Publications and Preprints

Luiz A. B. San Martin


Morse decomposition of semiflows on fiber bundles

with Mauro Patrão

October, 2005

Semiflows on topological spaces: Chain transitivity and shadowing semigroups

with Mauro Patrão

September, 2005

Lie algebras with complex structures having nilpotent eigenspaces

with Edson C. Licurgo dos Santos

August, 2995

Semigroups in Symmetric Lie Groups

with com Laércio J. dos Santos

June, 2005

Invariant cones and convex sets for bilinear control systems and  parabolic type of semigroups

with O. G. do Rocio e A. J. Santana

Journal of Dynamics and Control Systems, to appear.

May, 2005

Control sets and total positivity

with V. Ayala e W. Kliemann

Semigroup Forum 69(1) (2004), 113--140. MR 2 063 983

Invariant nearly-Kähler structures.

with Rita de Cássia Jesus Silva

September, 2003

Covering space for monotonic homotopy of trajectories of control systems

with Fritz Colonius and Eyup Kizil

 

J. Diff. Equations, 216, issue 2 (2005), 324-353. MR 2 162 339

f-Structures on the Classical Flag Manifold  which Admit (1,2)-Symplectic Metrics

with Nir Cohen, Caio J. C. Negreiros, Marlio Paredes e Sofia Pinzón

Tohoku Mathematical Journal (2) 57 (2005), no. 2, 261--271. MR 2137470

Maximal chain transitive sets for local groups

with Carlos J. Braga Barros

Boletim da Sociedade Paranaense de Matemática (3) 21 (2003), no 1-2, 113-125. MR 2 066 951

Geometric properties of invariant connections on Sl(n,R)/SO(n)

with Marco A. N. Fernandes

Journal of Geometry and Physics, Volume 47, Issues 2-3 (2003), 369-377. MR 1 991 481

Chain transitive sets for flows on flag bundles (pdf zip)

with Carlos J. Braga Barros

This paper applies the theory of semigroups in semi-simple Lie groups to a problem in dynamical systems a la Conley. Namely we give an algebraic description of the chain transitive component for a continuous-time flow evolving on a fiber bundle whose fibers are flag manifolds of semi-simple Lie groups.

-July, 2002. To appear in Forum Mathematicum http://www.degruyter.de/rs/278_3128_DEU_h.htm

Connected components of open semigroups in semi-simple Lie groups (ps zip)

with Osvaldo Germano do Rocio

-March, 2002.

A rank-three condition for invariant (1,2)-symplectic almost Hermitian structures on flag manifolds (dvi)

with Nir Cohen and Caio J. C. Negreiros

-September, 2001. To appear in Boletim da Sociedade Brasileira de Matemática (Bulletin of the Brazilian Math. Soc.).

Invariant almost Hermitian structures on flag manifolds

with Caio J. C. Negreiros

Given a complex semi-simple Lie group G its 'Fundamental homogeneous space' (maximal flag manifold) is the coset B = G/P modulo a Borel subgroup P of G. There exists an extensive literature about G/P for several reasons which range from questions in Differential Geometry to Representation Theory. In the context of compact Lie groups the spaces G/P are the given by cosets U/T where U is a compact real form of G and T is a maximal torus of U. These spaces are also known generically as 'flag manifolds' (or maximal flag manifolds), since G/P identifies with concrete space of (complete) flags of subspaces of an n-dimensional complex vector space when G is the special linear group Sl(n,C).

This paper studies U-invariant almost complex geometry on B = G/P. A basic issue are the invariant (1,2)-symplectic almost Hermitian structures. Two characterizations of these structures are given: 1) by means of the abelian ideals of a Borel subalgebra and 2) in terms of the alcoves of the corresponding affine Weyl group. The first characterization generalizes the results of the paper below, with independent proofs. In fact, instead of tournaments, that are linked only to the Sl(n,C) (concrete) flag manifolds, we use directly the algebra (combinatorics) of root systems, which gives life to the theory of semi-simple Lie algebras.

Having at hand the characterization of (1,2)-symplectic structures we give a classification of the invariant almost Hermitian structures into four classes.

Advances in Mathematics, 178 (2003), 277-310. (ps zip)

(1,2)-Symplectic metrics on flag manifolds and trournaments (dvi)

with Nir Cohen and Caio J. C. Negreiros

This paper is devoted to a problem in complex geometry, namely the characterization of almost complex structures on manifolds which admit Riemannian metrics which are (1,2)-symplectic (or as they are also called almost Kahler metrics). Here the combinatorics of tournaments is used to characterize those invariant almost complex structures in the complex full flag manifold which admit an invariant (1,2)-symplectic metrics.

Bulletin of London Mathematical Society 34 (2002), no. 6, 641-649. MR  1924350

The homotopy type of Lie semigroups in semi-simple Lie groups

with Alexandre J. Santana

A noncompact semi-simple Lie group G decomposes as the product of its corresponding symmetric space by the maximal compact subgroup K. This decompositions entails that the topology of G lives in K. In this paper its proved something similar holds for a semigroup S contained in G. Namely there is a compact subgroup K(S) of K giving the topology of S, in the sense that S and K(S) are homotopy equivalent (actually a stronger statement holds: A coset of K(S) is a deformation retract of the interior of S. These results are proved with the assumption that S is generated by a cone in the Lie algebra of G (since it is not assumed that S is closed this condition is a slight extension of the concept of Lie semigroup). Actually the proofs hold for semigroups having large subsemigroups generated by cones in Lie algebra. For instance, the semigroup of positive matrices is not generated by a cone in the Lie algebra but it contains a large such semigroup sothat the homotopy type of the semigroup of nxn positive matrices can be computed by the methods of this paper (it is the homotopy type of the orthogonal group SO(n-1).

Monattshefte für Mathematik, 136 (2002), 151-173. (ps zip)

Nonexistence of invariant semigroups in affine symmetric spaces

The semigroups that usually appear in the theory of causal symmetric spaces are infinitesimally generated by invariant cones. These cones occur only when the affine pair are of Hermitian type, so that infinitesimally generated invariant semigroups do not exist except in this case. This paper drops the term 'infinitesimally generated' from that statement: There exists an invariant semigroup with nonempty interior only if the affine symmetric space is of Hermitian type. This result was conjectured before by Joachim Hilgert and Karl-Hermann Neeb (Compression semigroups of open orbits on real flag manifolds. Monatsh. Math.,119, (1995), 187-214.)

Mathematische Annalen, 321, (2001), 587-600.

A family of maximal noncontrollable Lie wedges with empty interior

A natural question in Lie semigroup theory is whether the maximal Lie wedges in a Lie algebra are generating cones, i.e, have nonempty interior in the Lie algebra. This is true for solvable Lie algebras. Recently Dirk Mittenhuber built maximal Lie wedges in the rank one su(1,n). This paper gives other examples of such Lie wedges: The Lie wedge of the semigroup of dxd matrices having positive k-minors is maximal and have empty interior if d and k are even.

Systems & Control Letters (Special Issue on Lie Theory and Applications to Control) 43 (1) (2001) pp. 53-57.

Maximal semigroups in semi-simple Lie groups (dvi)

The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized. They are compression semigroups of specific type of sets, the B-convex sets.

Transactions of American Mathematical Ssociety, 353 (2001), 5165-5184.

Fisher Information and alfa-connections for a class of Transformational models

with Marco A. N. Fernandes

We characterize the invariant alfa-connections for statistical transformational models parameterized by homogeneous spaces. It turns out that in the semi-simple setting only the A_l series admit nontrivial invariant connections. Some properties of these connections are derived.

Differential Geometry and its Applications , 12 (2000), 165-184.

Lyapunov Exponents for Stochastic Differential Equations in Semi-simple Lie groups (dvi)

with Paulo R. C. Ruffino

We write an integral formula for the asymptotics of the A-part in the Iwasawa decomposition of the solution of an invariant stochastic equation in a semi-simple group. The integral is with respect to the invariant measure on the maximal flag manifold, the Furstenberg boundary. The integrand of the formula is related to the Takeuchi-Kobayashi Riemannian metric in the flag manifold - January/1998. To appear in Archivum Mathematicum.

Controllability of Discrete-time Control Systems on the Symplectic Group

with Carlos J. Braga Barros.

Gives sufficient conditions for the controllability of a 'bilinear' discrete-time control system when the system evolves in the symplectic group. It is similar in spirit and techniques to "On global controllability ..." (see below)

Systems & Control Letters, 42 (2001), 95-100. (ps zip)

Orders and Domains of Attraction of Control Sets in Flag Manifolds

Relates the Bruhat-Chevalley order in the Weyl group with the order of the control sets in the flag manifolds. Journal of Lie Theory (JoLT) , 8, (1998), 335-350. (ps zip)

Homogeneous Spaces Admitting Transitive Semigroups

The transitivity in a homogeneous space of a proper semigroup of a semi-simple Lie group is a rare event. Journal of Lie Theory (JoLT) , vol. 8 (1998), 111-128. (ps zip)

Transitive Actions of Semigroups in Semi-simple Lie Groups

with Pedro A. Tonelli.

Necessary conditions for an action of a semigroup to be transitive on a homogeneous space. Semigroup Forum, 58 (1999), 142-151.

On the Action of a Semigroup in a Fiber Bundle (dvi)

with Carlos J. Braga Barros.

Studies control sets for semigroups acting in a fiber bundle in a topological setting. The invariant control sets are described by invariant control sets in the base space and in the fibers. Matematica Contemporanea, 13 (1997), 1-19.

On Global Controllability of Discrete-Time Control Systems

Gives sufficient conditions in order that a discrete-time bilinear control system is controllable. The conditions are given in terms of the coefficients of the system and are similar to the conditions of Jurdjevic and Kupka in the continuous-time. Mathematics of Control, Signals, and Systems (MCSS) vol. 8 (1995) 279-297. (ps zip)

Controllability of two-dimensional bilinear systems: restricted controls and discrete-time.

with V. Ayala.

Provides necessary and sufficient algebraic conditions for the controllability of a bilinear system in dimension two. The approach is through Lie theory with the systems divided according to the transitive groups they generate. Proyecciones, vol. 18, 2 (1999), 207-223. MR 1 727 729. (ps zip)

Controllability of Two-Dimensional Bilinear Systems (dvi)

with Carlos J. Braga Barros, João R. Gonçalves Filho and Osvaldo G. do Rocio.

Provides necessary and sufficient algebraic conditions for the controllability of a bilinear system in dimension two. The approach is through Lie theory with the systems divided according to the transitive groups they generate. Proyecciones Revista de Matematica vol. 15 (1996), 111-139. (ps zip)

Chain Control Sets for Semigroup Actions

with Carlos J. Braga Barros.

Generalizes to general semigroup actions the concept of chain control set relating it to control sets for perturbed semigroups. Chain control sets for semigroups in flag manifolds are characterized. Computational and Applied Mathematics vol. 15 (1996), 257-276. (ps zip)

Semigroup Actions on Homogeneous Spaces

with Pedro A. Tonelli.

Builds a theory of semigroups in semi-simple Lie groups with finite center from the action on the flag manifolds of the group. Semigroup Forum vol. 50 (1995), 59-88. (ps zip)

On the number of control sets in projective spaces

with Carlos J. Braga Barros.

Counts the number of control sets of a linear semigroup acting in the projective space. Systems & Control Letters vol. 29 (1996), 21-26.

Discrete Semigroups in Nilpotent Groups

with Osvaldo G. do Rocio.

Shows that a proper semigroup in a lattice of a nilpotent group is contained in proper semigroup with interior points. In particular the maximal semigroups of a lattice are intersections of maximal semigroups with nonvoid interior with the lattice. Semigroup Forum vol. 51 (1995), 125-133.

Semigroups in Lattices of Solvable Groups

with Osvaldo G. do Rocio.

Generalizes the previous paper to solvable groups. The results here are not plain. They require an assumption about the immaginary roots of the adjoint representation of the Lie algebra of the group. Journal of Lie Theory (JOLT) (1995) 188-202.


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Last modified November 2005