On the K-theory of tubular algebras

Dirk Kussin

Colloquium Mathematicae (2000)

  • Volume: 86, Issue: 1, page 137-152
  • ISSN: 0010-1354

Abstract

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Let Λ be a tubular algebra over an arbitrary base field. We study the Grothendieck group K 0 ( Λ ) , endowed with the Euler form, and its automorphism group A u t ( K 0 ( Λ ) ) on a purely K-theoretical level as in [7]. Our results serve as tools for classifying the separating tubular families of tubular algebras as in the example [5] and for determining the automorphism group A u t ( D b Λ ) of the derived category of Λ.

How to cite

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Kussin, Dirk. "On the K-theory of tubular algebras." Colloquium Mathematicae 86.1 (2000): 137-152. <http://eudml.org/doc/210835>.

@article{Kussin2000,
abstract = {Let Λ be a tubular algebra over an arbitrary base field. We study the Grothendieck group $K_\{0\}(Λ)$, endowed with the Euler form, and its automorphism group $Aut(K_\{0\}(Λ))$ on a purely K-theoretical level as in [7]. Our results serve as tools for classifying the separating tubular families of tubular algebras as in the example [5] and for determining the automorphism group $Aut(D^\{b\}Λ)$ of the derived category of Λ.},
author = {Kussin, Dirk},
journal = {Colloquium Mathematicae},
keywords = {canonical algebras; tubular algebras; Grothendieck groups; Euler forms; separating tubular families; stable tubes; distinction lemma; automorphism groups},
language = {eng},
number = {1},
pages = {137-152},
title = {On the K-theory of tubular algebras},
url = {http://eudml.org/doc/210835},
volume = {86},
year = {2000},
}

TY - JOUR
AU - Kussin, Dirk
TI - On the K-theory of tubular algebras
JO - Colloquium Mathematicae
PY - 2000
VL - 86
IS - 1
SP - 137
EP - 152
AB - Let Λ be a tubular algebra over an arbitrary base field. We study the Grothendieck group $K_{0}(Λ)$, endowed with the Euler form, and its automorphism group $Aut(K_{0}(Λ))$ on a purely K-theoretical level as in [7]. Our results serve as tools for classifying the separating tubular families of tubular algebras as in the example [5] and for determining the automorphism group $Aut(D^{b}Λ)$ of the derived category of Λ.
LA - eng
KW - canonical algebras; tubular algebras; Grothendieck groups; Euler forms; separating tubular families; stable tubes; distinction lemma; automorphism groups
UR - http://eudml.org/doc/210835
ER -

References

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  1. [1] M. Barot, Representation-finite derived tubular algebras, Arch. Math. (Basel) 74 (2000), 83-94. Zbl0962.16010
  2. [2] M. Barot and J. A. de la Pe na, Derived tubular strongly simply connected algebras, Proc. Amer. Math. Soc. 127 (1999), 647-655. Zbl0940.16008
  3. [3] D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, London Math. Soc. Lecture Note Ser. 119, Cambridge Univ. Press, 1988. Zbl0635.16017
  4. [4] N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Grad. Texts in Math., 97, Springer, Berlin, 1984. Zbl0553.10019
  5. [5] D. Kussin, Non-isomorphic derived-equivalent tubular curves and their associated tubular algebras, J. Algebra 226 (2000), 436-450. Zbl0948.14003
  6. [6] D. Kussin, Graduierte Faktorialität und die Parameterkurven tubularer Familien, Ph.D. thesis, Universität Paderborn, 1997. 
  7. [7] H. Lenzing, A K-theoretic study of canonical algebras, in: Representation Theory of Algebras (Cocoyoc, 1994), R. Bautista, R. Mart-Villa, and J. A. de la Pe na (eds.), CMS Conf. Proc. 18, Amer. Math. Soc., Providence, RI, 1996, 433-473. 
  8. [8] H. Lenzing, Representations of finite dimensional algebras and singularity theory, in: Trends in Ring Theory (Miskolc, 1996) V. Dlab et al. (eds.), CMS Conf. Proc. 22, Amer. Math. Soc., Providence, RI, 1998, 71-97. Zbl0895.16003
  9. [9] H. Lenzing and H. Meltzer, The automorphism group of the derived category for a weighted projective line, Comm. Algebra 28 (2000), 1685-1700. Zbl0965.16008
  10. [10] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Representations of Algebras (Ottawa 1992), V. Dlab and H. Lenzing (eds.), CMS Conf. Proc. 14, Amer. Math. Soc., Providence, RI, 1993, 313-337. Zbl0809.16012
  11. [11] H. Lenzing and J. A. de la Pe na, Concealed-canonical algebras and separating tubular families, Proc. London Math. Soc. 78 (1999), 513-540. Zbl1035.16009
  12. [12] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984. 
  13. [13] C. M. Ringel, The canonical algebras, in: Topics in Algebra, Banach Center Publ. 26, 1990, with an appendix by William Crawley-Boevey, 407-432. 

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