Springer resolution

In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra,[1][2] or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969.[3] The fibers of this resolution are called Springer fibers.[4]

If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by the nilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebras.[5]

The Grothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or the whole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.[6][7]

Examples

When G=SL(2), the Lie algebra Springer resolution is T*P1 → n, where n are the nilpotent elements of sl(2). In this example, n are the matrices x with tr(x2)=0, which is a two dimensional conical subvariety of sl(2). n has a unique singular point 0, the fibre above which in the Springer resolution is the zero section P1 .

References

  1. ^ Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser Boston, Inc., ISBN 0-8176-3792-3, MR 1433132
  2. ^ Dolgachev, Igor; Goldstein, Norman (1984), "On the Springer resolution of the minimal unipotent conjugacy class", Journal of Pure and Applied Algebra, 32 (1): 33–47, doi:10.1016/0022-4049(84)90012-4, hdl:2027.42/24847, MR 0739636
  3. ^ Springer, Tonny A. (1969), "The unipotent variety of a semi-simple group", Algebraic Geometry (Internat. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, London, pp. 373–391, ISBN 978-0-19-635281-7, MR 0263830
  4. ^ Ginzburg, Victor (1998), "Geometric methods in the representation theory of Hecke algebras and quantum groups", Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 514, Kluwer Acad. Publ., Dordrecht, pp. 127–183, arXiv:math/9802004, Bibcode:1998math......2004G, ISBN 0-7923-5193-2, MR 1649626
  5. ^ Springer, Tonny A. (1976), "Trigonometric sums, Green functions of finite groups and representations of Weyl groups", Inventiones Mathematicae, 36: 173–207, Bibcode:1976InMat..36..173S, doi:10.1007/BF01390009, MR 0442103, S2CID 121820241
  6. ^ Steinberg, Robert (1974), Conjugacy classes in algebraic groups, Lecture Notes in Mathematics, vol. 366, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0067854, ISBN 978-3-540-06657-6, MR 0352279
  7. ^ Steinberg, Robert (1976), "On the desingularization of the unipotent variety", Inventiones Mathematicae, 36: 209–224, Bibcode:1976InMat..36..209S, doi:10.1007/BF01390010, MR 0430094, S2CID 120400717


Stub icon

This abstract algebra–related article is a stub. You can help Wikipedia by adding missing information.

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.