On central extensions of simple differential algebraic groups

Minchenko, Andrey

doi:10.1090/S0002-9939-2015-12639-1

On central extensions of simple differential algebraic groups
HTML articles powered by AMS MathViewer

by Andrey Minchenko

Proc. Amer. Math. Soc. 143 (2015), 2317-2330

DOI: https://doi.org/10.1090/S0002-9939-2015-12639-1

Published electronically: January 22, 2015

PDF | Request permission

Abstract:

We consider central extensions $Z\hookrightarrow E\twoheadrightarrow G$ in the category of linear differential algebraic groups. We show that if $G$ is simple non-commutative and $Z$ is unipotent with the differential type smaller than that of $G$, then such an extension splits. We also give a construction of central extensions illustrating that the condition on differential types is important for splitting. Our results imply that non-commutative almost simple linear differential algebraic groups, introduced by Cassidy and Singer, are simple.

References

Armand Borel and Jacques Tits, Homomorphismes “abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571 (French). MR 316587, DOI 10.2307/1970833
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Buium, A., Cassidy, P.: Differential algebraic geometry and differential algebraic groups. From algebraic differential equations to diophantine geometry. Selected works of Ellis Kolchin with commentary pp. 567–636 (1999)
P. J. Cassidy, Differential algebraic groups, Amer. J. Math. 94 (1972), 891–954. MR 360611, DOI 10.2307/2373764
Phyllis Joan Cassidy, The differential rational representation algebra on a linear differential algebraic group, J. Algebra 37 (1975), no. 2, 223–238. MR 409426, DOI 10.1016/0021-8693(75)90075-7
Phyllis Joan Cassidy, The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras, J. Algebra 121 (1989), no. 1, 169–238. MR 992323, DOI 10.1016/0021-8693(89)90092-6
Phyllis J. Cassidy and Michael F. Singer, A Jordan-Hölder theorem for differential algebraic groups, J. Algebra 328 (2011), 190–217. MR 2745562, DOI 10.1016/j.jalgebra.2010.08.019
J. Freitag, Indecomposability for differential algebraic groups, (2011). http://arxiv.org/abs/1106.0695
Henri Gillet, Differential algebra—a scheme theory approach, Differential algebra and related topics (Newark, NJ, 2000) World Sci. Publ., River Edge, NJ, 2002, pp. 95–123. MR 1921696, DOI 10.1142/9789812778437_{0}003
Sergey Gorchinskiy and Alexey Ovchinnikov, Isomonodromic differential equations and differential categories, J. Math. Pures Appl. (9) 102 (2014), no. 1, 48–78 (English, with English and French summaries). MR 3212248, DOI 10.1016/j.matpur.2013.11.001
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654
E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864
E. R. Kolchin, Constrained extensions of differential fields, Advances in Math. 12 (1974), 141–170. MR 340227, DOI 10.1016/S0001-8708(74)80001-0
E. R. Kolchin, Differential algebraic groups, Pure and Applied Mathematics, vol. 114, Academic Press, Inc., Orlando, FL, 1985. MR 776230
Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214
Andrey Minchenko and Alexey Ovchinnikov, Zariski closures of reductive linear differential algebraic groups, Adv. Math. 227 (2011), no. 3, 1195–1224. MR 2799605, DOI 10.1016/j.aim.2011.03.002
Andrey Minchenko and Alexey Ovchinnikov, Extensions of differential representations of $\textbf {SL}_2$ and tori, J. Inst. Math. Jussieu 12 (2013), no. 1, 199–224. MR 3001738, DOI 10.1017/S1474748012000692
A. Minchenko, A. Ovchinnikov, M. Singer, Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations (to appear), International Mathematics Research Notices http://arxiv.org/abs/1304.2693
A. Minchenko, A. Ovchinnikov, M. Singer, Unipotent differential algebraic groups as parameterized differential Galois groups (to appear), Journal of the Institute of Mathematics of Jussieu http://dx.doi.org/10.1017/S1474748013000200
A. Pillay, Differential algebraic groups and the number of countable differentially closed fields, Lecture Notes in Logic 5, 114–134 (1996). http://projecteuclid.org/euclid.lnl/1235423157
Anand Pillay, Some foundational questions concerning differential algebraic groups, Pacific J. Math. 179 (1997), no. 1, 179–200. MR 1452531, DOI 10.2140/pjm.1997.179.179
Joseph Fels Ritt, Differential Algebra, American Mathematical Society Colloquium Publications, Vol. XXXIII, American Mathematical Society, New York, N. Y., 1950. MR 0035763
Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin, 1979. MR 547117