Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Maximal semigroups in semi-simple Lie groups


Author: Luiz A. B. San Martin
Journal: Trans. Amer. Math. Soc. 353 (2001), 5165-5184
MSC (2000): Primary 20M20, 22E20, 22F30
DOI: https://doi.org/10.1090/S0002-9947-01-02868-9
Published electronically: June 14, 2001
MathSciNet review: 1852099
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here $\mathcal{B}$-convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a $\mathcal{B}$-convex set.


References [Enhancements On Off] (What's this?)

  • 1. Ando, T.: Totally positive matrices. Linear Algebra Appl. 90 (1987), 165-219. MR 88b:15023
  • 2. Colonius, K. and W. Kliemann: ``Dynamics and control''. Birkhäuser (2000). MR 2001e:93001
  • 3. Goodman, J.E.: When is a set of lines in space convex? Notices Amer. Math. Soc. 45 (1998), 222-232.
  • 4. Goodman, J.E. and R. Pollack: Foundations of a theory of convexity on affine Grassmann manifolds. Mathematika 42 (1995), 305-328. MR 97a:52002
  • 5. Hilgert, J., K.H. Hofmann and J. Lawson: ``Lie groups, convex cones, and semigroups''. Oxford University Press (1989). MR 91b:22008
  • 6. Hilgert, J. and K.-H. Neeb: ``Lie semigroups and their applications''. Lecture Notes in Math. 1552, Springer-Verlag (1993). MR 96j:22002
  • 7. Hilgert, J. and K.-H. Neeb: Maximality of compression semigroups. Semigroup Forum 50 (1995), 205-222. MR 96e:53069
  • 8. Lawson, J.D.: Maximal semigroups of Lie groups that are total. Proc. Edinburgh Math. Soc. 30 (1987) 479-501. MR 89e:22042
  • 9. Lusztig, G.: Introduction to total positivity. In ``Positivity in Lie Theory: Open Problems'' (Eds. J. Hilgert, J.D. Lawson, K.-H. Neeb and E.B. Vinberg). De Gruyter Expositions in Mathematics 26 (1998), 133-145. MR 99h:20077
  • 10. Mittenhuber, D.: Compression semigroups in semisimple Lie groups: a direct approach. In ``Positivity in Lie Theory: Open Problems'' (Eds. J. Hilgert, J.D. Lawson, K.-H. Neeb and E.B. Vinberg). De Gruyter Expositions in Mathematics 26 (1998), 165-183. MR 99j:22009
  • 11. do Rocio, O.G. and L.A.B. San Martin: Semigroups in lattices of solvable Lie groups. J. Lie Theory 51 (1995), 179-202. MR 97d:22007
  • 12. San Martin L.A.B.: Invariant control sets on flag manifolds. Math. Control Signals Systems 6 (1993), 41-61. MR 96i:93020
  • 13. -- Order and domains of attractions of control sets in flag manifolds. J. Lie Theory 8 (1998), 335-350. MR 99j:93024
  • 14. San Martin, L.A.B. and P.A. Tonelli: Semigroup actions on homogeneous spaces. Semigroup Forum 50 (1995), 59-88. MR 97a:22008
  • 15. Warner, G.: ``Harmonic analysis on semi-simple Lie groups I''. Springer-Verlag (1972). MR 58:16979

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20M20, 22E20, 22F30

Retrieve articles in all journals with MSC (2000): 20M20, 22E20, 22F30


Additional Information

Luiz A. B. San Martin
Affiliation: Instituto de Matemática, Universidade Estadual de Campinas, Cx.Postal 6065, 13083-970 Campinas SP, Brasil
Email: smartin@ime.unicamp.br

DOI: https://doi.org/10.1090/S0002-9947-01-02868-9
Keywords: Semigroups, semi-simple Lie groups, flag manifolds, convexity
Received by editor(s): March 18, 1999
Received by editor(s) in revised form: March 29, 2001
Published electronically: June 14, 2001
Additional Notes: Research partially supported by CNPq grant n$^{∘}$ $301060/94-0$.
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society