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 variety

References

[1] Chang H J. UberWittsche Lie-Ringe. Abh Math Semin Univ Hambg, 1941, 14: 151–184

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

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

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

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

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

[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

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

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

RIGHTS & PERMISSIONS

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