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Flag Type of Semigroups: A Survey

In: Advances in Mathematics and Applications

Author

Listed:
  • Luiz A. B. San Martin

    (University of Campinas, Institute of Mathematics, Statistics and Scientific Computing)

Abstract

In this chapter, we present an overview of the theory of semigroups in semi-simple Lie groups and its applications to dynamical systems, control systems, and random dynamical systems. A great deal of the results to be surveyed appeared first in Ph.D. theses by students of the Department of Mathematics of IMECC. The piece of semigroup theory to be discussed here was constructed with the purpose of understanding semigroups with non-empty interior in semi-simple Lie groups. A characteristic of this theory is that it is built upon the actions of the semigroups on the flag manifolds of the Lie groups. These actions contain crucial information about the semigroups due to the strong structural properties of the semi-simple Lie groups. The concept of flag type of a semigroup emerges as a synthesis of several results about the actions of the semigroups on the flag manifolds. This concept gives a classification of semigroups via block decompositions, much like the Jordan form of matrices. More than that, it provides key information about the structure of the semigroups in the semi-simple Lie groups. The results to be surveyed in this chapter exploit the concept of flag type to describe properties of the semigroups as well as to get applications to control and dynamical systems.

Suggested Citation

  • Luiz A. B. San Martin, 2018. "Flag Type of Semigroups: A Survey," Springer Books, in: Carlile Lavor & Francisco A. M. Gomes (ed.), Advances in Mathematics and Applications, pages 351-372, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-94015-1_14
    DOI: 10.1007/978-3-319-94015-1_14
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