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Luiz A. B. San Martin

Publications and source records attributed to Luiz A. B. San Martin.

At least 19 recordsLinked to original sources
arXiv2026

A description of the canonical symplectic form of the cotangent bundle of a generalized flag manifold

We investigate the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, with particular emphasis on cotangent bundles of generalized flag manifolds. Our approach gives an explicit description of the Liouville symplectic form on these spaces in terms of the connections and curvature forms of principal bundles naturally arising from the Lie group structure. This provides a geometric link between the Lie-theoretic properties of generalized flag manifolds and the symplectic structures carried by their cotangent bundles.

math.DG↗
arXiv2026

The Riemannian geometry of the probability space of the unit circle

This paper explores the Riemannian geometry of the Wasserstein space of the circle, namely $P(S^{1})$, the set of probability measures on the unit circle endowed with the 2-Wasserstein metric. Building on the foundational work of Otto, Lott, and Villani, the authors developed in another work an intrinsic framework for studying the differential geometry of Wasserstein spaces of compact Lie groups, making use of the Peter-Weyl Theorem. This formalism allowed them to explicit an example in this paper. Key contributions include explicit computations of the Riemannian metric matrix coefficients, Lie brackets, and the Levi-Civita connection, along with its associated Christoffel symbols. The geodesic equations and curves with constant velocity fields are analysed, expliciting their PDEs. Notably, the paper demonstrates that $P(S^{1})$ is flat, with vanishing curvature. These results provide a comprehensive geometric understanding of $P(S^{1})$, connecting optimal transport theory and differential geometry, with potential applications in dynamical systems.

math.DG↗
arXiv2026

On Differential and Riemannian Calculus on Wasserstein Spaces

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology via convergent sequences of the subjacent space. Applying it we also provide a new proof that Wasserstein spaces of closed manifolds are geodesically convex. Our framework is particularly handy to address the Wasserstein spaces of compact Lie groups, where we refine our formalism and present an explicit example.

math.DG↗
arXiv2026

The Wasserstein space of the circle is not flat

We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.

math.DG↗
arXiv2025

Semigroups in semi-simple Lie groups: Flag type and estimation of cocycles

The flag type of a semigroup S of a noncompact semisimple Lie group is an algebraic tool related to the geometry of the invariant control set determined by S on the flag manifolds of G. In the present paper we show that it is possible to recover the flag type by studying the existence of lower bounds for cocycles on the maximal flag manifold

math.RA↗
arXiv2022

Lagrangian submanifolds of Adjoint semisimple orbits given by real forms

We found some Lagrangian submanifolds of the adjoint semisimple orbit in two cases. For the first, the compact case, also known as the Generalized flag manifolds, we prove that the real flags can be seen as infinitesimally tight Lagrangian submanifolds with respect to the KKS symplectic form and we give a complete classification. And for the second, the complex case, we prove that the orbits of real forms are Lagrangian submanifolds with respect to the Hermitian symplectic form.

math.SG↗
arXiv2020

Morse functions and Real Lagrangian Thimbles on Adjoint Orbits

We compare Lagrangian thimbles for the potential of a Landau-Ginzburg model to the Morse theory of its real part. We explore Landau-Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.

math.SG↗
arXiv2020

Semigroups and Controllability of Invariant Control Systems on $\mathrm{Sl}\left(n,\mathbb{H}\right)$

Let $\mathrm{Sl}\left( n,\mathbb{H}\right)$ be the Lie group of $n\times n$ quaternionic matrices $g$ with $\left\vert \det g\right\vert =1$. We prove that a subsemigroup $S \subset \mathrm{Sl}\left( n,\mathbb{H}\right)$ with nonempty interior is equal to $\mathrm{Sl}\left( n,\mathbb{H}\right)$ if $S$ contains a subgroup isomorphic to $\mathrm{Sl}\left( 2,\mathbb{H}\right)$. As application we give sufficient conditions on $A,B\in \mathfrak{sl}\left( n,\mathbb{H}\right)$ to ensuring that the invariant control system $\dot{g}=Ag+uBg$ is controllable on $\mathrm{Sl}\left( n,\mathbb{H}\right)$. We prove also that these conditions are generic in the sense that we obtain an open and dense set of controllable pairs $\left( A,B\right)\in\mathfrak{sl}\left( n,\mathbb{H}\right)^{2}$.

math.OC↗
arXiv2020

Infinitesimally Tight Lagrangian Orbits

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples.

math.SG↗
arXiv2019

Invariant Generalized Complex Structures on Flag Manifolds

Let $G$ be a complex semi-simple Lie group and form its maximal flag manifold $\mathbb{F}=G/P=U/T$ where $P$ is a minimal parabolic subgroup, $U$ a compact real form and $T=U\cap P$ a maximal torus of $U$. The aim of this paper is to study invariant generalized complex structures on $\mathbb{F}$. We describe the invariant generalized almost complex structures on $\mathbb{F}$ and classify which one is integrable. The problem reduces to the study of invariant $4$-dimensional generalized almost complex structures restricted to each root space, and for integrability we analyse the Nijenhuis operator for a triple of roots such that its sum is zero. We also conducted a study about twisted generalized complex structures. We define a new bracket `twisted' by a closed $3$-form $Ω$ and also define the Nijenhuis operator twisted by $Ω$. We classify the $Ω$-integrable generalized complex structure.

math.DG↗
arXiv2019

De Rham $2$-cohomology of real flag manifolds

Let $\mathbb{F}_{Θ}=G/P_{Θ}$ be a flag manifold associated to a non-compact real simple Lie group $G$ and the parabolic subgroup $% P_{Θ}$. This is a closed subgroup of $G$ determined by a subset $% Θ$ of simple restricted roots of $\mathfrak{g}=Lie(G)$. This paper computes the second de Rham cohomology group of $\mathbb{F}_Θ$. We prove that it zero in general, with some rare exceptions. When it is non-zero, we give a basis of $H^2(\mathbb{F}_Θ,\mathbb{R})$ through the Weil construction of closed 2-forms as characteristic classes of principal fiber bundles. The starting point is the computation of the second homology group of $\mathbb{F}_{Θ}$ with coefficients in a ring $R$.

math.DG↗
arXiv2018

A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.

math.SG↗
arXiv2017

Invariant almost complex structures on real flag manifolds

In this work we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra $ C_{l}$ admit complex structures.

math.DG↗
arXiv2016

Control Systems on Flag Manifolds and their chain Control sets

In this paper we shown that the chain control sets of induced systems on flag manifolds coincides with their analogous one defined via semigroup actions. Consequently, any chain control set of the system contains a control set with nonempty interior and, if the number of the control sets with nonempty interior coincides with the number of the chain control sets, then the closure of any control set with nonempty interior is a chain control set.

math.DS↗
arXiv2015

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method started in Santos-San Martin \cite{ariasm} to get semigroup generators of semi-simple Lie groups. Consider a subset $Γ\subset G$ that contains a semi-simple subgroup $G_{1}$ of $G$. Then $Γ$ generates $G$ if $\mathrm{Ad}\left( Γ\right) $ generates a Zariski dense subgroup of the algebraic group $\mathrm{Ad}\left( G\right) $. The proof is reduced to check that some specific closed orbits of $G_{1}$ in the flag manifolds of $G$ are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups $G$ and subgroups $G_{1}\subset G$.

math.AT↗
arXiv2014

The Isotropy Representation of a Real Flag Manifold: Split Real Forms

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the decomposition into irreducible components is not in general unique, since there are cases with infinitely many invariant subspaces.

math.DG↗