arXiv research

The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.

Investigates solving curvature equations on special Lie groups.

Develops neural networks for reductive Lie groups, enhancing symmetry respect.

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for $n \ge 6$, by using a method of Riemannian submersions. In the pres…

Reduces equations for contact mechanical systems on Lie groups by exploiting symmetries.

Study on special Lie groups with Lorentzian metrics.

Study 2D viscoelastic equations using Lie group theory.

Study on $p$-Kähler structures on fibrations and Lie groups.

Given a compact Lie group $G$ with Lie algebra $\mathfrak{g}$, we consider its tangent Lie group $TG\cong G\ltimes_{\mathrm{Ad}} \mathfrak{g}$. In this short note, we prove that $TG$ admits a left-invariant naturally reductive Riemannian metric $g$ and a metric connection with skew torsion $\nabla$ such that $(TG,g,\na…

Investigates connections in Lie group bundles, focusing on geometric reduction.

Formula for sectional curvatures on matrix groups.

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.

We consider trivializations of second iterated bundles of a Lie group that preserve lifted group structures. With such a trivialization, we elaborate Hamiltonian dynamics on cotangent, Lagrangian dynamics on tangent bundles and, both Hamiltonian and Lagrangian dynamics on Tulczyjew's symplectic space which is tangent o…

Geometrically revisits and models homogeneous spaces of compact Lie group $G_2$.

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…

Investigates properties of moment maps and stratifications on Lie groups.

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-inv…

Introduces $b$-Lie groups and studies their symplectic structures and reductions.

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological spaces that admit functions similar to the Kempf-Ness function. The point of this…

Extending our reduction construction in \cite{Hu} to the Hamiltonian action of a Poisson Lie group, we show that generalized Kähler reduction exists even when only one generalized complex structure in the pair is preserved by the group action. We show that the constructions in string theory of the (geometrical) $T$-dua…

Based on the representation theory and the study on the involutions of compact simple Lie groups, we show that $F_4$ admits non-naturally reductive Einstein metrics.

We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of red…

Introduces derived Lie n-groupoids with shifted symplectic structures.

Two reduction schemes for symplectic manifolds are shown equivalent.

The paper proves convexity results for a specific type of Lie groups.

We provide examples of naturally reductive pseudo-Riemannian spaces, in particular an example of a naturally reductive pseudo-Riemannian 2-step nilpotent Lie group $(N, < \,,\,>_N)$, such that $< \,,\,>_N$ is invariant under a left action and for which the center is degenerate. The metric does not correspond to a bi-in…

This paper deals with naturally reductive pseudo-Riemannian 2-step nilpotent Lie groups $(N, \la \,,\,\ra_N)$, such that $\la \,,\,\ra_N$ is invariant under a left action. The case of nondegenerate center is completely characterized. In fact, whenever $\la \,,\, \ra_N$ restricts to a metric in the center it is proved h…

Criterion for polystability in Lie group actions on manifolds.

There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three appr…

We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.

Using a characterization of parabolics in reductive Lie groups due to Furstenberg, elementary properties of buildings, and some algebraic topology, we give a new proof of Tits' classification of 2-transitive Lie groups.

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

The paper studies symmetry reduction and optimal control on Riemannian manifolds.

Reduced sh-Lie structures have been studied for the case when a Lie group acts on the fibers of a vector bundle while preserving the base space of the bundle. In this paper we investigate how one obtains a reduced sh-Lie structure using the ideas of symmetry reduction where the action of the Lie group is transversal to…

In this paper we determine the at least $4$-dimensional affine reductive homogeneous manifolds for an at most $9$-dimensional simple Lie group or an at most $6$-dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable …

In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem…

Eldredge, Gordina and Saloff-Coste recently conjectured that, for a given compact connected Lie group $G$, there is a positive real number $C$ such that $λ_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all left-invariant metrics $g$ on $G$. In this short note, we establish the conjecture for the small subclass of natural…

We study the problem of prescribing the Ricci curvature in the class of naturally reductive metrics on a compact Lie group. We derive necessary as well as sufficient conditions for the solvability of the equations and provide a series of examples.

In this paper we develope a theory of reduction for classical systems with Poisson Lie groups symmetries using the notion of momentum map introduced by Lu. The local description of Poisson manifolds and Poisson Lie groups and the properties of Lu's momentum map allow us to define a Poisson reduced space.

This work concerns the non-flat metrics on the Heisenberg Lie group of dimension three $\Heis_3(\RR)$ and the bi-invariant metrics on the solvable Lie groups of dimension four. On $\Heis_3(\RR)$ we prove that the property of the metric being naturally reductive is equivalent to the property of the center being non-dege…

Paper shows spectra can't distinguish naturally reductive manifolds.

We prove a reduction theorem for the tangent bundle of a Poisson manifold $(M, π)$ endowed with a pre-Hamiltonian action of a Poisson Lie group $(G, π_G)$. In the special case of a Hamiltonian action of a Lie group, we are able to compare our reduction to the classical Marsden-Ratiu reduction of $M$. If the manifold $M…

The paper develops stability criteria for real reductive Lie groups acting on manifolds.

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…

Symplectic reduction by abelian subgroups coincides under specific conditions.

Study G2-instantons on specific Lie groups, finding conditions and structures.

Paper proves naturally reductive property is inaudible for certain manifolds.