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Let F Θ = G ∕ P Θ be a generalized flag manifold, where G is a real non-compact semi-simple Lie group and P Θ a parabolic subgroup. A classical result says the Schubert cells, which are the closure of the Bruhat cells, endow F Θ with a cellular CW structure. In this paper we exhibit explicit parametrizations of the Schubert cells by closed balls (cubes) in R n and use them to compute the boundary operator ∂ for the cellular homology. We recover the result obtained by Kocherlakota [1995], in the setting of Morse Homology, that the coefficients of ∂ are 0 or ± 2 (so that Z 2 -homology is freely generated by the cells). In particular, the formula given here is more refined in the sense that the ambiguity of signals in the Morse–Witten complex is solved.

The N-expansive systems have been recently studied in the literature [6] , [7] , [9] , [14] . Here we characterize them as those homeomorphisms for which every Borel probability measure is N-expansive. In particular, the strongly measure expansive homeomorphisms in the sense of [8] are precisely the homeomorphisms for which every invariant measure is 1-expansive. We also characterize the 1-expansive measures for equicontinuous homeomorphisms as the convex sum of finitely many Dirac measures supported on isolated points. In particular, such measures do not exist on metric spaces without isolated points. Furthermore, we consider N-expansive measure for flows and prove that a flow is N-expansive in the sense of [9] if and only if every Borel probability measure is N-expansive. Finally, we obtain a lower bound of the topological entropy of the N-expansive flows as the exponential growth rate of the number of periodic orbits.

Let G be a noncompact semi-simple Lie group with Iwasawa decomposition G = K ⁢ A ⁢ N {G=KAN} . For a semigroup S ⊂ G {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform I S ⁢ ( λ , u ) = ∫ S e λ ⁢ ( 𝖺 ⁢ ( g , u ) ) ⁢ 𝑑 g {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, u ∈ K {u\in K} , λ ∈ 𝔞 ∗ {\lambda\in\mathfrak{a}^{\ast}} , 𝔞 {\mathfrak{a}} is the Lie algebra of A and g ⁢ u = k ⁢ e 𝖺 ⁢ ( g , u ) ⁢ n ∈ K ⁢ A ⁢ N {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written 𝔽 Θ ⁢ ( S ) {\mathbb{F}_{\Theta(S)}} called the flag type of S, where Θ ⁢ ( S ) {\Theta(S)} is a subset of the simple system of roots. It is proved that I S ⁢ ( λ , u ) < ∞ {I_{S}(\lambda,u) if λ belongs to a convex cone defined from Θ ⁢ ( S ) {\Theta(S)} and u ∈ π - 1 ⁢ ( 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ) {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where 𝒟 Θ ⁢ ( S ) ⁢ ( S ) ⊂ 𝔽 Θ ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and π : K → 𝔽 Θ ⁢ ( S ) {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of I S ⁢ ( λ , u ) {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in 𝒟 Θ ⁢ ( S ) ⁢ ( S ) {\mathcal{D}_{\Theta(S)}(S)} invariant by the group S ∩ S - 1 {S\cap S^{-1}} of units of S.

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds., 23 pages

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. Fil: Freitas, Ana P. C.. Universidade Estadual de Campinas; Brasil Fil: del Barco, Viviana Jorgelina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidade Estadual de Campinas; Brasil Fil: San Martin, Luiz A. B.. Universidade Estadual de Campinas; Brasil

We count the geodesics of a given length connecting two points of a compact connected Lie group with a biinvariant metric. We reduce the question to the maximal torus by using the lattice, the diagram and the Weyl group to count the geodesics that occur outside the maximal torus. We apply our results to give short proofs of known results on conjugate and cut points of compact semisimple Lie groups.

To track the gradual change of the adjacency matrix of a simple graph $\mathcal{G}$ into the signless Laplacian matrix, V. Nikiforov in \cite{NKF} suggested the study of the convex linear combination $A_{\alpha }$ (\textit{$\alpha$-adjacency matrix}), \[A_{\alpha }\left( \mathcal{G}\right)=\alpha D\left( \mathcal{G}\right) +\left( 1-\alpha \right) A\left( \mathcal{G}\right),\] for $\alpha \in \left[ 0,1\right]$, where $A\left( \mathcal{G}\right)$ and $D\left( \mathcal{G}\right)$ are the adjacency and the diagonal vertex degrees matrices of $\mathcal{G}$, respectively. Taking this definition as an idea the next matrix was considered for $a,b \in \mathbb{R}$. The matrix $A_{a,b}$ defined by $$ A_{a,b}\left( \mathcal{G}\right) =a D\left( \mathcal{G}\right) + b A\left(\mathcal{G}\right),$$ extends the previous $\alpha$-adjacency matrix. This matrix is designated the \textit{$(a,b)$-adjacency matrix of $\mathcal{G}$}. Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the $(a,b)$-adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given.

Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.

A right-invariant control system \begin{document}$Σ$\end{document} on a connected Lie group \begin{document}$G$\end{document} induce affine control systems \begin{document}$Σ_{Θ}$\end{document} on every flag manifold \begin{document}$\mathbb{F}_{Θ}=G/P_{Θ}$\end{document} . In this paper we show that the chain control sets of the induced systems 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. Some relevant examples are included.

The energy of a symmetric matrix is the sum of the absolute values of its eigenvalues. We introduce a lower bound for the energy of a symmetric partitioned matrix into blocks. This bound is related to the spectrum of its quotient matrix. Furthermore, we study necessary conditions for the equality. Applications to the energy of the generalized composition of a family of arbitrary graphs are obtained. A lower bound for the energy of a graph with a bridge is given. Some computational experiments are presented in order to show that, in some cases, the obtained lower bound is incomparable with the well known lower bound $2\sqrt{m}$, where $m$ is the number of edges of the graph.

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., 22 pages. Final version to appear in Mathematische Zeitschrift

We give various realizations of the adjoint orbits of a semi-simple Lie group and describe their symplectic geometry. We then use these realizations to identify a family of Lagrangian submanifolds of the orbits.

Let G be a complex semi-simple Lie group and form its maximal flag manifold F = G ∕ P = U ∕ T where P is a minimal parabolic subgroup, U a compact real form and T = U ∩ P a maximal torus of U . The aim of this paper is to study invariant generalized complex structures on F . We describe the invariant generalized almost complex structures on 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 analyze 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.

We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We describe the topology of the regular and singular fibres, in particular we calculate their middle Betti numbers.

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset $$\Gamma \subset G$$ that contains a semi-simple subgroup $$G_{1}$$ of G. If one can show that $$ \Gamma $$ does not leave invariant a contractible subset on any flag manifold of G, then $$\Gamma $$ generates G if $$\mathrm {Ad}\left( \Gamma \right) $$ 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 $$G_{1}\subset G$$ .

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., Comment: 14 pages, accepted version