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Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras of Cartan type. I: Type \(W\). (English) Zbl 1479.17035

Summary: Let \((\mathfrak g,[p])\) be a finite-dimensional restricted Lie algebra over an algebraically closed field \(\mathbb{K}\) of characteristic \(p>0\), and \(G\) be the adjoint group of \(\mathfrak g\). We say that \(\mathfrak g\) satisfies the generic property if \(\mathfrak g\) admits generic tori introduced in [J.-M. Bois et al., Forum Math. 26, No. 5, 1333–1379 (2014; Zbl 1331.17017)]. In this paper, we first prove a generalized conjugacy theorem for Cartan subalgebras by means of the generic property. We then classify the \(G\)-conjugacy classes of homogeneous Borel subalgebras of the restricted simple Lie algebras \(\mathfrak g=W(n)\) when \(p>3\), and determine representatives of these classes. Here \(W(n)\) is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring \(\mathbb{K}[T_1,\ldots,T_n]/(T_1^p,\ldots,T_n^p)\). We also describe the closed connected solvable subgroups of \(G\) associated with those representative Borel subalgebras.

MSC:

17B50 Modular Lie (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
17B20 Simple, semisimple, reductive (super)algebras

Citations:

Zbl 1331.17017
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[1] R. Bezrukavnikov, I. Mirkovi´c and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic,Annals of Math.,167(2008), 945-991. · Zbl 1220.17009
[2] J.-M. Bois, R. Farnsteiner and B. Shu, Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem,Forum Math.,26(2014), 1333-1379. · Zbl 1331.17017
[3] S. P. Demu˘skin, Cartan subalgebras of the simple Liep-algebrasWnandSn,Siberian Math. J.,11(1970), 233-245. · Zbl 0215.09601
[4] S. P. Demu˘skin, Cartan subalgebras of simple nonclassical Liep-algebras,Math. USSR Izv.,6(1972), 905-924. · Zbl 0262.17004
[5] R. Farnsteiner, Varieties of tori and Cartan subalgebras of restricted Lie algebras, Trans. Amer. Math. Soc.,356(2004), 4181-4236. · Zbl 1106.17022
[6] R. Hartshorne,Algebraic Geometry, Springer-Verlag, New York, 1977. · Zbl 0367.14001
[7] S. Herpel and D. I. Stewart, On the smoothness of normalisers and the subalgebra structure of modular Lie algebras, arXiv:1402.6280v1 [math.GR]. · Zbl 1387.14121
[8] J. E. Humphreys,Algebraic groups and modular Lie algebras, Memoirs A. M. S. (71), Amer. Math. Soc., Providence, PI, 1967. · Zbl 0173.03001
[9] J. E. Humphreys,Linear Algebraic Groups, Springer-Verlag, New York, 1981. · Zbl 0471.20029
[10] J. C. Jantzen,Representations of algebraic groups, second edition, Amer. Math. Soc., Providence, PI, 2003. · Zbl 1034.20041
[11] J. C. Jantzen,Nilpotent orbits in representation theory, in “Lie Theory” PM 228, Birkh¨auser Boston, 2004, 1-206. · Zbl 1169.14319
[12] K. Ou, Weyl groups and Geometric setting of Lie algebras of Cartan type, Ph. D thesis, East China Normal University, 2016.
[13] K. Ou, B. Shu and H. Xiao, Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras of Cartan type (II), in preparation.
[14] A. A. Premet, On Cartan subalgebras of Liep-algebras,Math. USSR Izv.,29(1987), 145-157. · Zbl 0633.17011
[15] A. A. Premet, Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras,Math. USSR Sb.,66(1989), 555-570. · Zbl 0698.17008
[16] A. A. Premet, The theorem on restriction of invariants and nilpotent elements inWn, Math. USSR Sb.,73(1992), 135-159. · Zbl 0782.17012
[17] A. A. Premet and H. Strade,Classification of finite dimensional simple Lie algebras in prime characteristics, Representations of algebraic groups, quantum groups, and Lie algebras, 185-214, Contemp. Math., 413, Amer. Math. Soc., Providence, RI, 2006. · Zbl 1155.17007
[18] G. B. Seligman,Modular Lie Algebras, Springer-Verlag Berlin Heidelberg, 1967. · Zbl 0189.03201
[19] H. Strade,Simple Lie Algebras over Fields of Positive Charactersitic I. Structure Theory, Walter de Gruyter, Berlin, 2004. · Zbl 1074.17005
[20] H. Strade and R. Farnsteiner,Modular Lie Algebras and Their Representations, Marcel Dekker, New York, 1988. · Zbl 0648.17003
[21] R. L. Wilson, Automorphisms of graded Lie algebras of Cartan type,Comm. Algebra, 3(1975), 591-613. · Zbl 0318.17009
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