-
Previous Article
Tori can't collapse to an interval
- ERA Home
- This Issue
-
Next Article
Global stability of traveling waves for a spatially discrete diffusion system with time delay
On $ n $-slice algebras and related algebras
LCSM(Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China |
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) $.
References:
[1] |
M. Artin and J. J. Zhang,
Noncommutative projective schemes, Adv. Math., 109 (1994), 228-287.
doi: 10.1006/aima.1994.1087. |
[2] |
D. Baer, W. Geigle and H. Lenzing,
The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, 15 (1987), 425-457.
doi: 10.1080/00927878708823425. |
[3] |
A. Beilinson, V. Ginzburg and W. Soergel,
Koszul duality patterns in Representation theory, J. Amer. Math. Soc., 9 (1996), 473-527.
doi: 10.1090/S0894-0347-96-00192-0. |
[4] |
I. N. Bernšte$\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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Y. Guo,
On $n$-translation algebras, J. Algebra, 453 (2016), 400-428.
doi: 10.1016/j.jalgebra.2015.08.006. |
[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. |
[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. |
[17] |
J. Y. Guo and C. Xiao, $n$-APR tilting and $\tau$-mutations, J. Algebr. Comb., (2021).
doi: 10.1007/s10801-021-01015-z. |
[18] |
J. Y. Guo and Q. Wu,
Loewy matrix, Koszul cone and applications, Comm. Algebra, 28 (2000), 925-940.
doi: 10.1080/00927870008826869. |
[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. |
[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. |
[21] |
O. Iyama,
Cluster tilting for higher Auslander algebras, Adv. Math., 226 (2011), 1-61.
doi: 10.1016/j.aim.2010.03.004. |
[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. |
[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. |
[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. |
[25] |
R. Martńez-Villa,
Graded, selfinjective, and Koszul algebras, J. Algebra, 215 (1999), 34-72.
doi: 10.1006/jabr.1998.7728. |
[26] |
R. Martńez-Villa,
Skew Group Algebras and their Yoneda algebras, Math. J. Okayama Univ., 43 (2001), 1-16.
|
[27] |
J. McKay,
Graph, singularities and finite groups, Proc. Sympos. Pure Math., 37 (1980), 183-186.
|
[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. |
[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. |
[30] |
J. S. Vandergraft,
Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16 (1968), 1208-1222.
doi: 10.1137/0116101. |
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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Y. Guo,
On $n$-translation algebras, J. Algebra, 453 (2016), 400-428.
doi: 10.1016/j.jalgebra.2015.08.006. |
[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. |
[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. |
[17] |
J. Y. Guo and C. Xiao, $n$-APR tilting and $\tau$-mutations, J. Algebr. Comb., (2021).
doi: 10.1007/s10801-021-01015-z. |
[18] |
J. Y. Guo and Q. Wu,
Loewy matrix, Koszul cone and applications, Comm. Algebra, 28 (2000), 925-940.
doi: 10.1080/00927870008826869. |
[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. |
[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. |
[21] |
O. Iyama,
Cluster tilting for higher Auslander algebras, Adv. Math., 226 (2011), 1-61.
doi: 10.1016/j.aim.2010.03.004. |
[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. |
[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. |
[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. |
[25] |
R. Martńez-Villa,
Graded, selfinjective, and Koszul algebras, J. Algebra, 215 (1999), 34-72.
doi: 10.1006/jabr.1998.7728. |
[26] |
R. Martńez-Villa,
Skew Group Algebras and their Yoneda algebras, Math. J. Okayama Univ., 43 (2001), 1-16.
|
[27] |
J. McKay,
Graph, singularities and finite groups, Proc. Sympos. Pure Math., 37 (1980), 183-186.
|
[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. |
[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. |
[30] |
J. S. Vandergraft,
Spectral properties of matrices which have invariant cones, SIAM J. Appl. Math., 16 (1968), 1208-1222.
doi: 10.1137/0116101. |
[1] |
Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109 |
[2] |
Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020378 |
[3] |
Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1 |
[4] |
Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907 |
[5] |
Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379 |
[6] |
Bernold Fiedler, Carlos Rocha, Matthias Wolfrum. Sturm global attractors for $S^1$-equivariant parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 617-659. doi: 10.3934/nhm.2012.7.617 |
[7] |
Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic & Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045 |
[8] |
Ravi Anand, Dibyendu Roy, Santanu Sarkar. Some results on lightweight stream ciphers Fountain v1 & Lizard. Advances in Mathematics of Communications, 2020 doi: 10.3934/amc.2020128 |
[9] |
Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617 |
[10] |
Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021 doi: 10.3934/amc.2020133 |
Impact Factor: 0.263
Tools
Metrics
Other articles
by authors
[Back to Top]