Abstract
We extend a well-known result of R. Steinberg on the existence of an invariant maximal torus under a semisimple automorphism of an algebraic group over an algebraically closed field. We show that the same result holds when the underlying field is of characteristic zero, but not necessarily algebraically closed. We next study surjectivity of the power maps \({g\mapsto g^{n}}\) of disconnected algebraic groups of characteristic zero. In the case of disconnected real algebraic groups we apply our generalisation of Steinberg’s result to obtain results on the surjectivity of the power maps. We also extend a result of A. Borel on weak exponentiality in real Lie groups by relating it with the surjectivity of the square map.
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References
Borel A., Mostow G.D.: On semisimple automorphisms of Lie algebras. Ann. Math. 61, 389–405 (1955)
Borel A., Serre J.-P.: Sur certains sous-groupes des groupes de Lie compacts. Comment. Math. Helv. 27, 128–139 (1953)
Borel A., Tits J.: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27, 55–150 (1965)
Brown R.F.: On the power map in compact groups. Q. J. Math. Oxford Ser. 22(2), 395–400 (1971)
Chatterjee P.: On the surjectivity of power maps of solvable Lie groups. J. Algebra 248, 669–687 (2002)
Chatterjee P.: On the surjectivity of the power maps of algebraic groups in characteristic zero. Math. Res. Lett. 9, 741–756 (2002)
Chatterjee P.: On the surjectivity of the power maps of semisimple algebraic groups. Math. Res. Lett. 10, 625–633 (2003)
Chatterjee P.: On the power maps, orders and exponentiality of p-adic algebraic groups. J. Reine Angew. Math. 629, 201–220 (2009)
Chatterjee, P.: Surjectivity of power maps of real algebraic groups (preprint)
Collingwood D.H., McGovern W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Co., New York (1993)
Dani S.G., McCrudden M.: A criterion for exponentiality in certain Lie groups. J. Algebra 238, 82–98 (2001)
Djoković D.Z., Hofmann K.H.: The surjectivity questions for the exponential function of real Lie groups: a status report. J. Lie Theory 7, 171–197 (1997)
Djoković D.Z., Thang N.Q.: On the exponential map of almost simple real algebraic groups. J. Lie Theory 5, 275–291 (1995)
Hochschild G.: The Structure of Lie Groups. Holden-Day Inc, San Francisco (1965)
Hofmann K.H., Mukherjea A.: On the density of the image of the exponential function. Math. Ann. 234, 263–273 (1978)
Hopf H.: Über den Rang geschlossener Liescher Gruppen. Comment. Math. Helv. 13, 119–143 (1940)
McCrudden M.: On n-th roots and infinitely divisible elements in a connected Lie group. Math. Proc. Cambridge Philos. Soc. 89, 293–299 (1981)
Serre J.-P.: Bounds for the orders of the finite subgroups of G(k). In: Geck, M., Testerman, D., Thévenaz, J. (eds) Group Representation Theory, pp. 405–450. EPFL Press, Lausanne (2007)
Springer, T.A.: Linear Algebraic Groups, 2nd edn., PM-9. Birkhäuser (1998)
Steinberg R.: On the power maps in algebraic groups. Math. Res. Lett. 10, 621–624 (2003)
Steinberg, R.: Endomorphisms of linear algebraic groups, Memoirs of the Amer. Math. Soc., No. 80, American Mathematical Society, Providence, R.I. (1968)
Tauvel, P., Yu, R.W.T.: Lie algebras and algebraic groups. Springer Monographs in Mathematics, Springer, Berlin (2005)
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Chatterjee, P. Automorphism invariant Cartan subgroups and power maps of disconnected groups. Math. Z. 269, 221–233 (2011). https://doi.org/10.1007/s00209-010-0723-4
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DOI: https://doi.org/10.1007/s00209-010-0723-4