Action Of Lie Group Research Articles

FINITENESS PROPERTIES OF FORMAL LIE GROUP ACTIONS

Following the ideas of Arnold and Seigal–Yakovenko, we prove that the space of matrix coefficients of a formal Lie group action belongs to a Noetherian ring. Using this result, we extend the uniform intersection multiplicity estimates of these authors from the abelian case to general Lie groups. We also demonstrate a simple new proof for a jet-determination result of Baouendi et al.

An algorithm for determining actions of semisimple Lie groups on free resolutions

An algorithm for determining actions of semisimple Lie groups on free resolutions

A reconstruction theorem for Connes–Landi deformations of commutative spectral triples

We formulate and prove an extension of Connes’s reconstruction theorem for commutative spectral triples to so-called Connes–Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group G , also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontryagin dual of G , and then show that such spectral triples are well-behaved under further Connes–Landi deformation, thereby allowing for both quantisation from and dequantisation to G -equivariant abstract commutative spectral triples. We then use a refinement of the Connes–Dubois-Violette splitting homomorphism to conclude that suitable Connes–Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.

On Sobolev problems associated with actions of lie groups

We consider Sobolev problems with nonlocal boundary conditions associated with an action of a compact Lie group. We find a natural conditions of ellipticity of such problems, obtain the corresponding finiteness theorem, and give the index formula.

Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds

I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.

The action of a compact Lie group on nilpotent Lie algebras of type {{n,2}}

Abstract We classify finite-dimensional real nilpotent Lie algebras with 2-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to SO 2 ⁢ ( ℝ ) ${{\mathrm{SO}}_{2}(\mathbb{R})}$ . This is the first step to extend the class of nilpotent Lie algebras 𝔥 ${{\mathfrak{h}}}$ of type { n , 2 } ${\{n,2\}}$ to solvable Lie algebras in which 𝔥 ${{\mathfrak{h}}}$ has codimension one.

On a Remarkable Class of Left-symmetric Algebras and its Relationship with the Class of Novikov Algebras

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by translations. In other words, we consider left-symmetric algebras satisfying the identity [x, y]·z = 0. We derive some basic characterizations of such left-symmetric algebras, and we highlight their relationships with the so-called Novikov algebras and derivation algebras.

Free resolutions and modules with a semisimple Lie group action

We introduce the package HighestWeights for Macaulay2. This package provides tools to study the representation-theoretic structure of free resolutions and graded modules over a polynomial ring with the action of a semisimple Lie group. The methods of this package allow users to consider the free modules in a resolution, or the graded components of a module, as representations of a semisimple Lie group by means of their weights, and to obtain their decomposition into highest-weight representations.

On some geometric characteristics of the orbit foliations of the co-adjoint action of some 5-dimensional solvable Lie groups

In this paper, we discribe some geometric charateristics of the so-called MD(5,3C)-foliations and MD(5,4)- foliations, i.e., the foliations formed by the generic orbits of co-adjoint action of MD(5,3C)-groups and MD(5,4)-groups.

Proper actions of high-dimensional groups on complex manifolds

Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently high, all proper effective actions can be explicitly determined, and our principal goal is to provide a comprehensive exposition of known classification results in the complex setting. They include a complete description of Kobayashi-hyperbolic manifolds with high-dimensional automorphism group, which is a case of special interest.

Differential invariants and operators of invariant differentiation of the projectable action of Lie groups

We describe the relation between operators of invariant differentiation and invariant operators on orbits of Lie group actions. We propose a new effective method for finding differential invariants and operators of invariant differentiation and present examples.

The homotopy fixed point set of Lie group actions on elliptic spaces

Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$-space. In this paper we analyze the homotopy type of the homotopy fixed point set $X^{hG}$, and the natural injection $k\colon X^G\hookrightarrow X^{hG}$. We show that if $X$ is elliptic, that is, it has finite dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi_*(k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

Minimal linear Morse functions on the orbits in Lie algebras

It is proved that all Morse height functions on regular orbits of the adjoint action of compact semisimple Lie groups are perfect. In the case of an arbitrary linear representation of a compact Lie group it is proved that all height functions satisfy the Bott property on the orbits of the representation. The case of the group SO4 is considered in more details.

Realization of an equivariant holomorphic Hermitian line bundle as a Quillen determinant bundle

Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian line bundle $({\mathcal L}\,,h)$. This proves a conjecture of Dey and Mathai in \cite{DM}.

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.

Construction of spline curves on smooth manifolds by action of Lie groups

AbstractPolynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.

Weyl group symmetry on the GKM graph of a GKM manifold with an extended Lie group action

We consider the class of manifolds with compact Lie group actions which restrict to GKM-actions on the maximal torus. First, we see their GKM-graphs admit symmetry of the Weyl groups. And then, we study its combinatorial abstraction; starting with abstract GKM-graphs with symmetry, we derive certain properties which reflect topology in a purely combinatorial way.

Representing Analytic Cohomology Groups of Complex Manifolds

Consider a holomorphic vector bundle L→X and an open cover 𝔘={U a :a∈A} of X, parametrized by a complex manifold A. We prove that the sheaf cohomology groups H q (X,L) can be computed from the complex C hol • (𝔘,L) of cochains (f a 0 ...a q ) a 0 ,...,a q ∈A that depend holomorphically on the a j , provided S={(a,x)∈A×X:x∈U a } is a Stein open subset of A×X. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on L.

Stratifications of Inertia Spaces of Compact Lie Group Actions

We study the topology of the inertia space of a smooth $G$-manifold $M$ where $G$ is a compact Lie group. We construct an explicit Whitney stratification of the inertia space, demonstrating that the inertia space is a triangulable differentiable stratified space. In addition, we demonstrate a de Rham theorem for differential forms defined on the inertia space with respect to this stratification.

Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two

AbstractIf X is a connected complex manifold with dX = 2 that admits a (connected) Lie group G acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification Ĝ of G on X except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of X.