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Izv. Akad. Nauk SSSR Ser. Mat., 1970, Volume 34, Issue 3, Pages 523–531 (Mi izv2433)  

This article is cited in 23 scientific papers (total in 23 papers)

Stability criteria for the action of a semisimple group on a factorial manifold

V. L. Popov

Abstract: In this work it is proved that, for the regular action of a semisimple irreducible algebraic group $G$ on an affine space, the existence of a closed orbit of maximum dimension is equivalent to the existence of an invariant open set at any point of which the stationary subgroup is reductive. This result is established for the action of $G$ on manifolds of a special type (the so-called factorial manifolds). There are given several other conditions equivalent to the existence of a closed orbit of maximum dimension for the action of $G$ on an arbitrary affine manifold.

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English version:
Mathematics of the USSR-Izvestiya, 1970, 4:3, 527–535

Bibliographic databases:

UDC: 519.4
MSC: 17Bxx, 22E46, 17B40, 20E36
Received: 14.07.1969

Citation: V. L. Popov, “Stability criteria for the action of a semisimple group on a factorial manifold”, Izv. Akad. Nauk SSSR Ser. Mat., 34:3 (1970), 523–531; Math. USSR-Izv., 4:3 (1970), 527–535

Citation in format AMSBIB
\by V.~L.~Popov
\paper Stability criteria for the action of a semisimple group on a~factorial manifold
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1970
\vol 34
\issue 3
\pages 523--531
\jour Math. USSR-Izv.
\yr 1970
\vol 4
\issue 3
\pages 527--535

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. L. Popov, “On the stability of the action of an algebraic group on an algebraic variety”, Math. USSR-Izv., 6:2 (1972), 367–379  mathnet  crossref  mathscinet  zmath
    2. È. B. Vinberg, V. L. Popov, “On a class of quasihomogeneous affine varieties”, Math. USSR-Izv., 6:4 (1972), 743–758  mathnet  crossref  mathscinet  zmath
    3. A. G. Élashvili, “Canonical form and stationary subalgebras of points of general position for simple linear Lie groups”, Funct. Anal. Appl., 6:1 (1972), 44–53  mathnet  crossref  mathscinet  zmath
    4. V. L. Popov, “The classification of representations which are exceptional in the sense of Igusa”, Funct. Anal. Appl., 9:4 (1975), 348–350  mathnet  crossref  mathscinet  zmath
    5. Gerald W. Schwarz, “Representations of simple Lie groups with regular rings of invariants”, Invent math, 49:2 (1978), 167  crossref  mathscinet  zmath
    6. V. L. Popov, “Syzygies in the theory of invariants”, Math. USSR-Izv., 22:3 (1984), 507–585  mathnet  crossref  mathscinet  zmath
    7. D. I. Panyushev, “Semisimple automorphism groups of four-dimensional affine space”, Math. USSR-Izv., 23:1 (1984), 171–183  mathnet  crossref  mathscinet  zmath
    8. Michel Brion, “Surfaces quotients par un groupe unipotent”, Communications in Algebra, 11:10 (1983), 1011  crossref
    9. Michel Brion, “Surfaces quotients par un groupe unipotent”, Communications in Algebra, 11:9 (1983), 1011  crossref
    10. Muhammad A. Albar, “On presentation of group extensions”, Communications in Algebra, 12:23 (1984), 2967  crossref
    11. D. I. Panyushev, “Orbits of maximal dimension of solvable subgroups of reductive linear groups, and reduction for $U$-invariants”, Math. USSR-Sb., 60:2 (1988), 365–375  mathnet  crossref  mathscinet  zmath
    12. V. L. Popov, “Closed orbits of Borel subgroups”, Math. USSR-Sb., 63:2 (1989), 375–392  mathnet  crossref  mathscinet  zmath
    13. Gerald Schwarz, Chen-bo Zhu, “Invariant differential operators on symplectic Grassmann manifolds”, manuscripta math, 82:1 (1994), 191  crossref  mathscinet  zmath  isi
    14. V. É. Kordonskii, E. A. Tevelev, “Non-stable linear actions of connected semisimple complex algebraic groups”, Sb. Math., 186:1 (1995), 107–119  mathnet  crossref  mathscinet  zmath  isi
    15. V. L. Popov, “On polynomial automorphisms of affine spaces”, Izv. Math., 65:3 (2001), 569–587  mathnet  crossref  crossref  mathscinet  zmath  elib
    16. Vladimir L. Popov, “Tensor product decompositions and open orbits in multiple flag varieties”, Journal of Algebra, 313:1 (2007), 392  crossref
    17. Lex Renner, Alvaro Rittatore, “Observable subgroups of algebraic monoids”, Journal of Algebra, 323:12 (2010), 3202  crossref
    18. A. B. Anisimov, “On stability of diagonal actions and tensor invariants”, Sb. Math., 203:4 (2012), 500–513  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. V. L. Popov, “Tori in the Cremona groups”, Izv. Math., 77:4 (2013), 742–771  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    20. Mitsuyasu Hashimoto, “Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants”, Communications in Algebra, 43:4 (2015), 1524  crossref
    21. C. Procesi, “The geometry of polynomial identities”, Izv. Math., 80:5 (2016), 910–953  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    22. V. L. Popov, “On Conjugacy of Stabilizers of Reductive Group Actions”, Math. Notes, 105:3-4 (2019), 580–581  mathnet  crossref  crossref  mathscinet  isi  elib
    23. Bai Ch. Fu B. Manivel L., “On Fano Complete Intersections in Rational Homogeneous Varieties”, Math. Z., 295:1-2 (2020), 289–308  crossref  isi
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