We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test function groups, diffeomorphism groups, and weak direct products of Lie groups. The specific tools of differential calculus required for the Lie group constructions are developed. Notably, we establish differentiability properties of composition and evaluation, as well as exponential laws for function spaces. We also present techniques to deal with the subtle differentiability and continuity properties of non-linear mappings between spaces of test functions. Most of the results are independent of any specific properties of the topological vector spaces involved; in particular, we can deal with real and complex Lie groups modeled on non-locally convex spaces.
We completely characterize connected Lie groups all of whose countable subgroups are weakly amenable. We also provide a characterization of connected semisimple Lie groups that are weakly amenable. Finally, we show that a connected Lie group is weakly amenable if the group is weakly amenable as a discrete group.
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.
In this paper we extend and adapt several results on extensions of Lie algebras to topological Lie algebras over topological fields of characteristic zero. In particular we describe the set of equivalence classes of extensions of the Lie algebra by the Lie algebra a disjoint union of affine spaces with translation group H^2(\g,\z(\n))_{[S]}, where [S] denotes the equivalence class of the continuous outer action S \: \g \to \der \n. We also discuss topological crossed modules and explain how they are related to extensions of Lie algebras by showing that any continuous outer action gives rise to…
The optimal constants are found for Lebesgue norm multilinear inequalities of Holder-Brascamp-Lieb type for arbitrary discrete Abelian groups. Previously a criterion for finiteness of the constants had been established for finitely generated Abelian groups, and the optimal constant had been found in the torsion-free case. The main step here is the analysis of finite groups.
The discrete cocompact subgroups of the five-dimensional connected, simply connected nilpotent Lie groups are determined up to isomorphism. Moreover, we prove if $G=N\times A$ is a connected, simply connected, nilpotent Lie group with an Abelian factor $A$, then every uniform subgroup of $G$ is the direct product of a uniform subgroup of $N$ and ${\mathbb Z}^r$ where $r=\dim A$.
We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the discrete symmetries of the discrete Painlevé I equation and of the Toda lattice equation.
In this paper we consider the class of connected simple Lie groups equipped with the discrete topology. We show that within this class of groups the following approximation properties are equivalent: (1) the Haagerup property; (2) weak amenability; (3) the weak Haagerup property. In order to obtain the above result we prove that the discrete group GL(2,K) is weakly amenable with constant 1 for any field K.
We clarify the structure of nilpotent Lie groups which are multiplication groups of $3$-dimensional simply connected topological loops and prove that non-solvable Lie groups acting minimally on $3$-dimensional manifolds cannot be the multiplication group of $3$-dimensional topological loops. Among the nilpotent Lie groups for any filiform groups ${\mathcal F}_{n+2}$ and ${\mathcal F}_{m+2}$ with $n, m > 1$, the direct product ${\mathcal F}_{n+2} \times \mathbb R$ and the direct product ${\mathcal F}_{n+2} \times _Z {\mathcal F}_{m+2}$ with amalgamated center $Z$ occur as the multiplication gro…
We prove that if the multiplication group $Mult(L)$ of a connected $2$-dimensional topological loop is a Lie group, then $Mult(L)$ is an elementary filiform nilpotent Lie group of dimension at least $4$. Moreover, we describe loops having elementary filiform Lie groups $\mathbb F$ as the group topologically generated by their left translations and obtain a complete classification for these loops $L$ if $\hbox{dim} \ \mathbb F=3$. In this case necessary and sufficient conditions for $L$ are given that $Mult(L)$ is an elementary filiform Lie group for a given allowed dimension.
We classify all three-dimensional connected topological loops such that the group topologically generated by their left translations is the four-dimensional connected Lie group $G$ which has trivial center and precisely two one-dimensional normal subgroups. We show that $G$ is not the multiplication group of connected topological proper loops.
In this paper, we determine the topological entropy $h(ϕ)$ of a continuous endomorphism $ϕ$ of a Lie group $G$. This computation is a classical topic in ergodic theory which seemed to have long been solved. But, when $G$ is noncompact, the well known Bowen's formula for the entropy $h_{d}(ϕ)$ associated to a left invariant distance $d$ just provides an upper bound to $h(ϕ)$, which is characterized by the so called variational principle. We prove that \[ h\left(ϕ\right) = h\left(ϕ|_{T(G_ϕ)}\right) \] where $G_ϕ$ is the maximal connected subgroup of $G$ such that $ϕ(G_ϕ) = G_ϕ$, and $T(G_ϕ)$ is …
We prove that each $3$-dimensional connected topological loop $L$ having a solvable Lie group of dimension $\le 5$ as the multiplication group of $L$ is centrally nilpotent of class $2$. Moreover, we classify the solvable non-nilpotent Lie groups $G$ which are multiplication groups for $3$-dimensional simply connected topological loops $L$ and $\hbox{dim} \ G \le 5$. These groups are direct products of proper connected Lie groups and have dimension $5$. We find also the inner mapping groups of $L$.
Recall that a topological space is said to be a $k_ω$-space if it is the direct limit of an ascending sequence of compact Hausdorff topological spaces. If each point in a Hausdorff space $X$ has an open neighbourhood which is a $k_ω$-space, then $X$ is called locally $k_ω$. We show that a topological group is complete whenever the underlying topological space is locally $k_ω$. As a consequence, every infinite-dimensional Lie group modelled on a Silva space is complete.
Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.
We prove that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the group of its left translations is classical. We give a particular simple general construction for proper loops such that the compact group of their left translations is direct product of at least 3 factors.
For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …
We provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The "compact-like" properties we consider include (local) compactness, (local) omega-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is a sample of our characterizations: (i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups. (ii) An abelian topological group G is a Lie group if and only if G is l…
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie point symmetries and generalized symmetries are considered and applied to the discrete heat equation and to the integrable discrete time Toda lattice.
