr , Kim Yan

Identyfikatory

DOI

10.3233/FI-2020-1979

• Języki publikacji: EN

Abstrakty

EN

The effective method (based on Theorem 5.3) of classifying tubular algebras by the Cartan matrices of tilting sheaves over weighted projective lines with all indecomposable direct summands in some finite “fundamental domain” , by the reduction to the two elementary problems of discrete mathematics having algorithmic solutions is presented in details (see Problem A and B). The software package CART_TUB being an implementation of this method yields the precise classification of all up to isomorphism tubular algebras of a fixed tubular type p, by creating the complete lists of their Cartan matrices, and furnish their tilting realizations. In particular, the number of isomorphism classes of tubular algebras of the type p is determined (Theorem 2.3).

Słowa kluczowe

EN

tubular algebra; Cartan matrix; weighted projective line; tilting sheaf; Euler form; algorithm; clique; weighted graph; adjacency matrix; index renumbering relation

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2020
• Tom: Vol. 177, nr 1
• Strony: 39--67
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Dowbor Piotr

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

autor

Kim Yan

• Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg, Maison du Nombre, 6, Avenue de la Fonte, 4364 Esch-sur-Alzette, Luxembourg

Bibliografia

• [1] Ringel CM. Tame algebras and integral quadratic forms. Volume 1099 of Lecture Notes in Mathematics. Springer, 2006. ISBN:3540391274, 9783540391272.
• [2] Skowroński A. Selfinjective algebras of polynomial growth. Mathematische Annalen, 1989. 285(2): 177-199. doi:10.1007/BF01443513.
• [3] Bongartz K. Critical simply connected algebras. Manuscripta Mathematica, 1984. 46(1-3): 117-136. doi:10.1007/BF01185198.
• [4] Happel D, Vossieck D. Minimal algebras of infinite representation type with preprojective component. Manuscripta Mathematica, 1983. 42(2-3): 221-243. doi:10.1007/BF01169585.
• [5] Grzecza M, Kasjan S, Mróz A. Tree matrices and a matrix reduction algorithm of Belitskii. Fundamenta Informaticae, 2012. 118(3): 253-279. doi:10.3233/FI-2012-713.
• [6] Mróz A, Zwara G. Combinatorial algorithms for computing degenerations of modules of finite dimension. Fundamenta Informaticae, 2014. 132(4): 519-532. doi:10.3233/FI-2014-1057.
• [7] Dowbor P, Hübner T. A computer algebra approach to sheaves over weighted projective lines. In: Computational Methods for Representations of Groups and Algebras: Volume 173 of Progress in Mathematics. Birkhäser-Verlag, Basel-Boston, 1999. pp. 187-200. doi:10.1007/978-3-0348-8716-8_10.
• [8] Dowbor P, Hübner T. The TUBULAR package. 1997. pp. 17-43. Preprint, Universität Bielefeld, 97, URL http://www.math.uni-bielefeld.de/~fdlist.
• [9] Assem I, Skowroński A, Simson D. Elements of the Representation Theory of Associative Algebras: Vol. 1: Techniques of Representation Theory. Volume 65 of LMS Student Texts. Cambridge University Press, 2006. doi:10.1017/CBO9780511614309.
• [10] Geigle W, Lenzing H. A class of weighted projective curves arising in representation theory of finite dimensional algebras. In: Singularities, representation of algebras, and vector bundles. Volume 1273 of Lecture Notes in Mathematics. Springer, 1987. pp. 265-297. doi:10.1007/BFb0078849.
• [11] Lenzing H, Meltzer H. Tilting sheaves and concealed-canonical algebras. In: Representations of algebras, Seventh International Conference, Cocoyoc (Mexico) 1994. Volume 18 of CMS Conference Proceedings. American Mathematical Society, 1996. pp. 455-473.
• [12] Meltzer H. Exceptional vector bundles, tilting sheaves and tilting complexes for weighted projective lines. Volume 171 of Memoirs of the AMS. 2004. doi:10.1090/memo/0808.
• [13] Lenzing H, Meltzer H. Exceptional sequences determined by their Cartan matrix. Algebras and representation theory, 2002. 5(2): 201-209. doi:10.1023/A:1015646412663.
• [14] Simson D, Skowroński A. Elements of the representation theory of associative algebras: Vol. 3: Representation-infinite tilted algebras. Volume 72 of LMS Student Texts. Cambridge Univ. Press, 2007. ISSN:0963-163.
• [15] Lenzing H, Meltzer H. Sheaves on a weighted projective line of genus one and representations of a tubular algebra. In: Representations of Algebras, Sixth International Conference, Ottawa (Ontario) 1992. Volume 14 of CMS Conference Proceedings. American Mathematical Society, 1993. pp. 317-337.
• [16] Dowbor P, Meltzer H. Dimension vectors of indecomposable objects for nilpotent operators of degree 6 with one invariant subspace. Algebras and Representation Theory, 2019. 22(1): 99-140. doi:10.1007/s10468-017-9759-z.
• [17] Dowbor P, Meltzer H. On tilted realizations of tubular algebras. Preprint 2019.
• [18] Simson D. Mesh algorithms for solving principal Diophantine equations, sand-glass tubes and tori of roots. Fundamenta Informaticae, 2011. 109(4): 425-462. doi:10.3233/FI-2011-520.
• [19] Cormen TH, Leiserson CE, Rivest RL, Stein C. Introduction to algorithms. MIT press, 2009. ISBN-10:9780262033848, 13:978-0262033848.
• [20] Diestel R. Graph theory. Volume 173 of Graduate Texts in Mathematics. Springer, 1996.
• [21] West DB. Introduction to graph theory. Prentice Hall, New York, 1996. ISBN:0582249937, 9780582249936.
• [22] Bron C, Kerbosch J. Algorithm 457: Finding all cliques of an undirected graph. Communications of the ACM, 1973. 16(9): 575-577. doi:10.1145/362342.362367.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).