Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED MCKAY QUIVERS, ROOT SYSTEM AND KAC-MOODY ALGEBRAS

  • Hou, Bo (School of Mathematics and Statistics Henan University) ;
  • Yang, Shilin (College of Applied Science Beijing University of Technology)
  • Received : 2013.12.17
  • Published : 2015.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let Q be a finite quiver and $G{\subseteq}Aut(\mathbb{k}Q)$ a finite abelian group. Assume that $\hat{Q}$ and ${\Gamma}$ are the generalized Mckay quiver and the valued graph corresponding to (Q, G) respectively. In this paper we discuss the relationship between indecomposable $\hat{Q}$-representations and the root system of Kac-Moody algebra $g({\Gamma})$. Moreover, we may lift G to $\bar{G}{\subseteq}Aut(g(\hat{Q}))$ such that $g({\Gamma})$ embeds into the fixed point algebra $g(\hat{Q})^{\bar{G}}$ and $g(\hat{Q})^{\bar{G}}$ as a $g({\Gamma})$-module is integrable.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. I. Assem, D. Simson, and A. Skowronski, Elements of Representation Theory of Asso- ciative Algebras. Vol. 1, Cambridge University Press, 2006.
  2. M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, 511-531. https://doi.org/10.1090/S0002-9947-1986-0816307-7
  3. M. Auslander, I. Reiten, and S. O. Smalo, Representation Theory of Artin Algebras, Cambridge Stud. Adv. Math., Vol. 36, Cambridge University Press, 1995.
  4. W. Crawley-Boevey and M. P. Holland, Noncommutative deformations of Kleinian sin- gularities, Duke Math. J. 92 (1998), no. 3, 605-635. https://doi.org/10.1215/S0012-7094-98-09218-3
  5. L. Demonet, Skew group algebras of path algebras and preprojective algebras, J. Algebra 323 (2010), no. 4, 1052-1059. https://doi.org/10.1016/j.jalgebra.2009.11.034
  6. B. Deng, J. Du, B. Parshall, and J. Wang, Finite Dimensional Algebras and Puantum Groups, Math. Surveys and Monographs 150, American Mathematical Society, Providence, 2008.
  7. V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 6 (1976), no. 173, 57 pp.
  8. J. Guo, On the McKay quivers and m-Cartan Matrices, Sci. China Ser. A 52 (2009), no. 3, 511-516. https://doi.org/10.1007/s11425-008-0176-y
  9. J. Guo and Martiinez-Villa, Algebra pairs associated to McKay quivers, Comm. Algebra 30 (2002), no. 2, 1017-1032. https://doi.org/10.1081/AGB-120013196
  10. B. Hou and S. Yang, Skew group algebras of deformed preprojective algebras, J. Algebra 332 (2011), 209-228. https://doi.org/10.1016/j.jalgebra.2011.02.007
  11. A. Hubery, Representations of quiver respecting a quiver automorphism and a of Kac, Ph. D. thesis, Leeds Univeraity, 2002.
  12. A. Hubery, Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. 69 (2004), no. 1, 79-96. https://doi.org/10.1112/S0024610703004988
  13. V. G. Kac, Infinite Dimensional Lie Algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.
  14. V. G. Kac and S. P. Wang, On automorphisms of Kac-Moody algebras and groups, Adv. Math. 92 (1992), no. 2, 129-195. https://doi.org/10.1016/0001-8708(92)90063-Q
  15. G. Lusztig, Affine quivers and canonical bases, Inst. Hautes Etudes Sci. Publ. Math. 76 (1992), 111-163. https://doi.org/10.1007/BF02699432
  16. J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, pp. 183-186, Amer. Math. Soc., Providence, RI, 1980.
  17. M. Reid, McKay correspondance, arXiv:math.AG/9702016.
  18. I. Reiten and C. Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92 (1985), no. 1, 224-282. https://doi.org/10.1016/0021-8693(85)90156-5