Abstract
Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window
be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window
is reducible, and not equidimensional. Irreducible components of Open image in new window
and their dimensions are precisely given. As a consequence, the variety Open image in new window
is not normal.
Keywords
Witt algebra irreducible component dimension commuting varietyMSC
17B05 17B50 17B70Preview
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Notes
Acknowledgements
The authors would like to express their thanks to the referees for many useful suggestions and comments on the manuscript. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771279, 11801204) and the Natural Science Foundation of Shanghai (Grant No. 16ZR1415000).
References
- 1.Chang H J. Uber Wittsche Lie-Ringe. Abh Math Semin Univ Hambg, 1941, 14: 151–184CrossRefzbMATHGoogle Scholar
- 2.Demushkin S P. Cartan subalgebras of simple Lie p-algebras W n and S n. Sibirsk Mat Zh, 1970, 11(2): 310–325MathSciNetCrossRefGoogle Scholar
- 3.Humphreys J. Linear Algebraic Groups. Grad Texts in Math, Vol 21. New York: Springer-Verlag, 1975CrossRefzbMATHGoogle Scholar
- 4.Levy P. Commuting varieties of Lie algebras over fields of prime characteristic. J Algebra, 2002, 250: 473–484MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Mygind M. Orbit closure in the Witt algebra and its dual space. J Algebra Appl, 2014, 13(5): 1350146MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Ree R. On generalized Witt algebras. Trans Amer Math Soc, 1956, 83: 510–546MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Richardson R W. Commuting varieties of semisimple Lie algebras and algebraic groups. Compos Math, 1979, 38: 311–327MathSciNetzbMATHGoogle Scholar
- 8.Strade H, Farnsteiner R. Modular Lie Algebras and Their Representations. Monographs and Textbooks in Pure and Applied Mathematics, Vol 116. New York: Marcel Dekker Inc, 1988zbMATHGoogle Scholar
- 9.Tauvel P, Yu R. Lie Algebras and Algebraic Groups. Springer Monogr Math. Berlin: Springer, 2005zbMATHGoogle Scholar
- 10.Wilson R. Automorphisms of graded Lie algebras of Cartan type. Comm Algebra, 1975, 3(7): 591–613MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Yao Y-F, Chang H. The nilpotent commuting variety of the Witt algebra. J Pure Appl Algebra, 2014, 218(10): 1783–1791MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Yao Y-F, Shu B. Nilpotent orbits in the Witt algebra W 1. Comm Algebra, 2011, 39(9): 3232–3241MathSciNetCrossRefzbMATHGoogle Scholar