Commuting variety of Witt algebra
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.
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).
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