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doi: 10.3934/era.2021009

On $ n $-slice algebras and related algebras

Jin-Yun Guo , , Cong Xiao and  Xiaojian Lu

LCSM(Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China

* Corresponding author: Jin-Yun Guo

Received  April 2020 Revised  December 2020 Published  February 2021

Fund Project: The authors are supported by Natural Science Foundation of China grant 11671126, 12071120

The $ n $-slice algebra is introduced as a generalization of path algebra in higher dimensional representation theory. In this paper, we give a classification of $ n $-slice algebras via their $ (n+1) $-preprojective algebras and the trivial extensions of their quadratic duals. One can always relate tame $ n $-slice algebras to the McKay quiver of a finite subgroup of $ \mathrm{GL}(n+1, \mathbb C) $. In the case of $ n = 2 $, we describe the relations for the $ 2 $-slice algebras related to the McKay quiver of finite Abelian subgroups of $ \mathrm{SL}(3, \mathbb C) $ and of the finite subgroups obtained from embedding $ \mathrm{SL}(2, \mathbb C) $ into $ \mathrm{SL}(3,\mathbb C) $.

Keywords: n-slice algebra, (n+1)-preprojective algebra, McKay quiver, n-translation algebra.
Mathematics Subject Classification: Primary: 16G20, 16G60, 16S35; Secondary: 20C05.
Citation: Jin-Yun Guo, Cong Xiao, Xiaojian Lu. On $ n $-slice algebras and related algebras. Electronic Research Archive, doi: 10.3934/era.2021009
References:
[1]

M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math., 109 (1994), 228-287.  doi: 10.1006/aima.1994.1087.  Google Scholar

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D. Baer, W. Geigle and H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, 15 (1987), 425-457.  doi: 10.1080/00927878708823425.  Google Scholar

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A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in Representation theory, J. Amer. Math. Soc., 9 (1996), 473-527.  doi: 10.1090/S0894-0347-96-00192-0.  Google Scholar

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S. Brenner, M. C. R. Butler and A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory, 5 (2002), 331-367.  doi: 10.1023/A:1020146502185.  Google Scholar

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X.-W. Chen, Graded self-injective algebras "are" trivial extensions, J. Algebra, 322 (2009), 2601-2606.  doi: 10.1016/j.jalgebra.2009.05.034.  Google Scholar

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W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605-635.  doi: 10.1215/S0012-7094-98-09218-3.  Google Scholar

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V. Dlab and C. M. Ringel, Eigenvalues of Coxeter transformations and the Gelfand-Kirilov dimension of preprojective algebras, Proc. Amer. Math. Soc., 83 (1981), 228-232.  doi: 10.2307/2043500.  Google Scholar

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J. Y. Guo, On the McKay quivers and $m$-Cartan matrices, Sci. China Ser. A, 52 (2009), 511-516.  doi: 10.1007/s11425-008-0176-y.  Google Scholar

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J. Y. Guo, On McKay quivers and covering spaces (in Chinese), Sci. Sin. Math., 41 (2011), 393–402, (English version: arXiv: 1002.1768). Google Scholar

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J. Y. Guo, Coverings and truncations of graded self-injective algebras, J. Algebra, 355 (2012), 9-34.  doi: 10.1016/j.jalgebra.2012.01.009.  Google Scholar

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J. Y. Guo, On $n$-translation algebras, J. Algebra, 453 (2016), 400-428.  doi: 10.1016/j.jalgebra.2015.08.006.  Google Scholar

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J. Y. Guo, On trivial extensions and higher preprojective algebras, J. Algebra, 547 (2020), 379-397.  doi: 10.1016/j.jalgebra.2019.11.022.  Google Scholar

[15]

J. Y. Guo, $ {\mathbb Z} Q$ type constructions in higher representation theory, arXiv: 1908.06546. Google Scholar

[16]

J. Y. Guo and R. Martínez-Villa, Algebra pairs associated to McKay quivers, Comm. Algebra, 30 (2002), 1017–1032. doi: 10.1081/AGB-120013196.  Google Scholar

[17]

J. Y. Guo and C. Xiao, $n$-APR tilting and $\tau$-mutations, J. Algebr. Comb., (2021). doi: 10.1007/s10801-021-01015-z.  Google Scholar

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J. Y. Guo and Q. Wu, Loewy matrix, Koszul cone and applications, Comm. Algebra, 28 (2000), 925-940.  doi: 10.1080/00927870008826869.  Google Scholar

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D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119. Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511629228.  Google Scholar

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M. Herschend, O. Iyama and S. Oppermann, $n$-Representation infinite algebras, Adv. Math., 252 (2014), 292-342.  doi: 10.1016/j.aim.2013.09.023.  Google Scholar

[21]

O. Iyama, Cluster tilting for higher Auslander algebras, Adv. Math., 226 (2011), 1-61.  doi: 10.1016/j.aim.2010.03.004.  Google Scholar

[22]

O. Iyama and S. Oppermann, $n$-representation-finite algebras and $n$-APR tilting, Trans. Amer. Math. Soc., 363 (2011), 6575-6614.  doi: 10.1090/S0002-9947-2011-05312-2.  Google Scholar

[23]

O. Iyama and S. Oppermann, Stable categories of higher preprojective algebras, Adv. Math., 244 (2013), 23-68.  doi: 10.1016/j.aim.2013.03.013.  Google Scholar

[24]

K. Lee, L. Li, M. Mills, R. Schiffler and A. Seceleanu, Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers, Adv. Math., 367 (2020), 107130, 33 pp. doi: 10.1016/j.aim.2020.107130.  Google Scholar

[25]

R. Martńez-Villa, Graded, selfinjective, and Koszul algebras, J. Algebra, 215 (1999), 34-72.  doi: 10.1006/jabr.1998.7728.  Google Scholar

[26]

R. Martńez-Villa, Skew Group Algebras and their Yoneda algebras, Math. J. Okayama Univ., 43 (2001), 1-16.   Google Scholar

[27]

J. McKay, Graph, singularities and finite groups, Proc. Sympos. Pure Math., 37 (1980), 183-186.   Google Scholar

[28]

H. Minamoto and I. Mori, Structures of AS-regular algebras, Adv. Math., 226 (2011), 4061-4095. doi: 10.1016/j.aim.2010.11.004.  Google Scholar

[29]

I. Reiten and M. Van den Bergh, Two-dimensional tame and maximal orders of finite repre-sentation type, Mem. Amer. Math. Soc., 80 (1989), no. 408. doi: 10.1090/memo/0408.  Google Scholar

[30]

J. S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16 (1968), 1208-1222.  doi: 10.1137/0116101.  Google Scholar

show all references

References:
[1]

M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math., 109 (1994), 228-287.  doi: 10.1006/aima.1994.1087.  Google Scholar

[2]

D. Baer, W. Geigle and H. Lenzing, The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, 15 (1987), 425-457.  doi: 10.1080/00927878708823425.  Google Scholar

[3]

A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in Representation theory, J. Amer. Math. Soc., 9 (1996), 473-527.  doi: 10.1090/S0894-0347-96-00192-0.  Google Scholar

[4]

I. N. Bernšte$\mathop {\rm{i}}\limits^ \vee $n, I. M. Gel'fand and S. I. Gel'fand, Algebraic vector bundles on P$^{n}$ and problems of linear algebras, Funktsional. Anal. i Prilozhen., 12 (1978), 66-67.   Google Scholar

[5]

A. A. Be$\mathop {\rm{i}}\limits^ \vee $linson, Coherent sheaves on P$^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen., 12 (1978), 68-69.   Google Scholar

[6]

S. Brenner, M. C. R. Butler and A. D. King, Periodic algebras which are almost Koszul, Algebr. Represent. Theory, 5 (2002), 331-367.  doi: 10.1023/A:1020146502185.  Google Scholar

[7]

X.-W. Chen, Graded self-injective algebras "are" trivial extensions, J. Algebra, 322 (2009), 2601-2606.  doi: 10.1016/j.jalgebra.2009.05.034.  Google Scholar

[8]

W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J., 92 (1998), 605-635.  doi: 10.1215/S0012-7094-98-09218-3.  Google Scholar

[9]

V. Dlab and C. M. Ringel, Eigenvalues of Coxeter transformations and the Gelfand-Kirilov dimension of preprojective algebras, Proc. Amer. Math. Soc., 83 (1981), 228-232.  doi: 10.2307/2043500.  Google Scholar

[10]

J. Y. Guo, On the McKay quivers and $m$-Cartan matrices, Sci. China Ser. A, 52 (2009), 511-516.  doi: 10.1007/s11425-008-0176-y.  Google Scholar

[11]

J. Y. Guo, On McKay quivers and covering spaces (in Chinese), Sci. Sin. Math., 41 (2011), 393–402, (English version: arXiv: 1002.1768). Google Scholar

[12]

J. Y. Guo, Coverings and truncations of graded self-injective algebras, J. Algebra, 355 (2012), 9-34.  doi: 10.1016/j.jalgebra.2012.01.009.  Google Scholar

[13]

J. Y. Guo, On $n$-translation algebras, J. Algebra, 453 (2016), 400-428.  doi: 10.1016/j.jalgebra.2015.08.006.  Google Scholar

[14]

J. Y. Guo, On trivial extensions and higher preprojective algebras, J. Algebra, 547 (2020), 379-397.  doi: 10.1016/j.jalgebra.2019.11.022.  Google Scholar

[15]

J. Y. Guo, $ {\mathbb Z} Q$ type constructions in higher representation theory, arXiv: 1908.06546. Google Scholar

[16]

J. Y. Guo and R. Martínez-Villa, Algebra pairs associated to McKay quivers, Comm. Algebra, 30 (2002), 1017–1032. doi: 10.1081/AGB-120013196.  Google Scholar

[17]

J. Y. Guo and C. Xiao, $n$-APR tilting and $\tau$-mutations, J. Algebr. Comb., (2021). doi: 10.1007/s10801-021-01015-z.  Google Scholar

[18]

J. Y. Guo and Q. Wu, Loewy matrix, Koszul cone and applications, Comm. Algebra, 28 (2000), 925-940.  doi: 10.1080/00927870008826869.  Google Scholar

[19]

D. Happel, Triangulated Categories in the Representation Theory of Finite-Dimensional Algebras, London Math. Soc. Lecture Note Ser., vol. 119. Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9780511629228.  Google Scholar

[20]

M. Herschend, O. Iyama and S. Oppermann, $n$-Representation infinite algebras, Adv. Math., 252 (2014), 292-342.  doi: 10.1016/j.aim.2013.09.023.  Google Scholar

[21]

O. Iyama, Cluster tilting for higher Auslander algebras, Adv. Math., 226 (2011), 1-61.  doi: 10.1016/j.aim.2010.03.004.  Google Scholar

[22]

O. Iyama and S. Oppermann, $n$-representation-finite algebras and $n$-APR tilting, Trans. Amer. Math. Soc., 363 (2011), 6575-6614.  doi: 10.1090/S0002-9947-2011-05312-2.  Google Scholar

[23]

O. Iyama and S. Oppermann, Stable categories of higher preprojective algebras, Adv. Math., 244 (2013), 23-68.  doi: 10.1016/j.aim.2013.03.013.  Google Scholar

[24]

K. Lee, L. Li, M. Mills, R. Schiffler and A. Seceleanu, Frieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers, Adv. Math., 367 (2020), 107130, 33 pp. doi: 10.1016/j.aim.2020.107130.  Google Scholar

[25]

R. Martńez-Villa, Graded, selfinjective, and Koszul algebras, J. Algebra, 215 (1999), 34-72.  doi: 10.1006/jabr.1998.7728.  Google Scholar

[26]

R. Martńez-Villa, Skew Group Algebras and their Yoneda algebras, Math. J. Okayama Univ., 43 (2001), 1-16.   Google Scholar

[27]

J. McKay, Graph, singularities and finite groups, Proc. Sympos. Pure Math., 37 (1980), 183-186.   Google Scholar

[28]

H. Minamoto and I. Mori, Structures of AS-regular algebras, Adv. Math., 226 (2011), 4061-4095. doi: 10.1016/j.aim.2010.11.004.  Google Scholar

[29]

I. Reiten and M. Van den Bergh, Two-dimensional tame and maximal orders of finite repre-sentation type, Mem. Amer. Math. Soc., 80 (1989), no. 408. doi: 10.1090/memo/0408.  Google Scholar

[30]

J. S. Vandergraft, Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16 (1968), 1208-1222.  doi: 10.1137/0116101.  Google Scholar

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