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Lie algebras and separable morphisms in pro-affine algebraic groups

Author(s): Nazih Nahlus
Journal: Proc. Amer. Math. Soc. 131 (2003), 1321-1327.
MSC (2000): Primary 14L15, 16W30, 17B45, 20G15
Posted: December 16, 2002
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Abstract: Let $ K$ be an algebraically closed field of arbitrary characteristic, and let $ f:G\rightarrow H$ be a surjective morphism of connected pro-affine algebraic groups over $ K$. We show that if $ f$ is bijective and separable, then $ f$is an isomorphism of pro-affine algebraic groups. Moreover, $ f$ is separable if and only if (its differential) $ f^o$ is surjective. Furthermore, if $ f$ is separable, then $ {\mathcal L}(\operatorname{Ker}f)=\operatorname{Ker} f^o$.

References:

1.
A. Bialynicki-Birula, G.Hochschild and G.D. Mostow, Extensions of representations of algebraic groups, Amer. J. Math. 85 (1963), 131-144. MR 27:5871

2.
A. Borel, Linear algebraic groups, second edition, GTM 126, Springer-Verlag, 1991. MR 92d:20001

3.
N. Bourbaki, Algebra II, Chapters 4-7, Springer-Verlag, 1990. MR 91h:00003

4.
G. Hochschild, Coverings of pro-affine algebraic groups, Pacific J. Math. 35 (1970), 399-415. MR 43:4830

5.
-, Basic theory of algebraic groups and Lie algebras, GTM 75, Springer Verlag, 1981. MR 82i:20002

6.
G. Hochschild and G.D. Mostow, Pro-affine algebraic groups, Amer. J. Math. 91 (1969), 1127-1140. MR 41:350

7.
-, Complex analytic groups and Hopf algebras, Amer. J. Math. 91 (1969), 1141-1151. MR 40:5782

8.
J. E. Humphreys, Linear algebraic groups, GTM 21, Springer-Verlag, 1975. MR 53:633

9.
A. Lubotzky and A. R. Magid, Free prounipotent groups, J. Algebra 80 (1983), 323-349. MR 85b:14062

10.
A. R. Magid, The universal group cover of a pro-affine algebraic group, Duke Math. J. 42 (1975), 43-49. MR 51:8275

11.
-, Module categories of analytic groups, Cambridge Univ. Press, 1982. MR 84j:22008

12.
A. R. Magid, Prounipotent prolongation of algebraic groups, Contemp. Math. J. 224 (1999), 169-187. MR 99k:20097

13.
N. Nahlus, Representative functions on complex analytic groups, Amer. J. Math. 116 (1994), 621-636. MR 95g:22009

14.
-, Basic groups of Lie algebras and Hopf algebras, Pacific J. Math. 180 (1997), 135-151. MR 99g:17053

15.
T. A. Springer, Linear algebraic groups, second edition, Birkhäuser, 1998. MR 99h:20075

16.
W. Waterhouse, Introduction to affine group schemes, GTM 66, Springer-Verlag, 1979. MR 82e:14003

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Additional Information:

Nazih Nahlus
Affiliation: Department of Mathematics, American University of Beirut, Beirut, Lebanon
Email: nahlus@aub.edu.lb

DOI: 10.1090/S0002-9939-02-06963-0
PII: S 0002-9939(02)06963-0
Keywords: Lie algebras of pro-affine algebraic groups, separable morphisms
Received by editor(s): August 10, 2000
Posted: December 16, 2002
Communicated by: Dan M. Barbasch
Copyright of article: Copyright 2002, American Mathematical Society

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