A 2-cocycle on a group of symplectomorphisms
For a symplectic manifold $(M,\om)$ with exact symplectic form we construct a 2-cocycle on the group of symplectomorphisms and indicate cases when this cocycle is not trivial.
math.GR↗
arXiv subjects
Peter W. Michor
Publications and source records attributed to Peter W. Michor.
Cohomology for a group of diffeomorphisms of a manifold preserving an exact form
Let $M$ be a $G$-manifold and $\om$ a $G$-invariant exact $m$-form on $M$. We indicate when these data allow us to constract a cocycle on a group $G$ with values in the trivial $G$-module $\mathbb R$ and when this cocycle is nontrivial.
math.DG↗
The homotopy type of the space of degree 0 immersed plane curves
The space of all immersed closed curves of rotation degree 0 in the plane modulo reparametrizations has the same homotopy groups as the circle times the 2-sphere.
math.DG↗
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
The $L^2$-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type $M$ in a Riemannian manifold $(N,g)$ induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the $L^2$-metric.
math.DG↗
On polarizations in invariant theory
Given a reductive algebraic group $G$ and a finite dimensional algebraic $G$-module $V$, we study how close is the algebra of $G$-invariant polynomials on $V^{\oplus n}$ to the subalgebra generated by polarizations of $G$-invariant polynomials on $V$. We address this problem in a more general setting of $G$-actions on arbitrary affine varieties.
math.AG↗
Lifting smooth curves over invariants for representations of compact Lie groups, III
Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally bounded derivative. As a consequence any sufficiently often differentiable curve in the orbit space $V/G$ can be lifted twice differentiably. These results can be generalized to arbitrary polar representations. Finite reflection groups and finite rotation groups in dimensions two and three are discussed in detail.
math.RT↗
Homology and modular classes of Lie algebroids
For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.
math.DG↗
Lifting smooth curves over invariants for representations of compact Lie groups. II
Any sufficiently often differentiable curve in the orbit space of a compact Lie group representation can be lifted to a once differentiable curve into the representation space.
math.RT↗
Extensions of Lie algebras
We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out. In the new version references added: Most of the results are known. So this paper will not be submitted to a journal, it can be regarded as a review paper.
math.DG↗
Extensions of super Lie algebras
We study (non-abelian) extensions of a given super Lie algebra, identify a cohomological obstruction to the existence, parallel to the known one for Lie algebras. An analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out.
math.QA↗
Lifting mappings over invariants of finite groups
We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex representation of a finite group $G$ which admit a regular, holomorphic or formal lift to the representation space $V$. In particular, the case of complex reflection groups is investigated.
math.AG↗
Completing Lie algebra actions to Lie group actions
For a finite dimensional Lie algebra $\g$ of vector fields on a manifold $M$ we show that $M$ can be completed to a $G$-space in a unversal way, which however is neither Hausdorff nor $T_1$ in general. Here $G$ is a connected Lie group with Lie-algebra $\g$. For a transitive $\g$-action the completion is of the form $G/H$ for a Lie subgroup $H$ which need not be closed. In general the completion can be constructed by completing each $\g$-orbit.
math.DG↗
Choosing roots of polynomials smoothly, II
We show that the roots of any smooth curve of polynomials with only real roots can be parametrized twice differentiably (but not better).
math.CA↗
Reflection groups on Riemannian manifolds
We investigate discrete groups $G$ of isometries of a complete connected Riemannian manifold $M$ which are generated by reflections, in particular those generated by disecting reflections. We show that these are Coxeter groups, and that the the orbit space $M/G$ is isometric to a Weyl chamber $C$ which is a Riemannian manifold with corners and certain angle conditions along intersections of faces. We can also reconstruct the manifold and its action from the Riemannian chamber and its equipment of isotropy group data along the faces. We also discuss these results from the point of view of Riemannian orbifolds.
math.DG↗
Differentiable perturbation of unbounded operators
If $A(t)$ is a $C^{1,\al}$-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized $C^1$ in $t$. If $A$ is $C^\infty$ then the eigenvalues can be parameterized twice differentiable.
math.FA↗
The generalized Cayley map from an algebraic group to its Lie algebra
Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $Φ$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the representation space. The map $Φ$ identifies the field $Q(G)$ of rational functions on $G$ with an algebraic extension of the field $Q(\g)$ of rational functions on $\g$. For the spin representation of $\on{Spin}(V)$ the map $Φ$ essentially coincides with the classical Cayley transform. In general, properties of $Φ$ are established and these properties are applied to deal with a separation of variables (Richardson) problem for reductive algebraic groups: Find $\on{Harm}(G)$ so that for the coordinate ring $A(G)$ of $G$ we have $A(G) = A(G)^G\otimes \on{Harm}(G)$. As a consequence of a partial solution to this problem and a complete solution for SL(n) one has in general the equality $[Q(G):Q(\g)] = [Q(G)^G:Q(\g)^G]$ of the degrees of extension fields. Among other results, $Φ$ yields (for the complex case) a generalization, involving generic regular orbits, of the result of Richardson showing that the Cayley map, when $G$ is semisimple, defines an isomorphism from the variety of unipotent elements in $G$ to the variety of nilpotent elements in $\g$. In addition if $G$ is semisimple the Cayley map establishes a diffeomorphism between the real submanifold of hyperbolic elements in $G$ and the space of infinitesimal hyperbolic elements in $\g$. Some examples are computed in detail.
math.RT↗
Tensor fields and connections on holomorphic orbit spaces of finite groups
For a representation of a finite group $G$ on a complex vector space $V$ we determine when a holomorphic $\binom{p}{q}$-tensor field on the principle stratum of the orbit space $V/G$ can be lifted to a holomorphic $G$-invariant tensor field on $V$. This extends also to connections. As a consequence we determine those holomorphic diffeomorphisms on $V/G$ which can be lifted to orbit preserving holomorphic diffeomorphisms on $V$. This in turn is applied to characterize complex orbifolds.
math.DG↗
Invariant tensor fields and orbit varieties for finite algebraic transformation groups
Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $π_X:X\to X/G$. We characterize rational tensor fields $τ$ on $X/G$ such that the {\it pull back} of $τ$ is regular on $X$: these are precisely all $τ$ such that $\operatorname{div}_{R_{X/G}}(τ)\ge 0$ where $R_{X/G}$ is the {\it reflection divisor} of $X/G$ and $\operatorname{div}_{R_{X/G}}(τ)$ is the {\it $R_{X/G}$-divisor} of $τ$. We give some applications, in particular to the generalization of Solomon's theorem. In the last section we show that if $V$ is a finite dimensional vector space and $G$ a finite subgroup of $\operatorname{GL}(V)$, then each automorphism $ψ$ of $V/G$ admits a biregular lift $ϕ: V\to V$ provided that $ψ$ maps the regular stratum to itself and $ψ_*(R_{X/G})=R_{X/G}$.
math.AG↗