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Δ-tame quasi-hereditary algebras

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Abstract

Let (K, M, H) be an upper triangular biomodule problem. Brüstle and Hille showed that the opposite algebra A of the endomorphism algebra of a projective generator P of the matrices category of (K, M, H) is quasi-hereditary, and there is an equivalence between the category of Δ-good modules of A and Mat(K, M). In this note, based on the tame theorem for bimodule problems, we show that if the algebra A associated with an upper triangular bimodule problem is of Δ-tame representation type, then the category F(Δ) has the homogeneous property, i.e. almost all modules in F(Δ) are isomorphic to their Auslander-Reiten translations. Moreover, if (K, M, H) is an upper triangular bipartite bimodule problem, then A is of Δ-tame representation type if and only if F(Δ) is homogeneous.

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References

  1. Zhang Y B, Xu Y G. Tameness and homogeneity. arXiv:math. RT/0505630

  2. Drozd Yu A. Tame and wild matrix problems. In Representations and Quadratic Forms (in Russian). Kiev: Inst Math Akad Nauk Ukrain SSR, 1979, 39–74

    Google Scholar 

  3. Crawley-Boevey W W. On tame algebras and bocses. Proc London Math Soc, 56(3): 451–483 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Crawley-Boevey W W. Matrix problems and Drozd’s theorem. In: Balcerzyk S, ed. Topics in Algebra. Warsaw: Banach Center publications, PWN-Polish Scientific Publishers, 26(1): 199–222 (1990)

    Google Scholar 

  5. Rojter A V. Matrix problems and representations of BOCS’s. In: Dlab V, Gabriel P, eds. Representation Theory I, LNM 831. Berlin: Springer-Verlag, 1980, 288–324

    Chapter  Google Scholar 

  6. Sergeichuk V V. Canonical matrices for basic matrix problems. Linear Algebra and its Applications, 317(1/3): 53–102 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brüstle T, Sergeichuk V V. Estimate of the number of one-parameter families of modules over a tame algebra. Linear Algebra and its Application, 365(15): 115–133 (2003)

    Article  MATH  Google Scholar 

  8. Xu Y G, Zhang Y B. Indecomposability and the number of links. Sci China Ser A, 44(12): 1515–1522 (2001)

    MATH  MathSciNet  Google Scholar 

  9. Zhang Y B, Xu Y G. On tame and wild bocses. Sci Shina Ser A, 48(4): 456–468 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Zeng X Y, Zhang Y B. A correspondence of almost split sequences between some categories. Comm Algebra, 29(2): 1–26 (2001)

    Article  MathSciNet  Google Scholar 

  11. Cline E, Parshall B, Scoot L. Finite dimensional algebras and highest weight categories. J Reine Angew Math, 391: 85–99 (1988)

    MATH  MathSciNet  Google Scholar 

  12. Deng B M. Quasi-hereditary algebras and quadratic forms. J Algebra, 239: 438–459 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Deng B M, Xi C C. Quasi-hereditary algebras and Δ-good modules. In: Happel D, Zhang Y B, eds. Representations of Algebras, Proc 9-th International Conf, Vol. I Beijing: Beijing Normal Univ Press, 2000, 49–73

    Google Scholar 

  14. Brüstle T, Hille L. Finite, tame and wild actions of parabolic subgroups in GL(V) on certain unipotent subgroups. J Algebra, 226: 347–360 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  15. Brüstle T, Hille L. Actions of parabolic subgroups in GL(V) on unipotent normal subgroups and quasihereditary algebras. Colloquium Mathematicum, 83: 281–294 (2000)

    MATH  MathSciNet  Google Scholar 

  16. Brüstle T, Hille L. Matrices over upper triangular bimodules and Δ-filtered modules over quasi-hereditary algebras. Colloquium Mathematicum, 83: 295–303 (2000)

    MATH  MathSciNet  Google Scholar 

  17. Deng B M. Quasi-hereditary algebras associated with upper triangular bimodules. Comm Algebra, 29(8): 3581–3591 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Xu Y G, Li L C. Bimodule problems radt (−,−) and quasi-hereditary algebras. Acta Math Sinica Chinese Series, 45(3): 605–616 (2002)

    MATH  MathSciNet  Google Scholar 

  19. Hille L, Vossieck D. The quasi-hereditary algebra associated with the radical bimodule over a hereditary algebra. Coll Math, 98: 201–211 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ringel C M. The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences. Math Zeit, 208: 209–223 (1991)

    MathSciNet  Google Scholar 

  21. Zeng Z T. On preprojective and preinjective partitions of a Δ-good module category. J Algebra, 285: 608–622 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Bauch M. Subspace categories as categories of good modules over quasi-hereditary algebras. Arch Math (Basel), 62(2): 112–115 (1994)

    MATH  MathSciNet  Google Scholar 

  23. Ovsienko S. A characterization of quasi-hereditary algebras. preprint

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Hubei University, Wuhan, 430062, China

    Yun-ge Xu

  2. School of Mathematics Sciences, Beijing Normal University, Beijing, 100875, China

    Ying-bo Zhang

Authors
  1. Yun-ge Xu
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  2. Ying-bo Zhang
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Correspondence to Ying-bo Zhang.

Additional information

This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10426014, 10501010 and 19331030) and the Foundation of Hubei Provincial Department of Education (Grant No. D200510005).

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Xu, Yg., Zhang, Yb. Δ-tame quasi-hereditary algebras. SCI CHINA SER A 50, 240–252 (2007). https://doi.org/10.1007/s11425-007-2074-3

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  • DOI: https://doi.org/10.1007/s11425-007-2074-3

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