Algebra Of Vector Fields Research Articles

ANDERSÉN–LEMPERT-THEORY WITH PARAMETERS: A REPRESENTATION THEORETIC POINT OF VIEW

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂnon the vector space of algebraic vector fields on ℂnand more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂkinto ℂnare announced.

Complete algebraic vector fields on affine surfaces–Part II

In this work we finish off the classification of meromorphic semi-complete vector fields announced in Rebelo [J. Geom. Anal 13(4) (2003) 669–696]. As an application of our results, we give a simple and more geometric proof of the classification of complete polynomial vector fields on C 2 recently obtained through the works of Brunella and McQuillan.

On extensions of superconformal algebras

Starting from vector fields that preserve a differential form on a Riemann sphere with Grassmann variables, one can construct a superconformal algebra by considering central extensions of the algebra of vector fields. In this paper, the N=4 case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of noncentral extensions. How this modifies the existing field theory, representation theory, and Gelfand–Fuchs constructions is discussed. It is also discussed how graded Riemann sphere geometry can be used to give a geometrical description of the central charge in the N=1 theory.

Magnetic hydrodynamics with asymmetric stress tensor

In this paper we study equations of magnetic hydrodynamics with a stress tensor. We interpret this system as the generalized Euler equation associated with an Abelian extension of the Lie algebra of vector fields with a nontrivial 2-cocycle. We use the Lie algebra approach to prove the energy conservation law and the conservation of cross-helicity.

Affine representations of Lie algebras and geometric interpretation in the case of smooth manifolds

In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations and 1-cohomology classes of the algebra. We classify the affine representations of the Lie algebra of vector fields of a smooth manifold associated to its action on symmetric tensor fields of type (1,2). Among them, we recover the space of symmetric affine linear connections and that of projective structures of the manifold. We compute some of the associated cohomologies.

Equivariant Operators Between Some Modules of the Lie Algebra of Vector Fields#

The space of differential operators of order ≤ k, from the differential forms of degree p of a smooth manifold M into the functions of M, is a module over the Lie algebra of vector fields of M, when it is equipped with the natural Lie derivative. In this paper, we compute all equivariant i.e., intertwining operators and conclude that the preceding modules of differential operators are never isomorphic. We also answer a question of Lecomte, who observed that the restriction of some homotopy operator – introduced in Lecomte [Lecomte, P. (1994). On some sequence of graded Lie algebras associated to manifolds. Ann. Glob. Ana. Geo. 12:183–192] – to is equivariant for small values of k and p.

Lie Superalgebra Structures in H? (g;g)

Let g=vect(M) be the Lie (super)algebra of vector fields on any connected (super)manifold M; let “-” be the change of parity functor, C i and H i the space of i-chains and i-cohomology. The Nijenhuis bracket makes into a Lie superalgebra that can be interpreted as the centralizer of the exterior differential considered as a vector field on the supermanifold associated with the de Rham bundle on M. A similar bracket introduces structures of DG Lie superalgebra in L * and for any Lie superalgebra g. We use a Mathematica-based package SuperLie (already proven useful in various problems) to explicitly describe the algebras l * for some simple finite dimensional Lie superalgebras g and their “relatives” - the nontrivial central extensions or derivation algebras of the considered simple ones.

Relative Poincaré lemma, contractibility, quasi-homogeneity and vector fields tangent to a singular variety

We study the interplay between the properties of the germ of a singular variety $N\subset \mathbb R^n$ given in the title and the algebra of vector fields tangent to $N$. The Poincare lemma property means that any closed differential $(p+1)$-form vanishing at any point of $N$ is a differential of a $p$-form which also vanishes at any point of $N$. In particular, we show that the classical quasi-homogeneity is not a necessary condition for the Poincare lemma property; it can be replaced by quasi-homogeneity with respect to a smooth submanifold of $\mathbb R^n$ or a chain of smooth submanifolds. We prove that $N$ is quasi-homogeneous if and only if there exists a vector field $V, V(0)=0,$ which is tangent to $N$ and has positive eigenvalues. We also generalize this theorem to quasi-homogeneity with respect to a smooth submanifold of $\mathbb R^n$.

A classification of locally homogeneous connections on 2-dimensional manifolds via group-theoretical approach

The aim of this paper is to classify (locally) all torsion-less locally homogeneous affine connections on two-dimensional manifolds from a group-theoretical point of view. For this purpose, we are using the classification of all non-equivalent transitive Lie algebras of vector fields in ℝ2 according to P.J. Olver [7].

On the geometry of Riemannian manifolds with a Lie structure at infinity

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

Virasoro algebra with central charge c=1 on the horizon of a two-dimensional-Rindler space–time

Using the holographic machinery built up in a previous work, we show that the hidden SL(2,R) symmetry of a scalar quantum field propagating in a Rindler space–time admits an enlargement in terms of a unitary positive-energy representation of Virasoro algebra defined in the Fock representation. That representation has central charge c=1. The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated quantum field theory on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy h≔μ2/2 vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for h=1/2, the ground state of that generator seems to correspond to a thermal state when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. Finally, under Wick rotation in Rindler time, the pair of quantum field theories which are built up on the future and past horizon defines a proper two-dimensional conformal quantum field theory on a cylinder.

On the cohomology of the spaces of differential operators acting on skewsymmetric tensor fields or on forms, as modules of the Lie algebra of vector fields

The spaces of differential operators acting on skewsymmetric contravariant tensor fields or on smooth forms of a smooth manifold are representations of its Lie algebra of vector fields. We compute the first cohomology spaces of these representations and show how they are related to the cohomology with coefficients in the space of smooth functions of the manifold.

Complete algebraic vector fields on affine surfaces, Part I

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].

The algebra of K-invariant vector fields on a symmetric space G/K

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra $\X(G/K)^K$ of $K$-invariant regular algebraic vector fields using the geometry of $G/K$ and the $K$-spherical representations of $G$. Assume $G$ is semisimple and simply-connected and let $\J$ be the algebra of $K$ biinvariant functions on $G$. An explicit set of free generators for the localization $ \X(G/K)^K_{\psi}$ is constructed for a suitable $\psi \in \J$. A commutator formula is obtained for $K$-invariant vector fields in terms of the corresponding $K$-covariant maps from $G$ to the isotropy representation of $G/K$. Vector fields on $G/K$ whose horizontal lifts to $G$ are tangent to the Cartan embedding of $G/K$ into $G$ are called \emph{flat}. When $G$ is simple and simply connected, it is shown that every element of $\X(G/K)^K$ is flat if and only if $K$ is semisimple. The gradients of the fundamental characters of $G$ are shown to generate all conjugation-invariant vector fields on $G$. These results are applied in the case of the adjoint representation of $G = \SL(2,\C)$ to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of $\C[G]$.

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

Local cocycles and central extensions for multipoint algebras of Krichever-Novikov type

Multi-point algebras of Krichever Novikov type for higher genus Riemann surfaces are generalisations of the Virasoro algebra and its related algebras. Complete existence and uniqueness results for local 2-cocycles defining almost-graded central ex- tensions of the functions algebra, the vector field algebra, and the differential operator algebra (of degree ≤ 1) are shown. This is applied to the higher genus, multi-point affine algebras to obtain uniqueness for almost-graded central extensions of the current algebra of a simple finite-dimensional Lie algebra. An earlier conjecture of the author concerning the central extension of the differential operator algebra induced by the semi-infinite wedge representations is proved.

On the Leibniz cohomology of vector fields

Gelfand and Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.

Geodesics on extensions of the Lie algebra of vector fields on the circle

We consider Lie algebra 2-cocycles on the Lie algebra of vector fields on the circle with values in the space of tensor densities on the circle, whose integral over the circle is the Virasoro cocycle. Geodesic equations on the abelian extensions defined b

Weighted Decay Estimates for the Wave Equation

In this work we study weighted Sobolev spaces in Rn generated by the Lie algebra of vector fields (1+|x|2)1/2∂xj,j=1, …, n. Interpolation properties and Sobolev embeddings are obtained on the basis of a suitable localization in Rn. As an application we derive weighted Lq estimates for the solution of the homogeneous wave equation. For the inhomogeneous wave equation we generalize the weighted Strichartz estimate established by V. Georgiev (1997, Amer. J. Math.119, 1291–1319) and establish global existence results for the supercritical semilinear wave equation with non-compact small initial data in these weighted Sobolev spaces.

Quantum spin dynamics (QSD): VII.Symplectic structures andcontinuum lattice formulations of gauge field theories

Interesting nonlinear functions on the phase spaces ofclassical field theories can never be quantizedimmediately because the basic fields of the theory becomeoperator-valued distributions. Therefore, one is usuallyforced to find a smeared substitute for such a functionwhich corresponds to a regularization. The smearedfunctions define a new symplectic manifold of their ownwhich is easy to quantize. Finally, one must remove theregulator and establish that the final operator, if itexists, has the correct classical limit.In this paper we begin the investigation of these steps fordiffeomorphism-invariant quantum field theories ofconnections. We introduce a (generalized) projectivefamily of symplectic manifolds, coordinatized by thesmeared fields, which is labelled by a pair consisting ofa graph and another graph dual to it. We show that asubset of the corresponding projective limit can beidentified with the symplectic manifold that one startedfrom. Then we illustrate the programme outlined above byapplying it to the Gauss constraint.This paper also complements, as a side result, earlier work by Ashtekar,Corichi and Zapata who observed that certain operators arenon-commuting on certain states, although the Poissonbrackets between the corresponding classical functionsvanish. These authors showed that this is not acontradiction provided that one refrains from a phasespace quantization but rather applies a quantizationbased on the Lie algebra of vector fields on theconfiguration space of the theory. Here we show that onecan provide a phase space quantization, that is,one can find other functions on the classical phase spacewhich give rise to the same operators but whose Poissonalgebra precisely mirrors the quantum commutator algebra.The framework developed here is the classical cornerstone on which thesemiclassical analysis in a new series of papers called`gauge theory coherent states' is based.