Algebras with finitely generated invariant subalgebras
Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 379-398.

We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

Nous classifions des algèbres intègres finiment engendrées munies d’une action rationnelle d’un groupe réductif connexe G avec la propriété suivante : toute sous- algèbre G-invariante est finiment engendrée. De plus nous obtenons quelques résultats sur les plongements affines des espaces homogènes.

DOI: 10.5802/aif.1947
Classification: 13A50,  13E15,  14L17,  14L30,  14M17,  14R20
Keywords: algebraic groups, rational G-algebras, quasi-affine homogeneous spaces, affine embeddings
Arzhantsev, Ivan V. 1

1 Moscow State University, Department of Mathematics and Mechanics, Chair of Higher Algebra, Vorobievy Gory, GSP-2, Moscow 119992 (Russie)
@article{AIF_2003__53_2_379_0,
     author = {Arzhantsev, Ivan V.},
     title = {Algebras with finitely generated invariant subalgebras},
     journal = {Annales de l'Institut Fourier},
     pages = {379--398},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     doi = {10.5802/aif.1947},
     zbl = {1099.13500},
     mrnumber = {1990001},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1947/}
}
TY  - JOUR
AU  - Arzhantsev, Ivan V.
TI  - Algebras with finitely generated invariant subalgebras
JO  - Annales de l'Institut Fourier
PY  - 2003
DA  - 2003///
SP  - 379
EP  - 398
VL  - 53
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1947/
UR  - https://zbmath.org/?q=an%3A1099.13500
UR  - https://www.ams.org/mathscinet-getitem?mr=1990001
UR  - https://doi.org/10.5802/aif.1947
DO  - 10.5802/aif.1947
LA  - en
ID  - AIF_2003__53_2_379_0
ER  - 
%0 Journal Article
%A Arzhantsev, Ivan V.
%T Algebras with finitely generated invariant subalgebras
%J Annales de l'Institut Fourier
%D 2003
%P 379-398
%V 53
%N 2
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1947
%R 10.5802/aif.1947
%G en
%F AIF_2003__53_2_379_0
Arzhantsev, Ivan V. Algebras with finitely generated invariant subalgebras. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 379-398. doi : 10.5802/aif.1947. http://www.numdam.org/articles/10.5802/aif.1947/

[Ak77] D. N. Akhiezer Dense orbits with two ends, Izv. Akad. Nauk SSSR, Ser. Mat (in Russian), Volume 41 (1977) no. 2, pp. 308-324 | MR | Zbl

[Ak77] D. N. Akhiezer Dense orbits with two ends, Math. USSR-Izv. (English trans.), Volume 11 (1977) no. 2, pp. 293-307 | Zbl

[AT01] I. V. Arzhantsev; D. A. Timashev Affine embeddings with a finite number of orbits, Transformation Groups, Volume 6 (2001) no. 2, pp. 101-110 | DOI | MR | Zbl

[BB92] F. Bien; A. Borel Sous-groupes épimorphiques de groupes linéaires algébriques I, C. R. Acad. Sci. Paris, Série I, Volume 315 (1992), pp. 649-653 | MR | Zbl

[Br89] M. Brion Groupe de Picard et nombres caractéristiques des variétés sphériques, Duke Math. J, Volume 58 (1989) no. 2, pp. 397-424 | MR | Zbl

[Gr97] F. D. Grosshans Algebraic Homogeneous Spaces and Invariant Theory, LNM, 1673, Springer-Verlag, Berlin, 1997 | MR | Zbl

[Ho69] G. Horroks Fixed point schemes of additive group actions, Topology, Volume 8 (1969), pp. 233-242 | DOI | MR | Zbl

[Hu75] J. E. Humphreys Linear Algebraic Groups, Grad. Texts in Math, 21, Springer-Verlag, New-York, 1975 | MR | Zbl

[Ke78] G. Kempf Instability in invariant theory, Ann. of Math, Volume 108 (1978) no. 2, pp. 299-316 | DOI | MR | Zbl

[La99] I. A. Latypov Homogeneous spaces of compact connected Lie groups which admit nontrivial invariant algebras, Journal of Lie Theory, Volume 9 (1999), pp. 355-360 | MR | Zbl

[LR79] D. Luna; R. W. Richardson A generalization of the Chevalley restriction theorem, Duke Math. J, Volume 46 (1979) no. 3, pp. 487-496 | DOI | MR | Zbl

[LS03] M. W. Liebeck; G. M. Seitz Variations on a theme of Steinberg, Journal of Algebra, Volume 260 (2003), pp. 261-297 | DOI | MR | Zbl

[Lu73] D. Luna Slices étales, Bull. Soc. Math. France, Paris, Volume Mémoire 33 (1973), pp. 81-105 | Numdam | MR | Zbl

[Lu75] D. Luna Adhérences d'orbite et invariants, Invent. Math, Volume 29 (1975), pp. 231-238 | DOI | MR | Zbl

[McN98] G. J. McNinch Dimensional criteria for semisimplicity of representations, Proc. London Math. Soc (3), Volume 76 (1998), pp. 95-149 | DOI | MR | Zbl

[Po75] V. L. Popov Classification of three-dimensional affine algebraic varieties that are quasihomogeneous with respect to an algebraic group, Izv. Akad. Nauk SSSR, Ser. Mat. (in Russian), Volume 39 (1975) no. 3, pp. 566-609 | MR | Zbl

[Po75] V. L. Popov Classification of three-dimensional affine algebraic varieties that are quasihomogeneous with respect to an algebraic group, Math. USSR-Izv. (English trans.), Volume 9 (1975), pp. 535-576 | DOI | MR | Zbl

[PV72] V. L. Popov; E. B. Vinberg A certain class of quasihomogeneous affine algebraic varieties, Izv. Akad. Nauk SSSR, Ser. Mat (in Russian), Volume 36 (1972), pp. 749-764 | MR | Zbl

[PV72] V. L. Popov; E. B. Vinberg A certain class of quasihomogeneous affine algebraic varieties, Math. USSR-Izv. (English trans.), Volume 6 (1972), pp. 743-758 | Zbl

[PV89] V. L. Popov; E. B. Vinberg Invariant Theory, VINITI, Moscow, 1989 (Itogy Nauki i Tekhniki, Sovr. Problemy Mat. Fund. Napravlenia (in Russian)), Volume vol. 5 (1989), pp. 137-309 | Zbl

[PV89] V. L. Popov; E. B. Vinberg Invariant Theory, Algebraic Geometry IV (Encyclopaedia of Math. Sciences (English trans.)), Volume vol. 55 (1994), pp. 123-278 | Zbl

[Ri77] R. W. Richardson Affine coset spaces of reductive algebraic groups, Bull. London Math. Soc, Volume 9 (1977), pp. 38-41 | DOI | MR | Zbl

[Su88] A. A. Sukhanov Description of the observable subgroups of linear algebraic groups, Mat. Sbornik (in Russian), Volume 137 (1988) no. 1, pp. 90-102 | MR | Zbl

[Su88] A. A. Sukhanov Description of the observable subgroups of linear algebraic groups, Math. USSR-Sb. (English trans.), Volume 65 (1990) no. 1, pp. 97-108 | DOI | MR | Zbl

Cited by Sources: