Frontiers of Mathematics in China >

2018 , Vol. 13 >Issue 5: 1179 - 1187

DOI: https://doi.org/10.1007/s11464-018-0725-9

RESEARCH ARTICLE

Commuting variety of Witt algebra

  • Yu-Feng YAO 1 ,
  • Hao CHANG , 2
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  • 1. Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
  • 2. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received date: 15 May 2016

Accepted date: 27 Jul 2018

Published date: 29 Oct 2018

Copyright

2018 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature
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Abstract

Let g= W1 be the Witt algebra over an algebraically closed field k of characteristic p >3, and let C (g) = {(x, y) ∈g ×g | [x, y] = 0}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 C (g) is reducible, and not equidimensional. Irreducible components of C (g) and their dimensions are precisely given. As a consequence, the variety C (g) is not normal.

Key words： Witt algebra; irreducible component; dimension; commuting variety

Cite this article

Yu-Feng YAO , Hao CHANG . Commuting variety of Witt algebra[J]. Frontiers of Mathematics in China, 2018 , 13(5) : 1179 -1187 . DOI: 10.1007/s11464-018-0725-9

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Chang H J. UberWittsche Lie-Ringe. Abh Math Semin Univ Hambg, 1941, 14: 151–184

DOI

2
Demushkin S P. Cartan subalgebras of simple Lie p-algebras Wn and Sn. Sibirsk Mat Zh, 1970, 11(2): 310–325

DOI

3
Humphreys J. Linear Algebraic Groups. Grad Texts in Math, Vol 21. New York: Springer-Verlag, 1975

DOI

4
Levy P. Commuting varieties of Lie algebras over fields of prime characteristic. J Algebra, 2002, 250: 473–484

DOI

5
Mygind M. Orbit closure in the Witt algebra and its dual space. J Algebra Appl, 2014, 13(5): 1350146

DOI

6
Ree R. On generalized Witt algebras. Trans Amer Math Soc, 1956, 83: 510–546

DOI

7
Richardson R W. Commuting varieties of semisimple Lie algebras and algebraic groups. Compos Math, 1979, 38: 311–327

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, 1988

9
Tauvel P, Yu R. Lie Algebras and Algebraic Groups. Springer Monogr Math. Berlin: Springer, 2005

10
Wilson R. Automorphisms of graded Lie algebras of Cartan type. Comm Algebra, 1975, 3(7): 591–613

DOI

11
Yao Y-F, Chang H. The nilpotent commuting variety of the Witt algebra. J Pure Appl Algebra, 2014, 218(10): 1783–1791

DOI

12
Yao Y-F, Shu B. Nilpotent orbits in the Witt algebra W1. Comm Algebra, 2011, 39(9): 3232–3241

DOI

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