n-APR tilts of a class of Koszul algebra realized by τ_[n]-mutations

Abstract

In this paper, we introduce the τ_[n]-mutations of a class of Koszul algebra and prove that for the Koszul algebra with the global dimension ≤ n, if its Koszul dual is an admissible (n-1)-translation algebra, then the quiver of endomorphism algebra of n-APR tilting module can be realized by τ_[n]-mutation.

Graphical abstract

Keywords

n-APR tilting module / n-translation algebra / τ_[n]-mutation

References

[1] Auslander M., Platzeck, M.I. , Reiten, I.. Coxeter functors without diagrams. Trans. Amer. Math. Soc. 1979; 250: 1–46

[2] AuslanderM., Reiten, I.. and Smalø, S.O., Representation Theory of Artin algebras, Corrected Reprint of the 1995 Original, Cambridge Stud. Adv. Math., Vol. 36, Cambridge: Cambridge Univ. Press, 1997

[3] Beilinson A., Ginzburg, V. , Soergel, W.. Koszul duality patterns in representation theory. J. Amer. Math. Soc. 1996; 9(2): 473–527

[4] Bernstein I.N., Gel’fand, I.M., Ponomarev, V.A.. Coxeter functors and Gabriel’s theorem. Russian Math. Surveys 1973; 28(2): 17–32

[5] Brenner S., Butler, M.C.R. , King, A.D.. Periodic algebras which are almost Koszul. Algebr. Represent. Theory 2002; 5(4): 331–367

[6] Guo J.Y.. Translation algebras and their applications. J. Algebra 2002; 255: 1–21

[7] Guo J.Y.. Coverings and truncations of graded self-injective algebras. J. Algebra 2012; 355: 9–34

[8] Guo J.Y.. On n-translation algebras. J. Algebra 2016; 453: 400–428

[9] Guo J.Y. , Luo, D.R.. On n-cubic pyramid algebras. Algebr. Represent. Theory 2016; 19(4): 991–1016

[10] Guo J.Y., Yin, Y. , Zhu, C.. Returning arrows for self-injective algebras and Artin-Schelter regular algebras. J. Algebra 2014; 397: 365–378

[11] Iyama O.. Cluster tilting for higher Auslander algebras. Adv. Math. 2011; 226(1): 1–61

[12] Iyama O. , Oppermann, S.. n-representation-finite algebras and n-APR tilting. Trans. Amer. Math. Soc. 2011; 363(12): 6575–6614

[13] WeibelC.A., An Introduction to Homological Algebra, Cambridge Stud. Adv. Math., Vol. 38, Cambridge: Cambridge Univ. Press, 1994

[14] MizunoY., Studies on mutation and tilting theory, Ph.D. Thesis, Nagoya: Nagoya University, 2013

[15] Mizuno Y. , Yamaura, K.. Higher APR tilting preserves n-representation infiniteness. J. Algebra 2016; 447: 56–73

[16] Zheng L.J. , Wu, C.S.. A note on n-BB-tilting modules. Adv. Math. (China) 2014; 43(3): 329–334

RIGHTS & PERMISSIONS

Higher Education Press 2025