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, Volume 156, Issue 1–2, pp 215–240 | Cite as

Graded simple Lie algebras and graded simple representations

  • Volodymyr Mazorchuk
  • Kaiming Zhao
Open Access


Let Q be an abelian group and \(\Bbbk \) a field. We prove that any Q-graded simple Lie algebra \(\mathfrak {g}\) over \(\Bbbk \) is isomorphic to a loop algebra in case \(\Bbbk \) has a primitive root of unity of order |Q|, if Q is finite, or \(\Bbbk \) is algebraically closed and \(\dim \mathfrak {g}<|\Bbbk |\) (as cardinals). For Q-graded simple modules over any Q-graded Lie algebra \(\mathfrak {g}\), we propose a similar construction of all Q-graded simple modules over any Q-graded Lie algebra over \(\Bbbk \) starting from nonextendable gradings of simple \(\mathfrak {g}\)-modules. We prove that any Q-graded simple module over \(\mathfrak {g}\) is isomorphic to a loop module in case \(\Bbbk \) has a primitive root of unity of order |Q| if Q is finite, or \(\Bbbk \) is algebraically closed and \(\dim \mathfrak {g}<|\Bbbk |\) as above. The isomorphism problem for simple graded modules constructed in this way remains open. For finite-dimensional Q-graded semisimple algebras we obtain a graded analogue of the Weyl Theorem.

Mathematics Subject Classification

17B05 17B10 17B20 17B65 17B70 



The research presented in this paper was carried out during the visit of both authors to the Institute Mittag-Leffler. V.M. is partially supported by the Swedish Research Council, Knut and Alice Wallenbergs Stiftelse and the Royal Swedish Academy of Sciences. K.Z. is partially supported by NSF of China (Grant 11271109) and NSERC. We thank Alberto Elduque and Mikhail Kochetov for comments on the original version of the paper, in particular, for pointing out two subtle mistakes.


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden
  2. 2.Department of MathematicsWilfrid Laurier UniversityWaterlooCanada
  3. 3.College of Mathematics and Information ScienceHebei Normal (Teachers) UniversityShijiazhuangPeople’s Republic of China

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