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On Connected Components of Skew Group Algebras

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Abstract

Let A be a basic connected finite dimensional associative algebra over an algebraically closed field k and G be a cyclic group. There is a quiver QG with relations ρG such that the skew group algebras A[G] is Morita equivalent to the quotient algebra of path algebra kQG modulo ideal (ρG). Generally, the quiver QG is not connected. In this paper we develop a method to determine the number of connect components of QG. Meanwhile, we introduce the notion of weight for underlying quiver of A such that A is G-graded and determine the connect components of smash product A#kG*.

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References

  1. Cohen, M., Montgomery, S.: Group-graded rings, smash products, and group actions. Transactions of the American Mathematical Society, 282(1), 237–258 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Demonet, L.: Skew group algebras of path algebras and preprojective algebras. Journal of Algebra, 323(4), 1052–1059 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, J. Y.: Coverings and truncations of graded self-injective algebras. Journal of Algebra, 355(1), 9–34 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Guo, J. Y.: On n-translation algebras. Journal of Algebra, 453, 400–428 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Happel, D.: Triangulated Categories in the Representation Theory of Finite-dimensional Algebras, Cambridge University Press, Cambridge, 1988

    Book  MATH  Google Scholar 

  6. Noether, E.: Hyperkomplexe systeme in ihren beziehungen zur kommutativen algebra und zahlentheorie. Verhandlungen des Internationalen Mathematiker Kongresses Zürich, 1, 189–194 (1932)

    MATH  Google Scholar 

  7. Reiten, I., Riedtmann, C.: Skew group algebras in the representation theory of artin algebras. Journal of Algebra, 92(1), 224–282 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their helpful comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Jian Min Chen, Qiang Dong & Ya Nan Lin

Authors
  1. Jian Min Chen
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  2. Qiang Dong
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  3. Ya Nan Lin
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Corresponding author

Correspondence to Qiang Dong.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 11871404, 11971398 and 12131018)

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Chen, J.M., Dong, Q. & Lin, Y.N. On Connected Components of Skew Group Algebras. Acta. Math. Sin.-English Ser. 39, 799–813 (2023). https://doi.org/10.1007/s10114-022-1237-9

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  • DOI: https://doi.org/10.1007/s10114-022-1237-9

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