arXiv subjects
Publications and source records attributed to Jinpeng An.
We prove a generalized implicit function theorem for Banach spaces, without the usual assumption that the subspaces involved being complemented. Then we apply it to the problem of parametrization of fibers of differentiable maps, the Lie subgroup problem for Banach-Lie groups, as well as Weil's local rigidity for homomorphisms from finitely generated groups to Banach-Lie groups.
We prove that Ad-semisimple conjugacy classes in a connected Lie group $G$ are closed embedded submanifolds of $G$. We also prove that if $α:H\to G$ is a homomorphism of connected Lie groups such that the kernel of $α$ is discrete in $H$, then for an Ad-semisimple conjugacy class $C$ in $G$, every connected component of $α^{-1}(C)$ is a conjugacy class in $H$. Corresponding results for adjoint orbits in real Lie algebras are also proved.
For a cyclic group $A$ and a connected Lie group $G$ with an $A$-module structure (with the additional conditions that $G$ is compact and the $A$-module structure on $G$ is 1-semisimple if $A\cong\ZZ$), we define the twisted Weyl group $W=W(G,A,T)$, which acts on $T$ and $H^1(A,T)$, where $T$ is a maximal compact torus of $G_0^A$, the identity component of the group of invariants $G^A$. We then prove that the natural map $W\backslash H^1(A,T)\to H^1(A,G)$ is a bijection, reducing the calculation of $H^1(A,G)$ to the calculation of the action of $W$ on $T$. We also prove some properties of the twisted Weyl group $W$, one of which is that $W$ is a finite group. A new proof of a known result concerning the ranks of groups of invariants with respect to automorphisms of a compact Lie group is also given.
In this paper we generalize a result in [1], showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.
In this paper we prove some properties of the nonabelian cohomology $H^1(A,G)$ of a group $A$ with coefficients in a connected Lie group $G$. When $A$ is finite, we show that for every $A$-submodule $K$ of $G$ which is a maximal compact subgroup of $G$, the canonical map $H^1(A,K)\to H^1(A,G)$ is bijective. In this case we also show that $H^1(A,G)$ is always finite. When $A=\ZZ$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^\ZZ$ of the group of invariants $G^\ZZ$, $H^1(\ZZ,T)\to H^1(\ZZ,G)$ is surjective if and only if the $\ZZ$-action on $G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(\ZZ,T)\to H^1(\ZZ,G)$ are finite. When $A=\Zn$, we show that $H^1(\Zn,T)\to H^1(\Zn,G)$ is always surjective, where $T$ is a maximal compact torus of the identity component $G_0^{\Zn}$ of $G^{\Zn}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.
In this paper we present a criterion for the covering condition of the generalized random matrix ensemble, which enable us to verify the covering condition for the seven classes of generalized random matrix ensemble in an unified and simpler way.
Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group $G$, each connected component of the set $E_n(G)$ of all elements of order $n$ in $G$ is a conjugacy class in $G$. In particular, all conjugacy classes of finite order in $G$ are closed. Some properties of connected components of $E_n(G)$ are also given.
In this paper we give a realization of some symmetric space G/K as a closed submanifold P of G. We also give several equivalent representations of the submanifold P. Some properties of the set gK\cap P are also discussed, where gK is a coset space in G.
We give a generalization of the random matrix ensembles, including all lassical ensembles. Then we derive the joint density function of the generalized ensemble by one simple formula, which give a direct and unified way to compute the density functions for all classical ensembles and various kinds of new ensembles. An integration formula associated with the generalized ensemble is also given. We also give a classification scheme of the generalized ensembles, which will include all classical ensembles and some new ensembles which were not considered before.
According to the classification scheme of the generalized random matrix ensembles, we present various kinds of concrete examples of the generalized ensemble, and derive their joint density functions in an unified way by one simple formula which was proved in [2]. Particular cases of these examples include Gaussian ensemble, chiral ensemble, new transfer matrix ensembles, circular ensemble, Jacobi ensembles, and so on. The associated integration formulae are also given, which are just many classical integration formulae or their variation forms.