The paper studies automorphisms of 2-step nilpotent Lie groups, showing continuity up to center and field automorphisms.
problem Investigating the continuity of abstract automorphisms in 2-step nilpotent Lie groups.
method Analyzes various types of 2-step nilpotent Lie groups, using tools from Riemannian geometry.
result Abstract automorphisms are continuous 'up to discontinuity due to the center and field automorphisms of C' for many 2-step nilpotent Lie groups. Classifies metrics on specific Lie groups.
problem Classifying Riemannian metrics on nonunimodular Lie groups.
method Automorphism classification of inner products on Lie algebras.
result Classification of metrics on 4D nonunimodular Lie groups.
Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.
problem Characterizing the basic automorphism groups of Cartan foliations.
method Analyzes sufficient conditions and estimates dimensions for basic automorphism groups of Cartan foliations covered by fibrations.
result Identifies sufficient conditions for the existence of a finite-dimensional Lie group structure in basic automorphism groups.
The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.
problem Formulating Hamiltonian actions of Lie 2-groups on orbifolds.
method Constructing automorphisms of Lie groupoids, using 2-group actions and Kan fibrations.
result Symplectic reductions of Lie 2-group actions on orbifolds are Lie 2-groupoids under certain conditions.
Compact Lie groups can be realized as automorphism groups of Riemannian manifolds.
problem Realizing compact Lie groups as automorphism groups of Riemannian manifolds.
method Analyzing invariant metrics and their automorphism groups.
result The space of G-invariant metrics whose automorphism groups preserve G-orbits is dense Gδ in the space of all G-invariant metrics. We consider a family of 2-step nilpotent Lie algebras associated to uniform complete graphs on odd number of vertices. We prove that the symmetry group of such a graph is the holomorph of the additive cyclic group Zn. Moreover, we prove that the (Lie) automorphism group of the corresponding nilpotent Lie algebra co…
The paper classifies left-invariant pseudo-Riemannian metrics on specific Lie groups.
problem Classifying left-invariant pseudo-Riemannian metrics on Lie groups.
method Analyzing left-invariant metrics on specific Lie groups with n≥4. result A complete classification of left-invariant pseudo-Riemannian metrics for Lie groups of dimension n≥4. In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invarian…
This paper proves all endomorphisms of framed little disk operad are automorphisms.
problem Proving all endomorphisms of framed little disk operad are automorphisms.
method Using the classification of self maps of simple Lie groups.
result All endomorphisms of the framed little disk operad are automorphisms.
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…
In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism g…
New method computes automorphisms of surface groups using skein algebras.
problem Computing automorphisms of surface groups.
method Using skein algebras and Goldman Lie algebra.
result Refined formula for automorphisms of homology cylinders.
Complete classification of para-Kähler structures on Lie groups.
problem Classifying para-Kähler structures on Lie groups.
method Complete classification via automorphism consideration.
result Classification of para-Kähler structures on four-dimensional Lie groups.
We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse Cartan geometry may not be effective. Some estimates of the dimension o…
In this article we define the twisted product of groups as the generalization of the semidirect product of groups. We will find the necessary and sufficient condition in order that the twisted product of groups to be a group. In particular, for two copies of the same group, the twisted product of group by itself throug…
Classifies complex structures on SL(2,C), finding new non-regular ones.
problem Classifying complex structures on SL(2,C) up to automorphisms.
method Classification via Lie group automorphisms and topological analysis.
result Found one new, non-regular complex structure.
We prove for the automorphism group of an arbitrary parabolic geometry that the C0 and C∞ topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
Given any real-analytic CR manifold M, we provide general conditions on M guaranteeing that the group of all its global real-analytic CR automorphisms is a Lie group (in an appropriate topology). Our conditions are in particular satisfied when M is an arbitrary compact real-analytic hypersurface embedded in some Stein …
Vanishing of equivariant cohomology groups for proper Lie group actions.
problem Vanishing of equivariant differentiable cohomology groups for proper Lie group actions.
method Establishing vanishing of equivariant differentiable cohomology groups with coefficients in C∞-functions. result The canonical class in the first differential cohomology of G with coefficients in C∞-functions on M vanishes if and only if G acts properly on M. We show that the isotropy action of a homogeneous space G/K, where G and K are compact, connected Lie groups and K is defined by an automorphism on G, is equivariantly formal and that (G,K) is a Cartan pair.
Schenkel proved that the automorphism group of a flat Minkowski plane is a Lie group of dimension at most 6 and described planes whose automorphism group has dimension at least 4 or one of whose kernels has dimension 3. We extend these results to the case of toroidal circle planes.
In many Lagrangian field theories, there is a Poisson bracket on the space of local functionals. One may identify the fields of such theories as sections of a vector bundle. It is known that the Poisson bracket induces an sh-Lie structure on the graded space of horizontal forms on the jet bundle of the relevant vector …
Completed realizations of automorphisms and subgroups in exceptional Lie group E8.
problem Classifying automorphisms and subgroups of the exceptional Lie group E8. method Detailed analysis of automorphisms and subgroups induced by specific transformations.
result Explicit forms and structures of automorphisms and subgroups in E8. Study projective representations of infinite-dimensional Hilbert-Lie groups.
problem Characterize and classify representations of Hilbert-Lie groups.
method Use covariance with respect to one-parameter groups of automorphisms and implement perturbation theory.
result Explicit determination of central extensions for projective representations.
Study formalities on closed surfaces using connections.
problem Formalities of Goldman-Turaev Lie bialgebra on closed surfaces.
method Reformulated Kashiwara-Vergne groups and associators in higher genera using non-commutative connections.
result Determined pro-unipotent automorphism group of associated graded.
In this paper we treat the intersection of fixed point subgroups by the involutive automorphisms of exceptional Lie group G=F4,E6,E7. We shall find involutive automorphisms of G such that the connected component of the intersection of those fixed point subgroups coincides with the maximal torus of G.
Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a C∞ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…
Study automorphism group actions on Jacobi diagrams spaces.
problem Understanding automorphism group actions on Jacobi diagrams.
method Using actions of GL(n,Z) and IA-automorphism group Lie algebra, extend to Andreadakis filtration.
result Obtained indecomposable decomposition and radical filtration of Jacobi diagrams spaces.
For a real, non-singular, 2-step nilpotent Lie algebra n, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n}),where\Aut_0(\mathfrak{n})$ is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some …
We endow the group of automorphisms of an exact Courant algebroid over a compact manifold with an infinite dimensional Lie group structure modelled on the inverse limit of Hilbert spaces (ILH). We prove a slice theorem for the action of this Lie group on the space of generalized metrics. As an application, we show that…
Develops Lie algebraic approach for compact complex homogeneous manifolds.
problem Proves important results on compact complex homogeneous manifolds.
method Uses standard results in Lie theory to associate a canonical abelian Lie algebra with a given integrable complex structure.
result Provides a new method of associating a canonical abelian Lie algebra with a given integrable complex structure.
The paper classifies path structures on 3D Lie groups and reduces non-flat ones to Z/2Z-structures.
problem Classifying and reducing path structures on 3D Lie groups.
method Analyzes curvature and automorphism groups to reduce path structures to simpler forms.
result Automorphism groups of non-flat path structures are maximal dimension 3.
The paper classifies isometries on specific Lie groups.
problem Classifying isometries on nonnilpotent, solvable 3D Lie groups.
method Proving automorphisms are the only isometries for rank two Almost-Riemannian Structures.
result A classification result for rank two ARSs on nonnilpotent, solvable 3D Lie groups.
We study the geodesics problem in Heisenberg group H (case SR and riemannian). The sheaf of infinitesimal automorphisms of the (2n,2n+1) distribution D over H is an infinite, transitive Lie algebra sheaf.
Study on special Lie groups with Lorentzian metrics.
problem Characterize structure of 2-step nilpotent Lorentzian naturally reductive Lie groups. method Develop framework for naturally reductive Lie groups, extend to Lorentzian context, analyze degenerate and non-degenerate cases.
result Complete structural description of naturally reductive 2-step Lorentzian nilpotent Lie groups. Lie groups of automorphisms of cotangent bundles of Lie groups are completely characterized and interesting results are obtained. We give prominence to the fact that the Lie groups of automorphisms of cotangent bundles of Lie groups are super symmetric Lie groups. In the cases of orthogonal Lie lgebras, semi-simple Lie…
We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterisation of low dimensional cohomology spaces i…
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-gr…
In this paper, for a Lie 2-algebra $\g$, we construct the automorphism 2-group $\Aut(\g)$, which turns out to be an integration of the derivation Lie 2-algebra $\Der(\g)$.
We prove that the automorphism group of a compact 6-manifold M endowed with a symplectic half-flat SU(3)-structure has abelian Lie algebra with dimension bounded by min{5,b1(M)}. Moreover, we study the properties of the automorphism group action and we discuss relevant examples. In particular, we provide new com…
We characterize co-Hopfian finitely generated torsion free nilpotent groups in terms of their Lie algebra automorphisms, and construct many examples of such groups.
The paper explores automorphism groups of parabolic structures on aspherical manifolds.
problem Characterizing the automorphism groups of parabolic structures on aspherical manifolds.
method Analyzing properties of closed aspherical parabolic ${\sfG}$-manifolds and their automorphism groups.
result Certain parabolic ${\sfG}$-structures impose strong restrictions on the topology of compact aspherical manifolds.
We describe explicitly Lie superalgebra isomorphisms between the Lie superalgebras of first-order superdifferential operators on supermanifolds, showing in particular that any such isomorphism induces a diffeomorphism of the supermanifolds. We also prove that the group of automorphisms of such a Lie superalgebra is a s…
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This implies that all algebraic groups (not necessarily affine) over fields of characteristic zero and some transformation groups of complex spaces and Riemannian manifods are Jordan.
Given a group automorphism φ:Γ→Γ, one has an action of Γ on itself by φ-twisted conjugacy, namely, g.x=gxφ(g−1). The orbits of this action are called φ-conjugacy classes. One says that Γ has the R∞-property if there are infinitely many φ-conjugacy classes for every automorphism φ of Γ. I…
For bounded pseudoconvex domains with finite type we give a precise description of the automorphism group: if an orbit of the automorphism group accumulates on at least two different points of the boundary, then the automorphism group has finitely many components and is the almost direct product of a compact group and …
The paper classifies metrics on specific Lie groups and finds unique properties of these metrics.
problem Classifying left-invariant Lorentzian metrics on specific Lie groups.
method Analyzing the three-dimensional Heisenberg group and its direct product with Euclidean space.
result There are exactly six left-invariant Lorentzian metrics on the Lie group, one of which is flat and the others are Ricci solitons but not Einstein.
We contribute to the classification of toroidal circle planes and flat Minkowski planes possessing three-dimensional connected groups of automorphisms. When such a group is an almost simple Lie group, we show that it is isomorphic to PSL(2,R). Using this result, we describe a framework for the full cl…