Lie Algebra Of Vector Fields Research Articles

Ternary invariant differential operators acting on spaces of weighted densities

We classify ternary differential operators over n-dimensional manifolds. These operators act on the spaces of weighted densities and are invariant with respect to the Lie algebra of vector fields. For n = 1, some of these operators can be expressed in terms of the de Rham exterior differential, the Poisson bracket, the Grozman operator, and the Feigin-Fuchs antisymmetric operators; four of the operators are new up to dualizations and permutations. For n > 1, we list multidimensional conformal tranvectors, i.e., operators acting on the spaces of weighted densities and invariant with respect to o(p + 1, q + 1), where p + q = n. With the exception of the scalar operator, these conformally invariant operators are not invariant with respect to the whole Lie algebra of vector fields.

COHOMOLOGY OF THE LIE SUPERALGEBRA OF CONTACT VECTOR FIELDS ON 1|1 AND DEFORMATIONS OF THE SUPERSPACE OF SYMBOLS

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra of contact vector fields on the (1, 1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but (1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal (1|2)-trivial deformations of the -module structure on the superspaces of symbols of differential operators. We prove that any generic formal (1|2)-trivial deformation of this -module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On (2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys. No. 1 (2007) 112–127]. Further superizations correspond to (N|2)-relative cohomology of the Lie superalgebras of contact vector fields on 1|N-dimensional superspace.

Special polynomials in free framed Lie algebra

A framed Lie algebra is an algebra with two operations which is a Lie algebra with respect to one of these operations. A basic example is a Lie algebra of vector fields on a manifold with connection where the covariant derivative serves as an additional operation. In a free framed Lie algebra, we distinguish a set of special polynomials that geometrically correspond to invariantly defined tensors. A necessary condition of being special is derived, and we presume that this condition is also sufficient.

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. I

In this paper we construct and study the actions of certain deformations of the Lie algebra of Hamiltonians on the plane on the Chow groups (resp., cohomology) of the relative symmetric powers $${\mathcal{C}}^{[\bullet]}$$ and the relative Jacobian $${\mathcal{J}}$$ of a family of curves $${\mathcal{C}}/S$$ . As one of the applications, we show that in the case of a single curve C this action induces a $${\mathbb{Z}}$$ -form of a Lefschetz sl2- action on the Chow groups of C [N]. Another application gives a new grading on the ring CH 0(J) of 0-cycles on the Jacobian J of C (with respect to the Pontryagin product) and equips it with an action of the Lie algebra of vector fields on the line. We also define the groups of tautological classes in CH $$^{*}({\mathcal{C}}^{[\bullet]})$$ and in CH $$^{*}({\mathcal{J}})$$ and prove for them analogs of the properties established in the case of the Jacobian of a single curve by Beauville in [5]. We show that our algebras of operators preserve the subrings of tautological cycles and act on them as some explicit differential operators.

Integration of locally exponential Lie algebras of vector fields

A locally convex Lie algebra \({\mathfrak g}\) is said to be locally exponential if it belongs to some local Lie group in canonical coordinates. In this note we give criteria for locally exponential Lie algebras of vector fields on an infinite-dimensional manifold to integrate to global Lie group actions. Moreover, we show that all necessary conditions are satisfied if the manifold is finite-dimensional connected and σ-compact, which leads to a generalization of Palais’ Integrability Theorem.

Deformations of Lie algebras of vector fields arising from families of schemes

Fialowski and Schlichenmaier constructed global deformations of the infinitesimally and formally rigid Lie algebra of polynomial vector fields on the circle from families of projective curves with marked points. In the present article, we show how to obtain these examples in a conceptual way. For this, we define a stack of deformations of Lie algebras and study the morphism from the moduli stack to this deformation stack which associates to a family of marked curves the Lie algebra of vertical vector fields on the punctured family.

Superposition rules, lie theorem, and partial differential equations

A rigorous geometric proof of the Lie theorem on nonlinear superposition rules for solutions of nonautonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an alternative but equivalent definition of a superposition rule: it is considered as a foliation with some suitable properties. The problem of uniqueness of the superposition function is solved, the key point being the codimension of the foliation constructed from the given Lie algebra of vector fields. Finally, as a more convincing argument supporting the use of this alternative definition of superposition rule, it is shown that this definition allows an immediate generalization of the Lie theorem for the case of systems of partial differential equations.

Jet Modules

AbstractIn this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of suchmodules is given by spaces of jets of tensor fields.

Pseudodifferential operators on manifolds with a Lie structure at infinity

We define and study an algebra Ψ ∞,0,V (M0) of pseudodifferential operators canonically associated to a noncompact, Riemannian manifold M0 whose geometry at infinity is described by a Lie algebra of vector fields V on a compactification M of M0 to a compact manifold with corners. We show that the basic properties of the usual algebra of pseudodifferential operators on a compact manifold extend to Ψ ∞,0,V (M0). We also consider the algebra Diff ∗ (M0) of differential operators on M0 generated by V and C ∞ (M), and show that Ψ ∞,0,V (M0) is a microlocalization of Diff ∗ (M0). Our construction solves a problem posed by Melrose in 1990. Finally, we introduce and study semi-classical and “suspended” versions of the algebra Ψ ∞,0,V (M0).

Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields of the circle and periodic shallow water equations

This paper is a survey article on bi-Hamiltonian systems on the dual of the Lie algebra of vector fields on the circle. Here, we investigate the special case where one of the structures is the canonical Lie-Poisson structure and the second one is constant. These structures, called affine or modified Lie-Poisson structures, are involved in the integrability of certain Euler equations that arise as models for shallow water waves.

Simple Lie superalgebras and nonintegrable distributions in characteristic p

Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p ≤ 5) and several other little-known simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4), and Sp(10). The description was performed in terms of Cartan-Tanaka-Shchepochkina prolongs using Shchepochkina’s algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite-dimensional Lie superalgebras overℂ; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one. Bibliography: 30 titles.

The space of [formula omitted]-ary differential operators as a module over the Lie algebra of vector fields

The space D λ ¯ ; μ , where λ ¯ = ( λ 1 , … , λ m ) , of m -ary differential operators acting on weighted densities is a ( m + 1 ) -parameter family of modules over the Lie algebra of vector fields. For almost all the parameters, we construct a canonical isomorphism between the space D λ ¯ ; μ and the corresponding space of symbols as s l ( 2 ) -modules. This yields to the notion of the s l ( 2 ) -equivariant symbol calculus for m -ary differential operators. We show, however, that these two modules cannot be isomorphic as s l ( 2 ) -modules for some particular values of the parameters. Furthermore, we use the symbol map to show that all modules D λ ¯ ; μ 2 (i.e., the space of second-order operators) are isomorphic to each other, except for a few modules called singular.

On (2)-relative cohomology of the Lie algebra of vector fields and differential operators

Let Vect(ℝ) be the Lie algebra of smooth vector fields on ℝ. The space of symbols Pol(T*ℝ) admits a non-trivial deformation (given by differential operators on weighted densities) as a Vect(ℝ)-module that becomes trivial once the action is restricted to (2) ⊂ Vect(ℝ). The deformations of Pol(T*ℝ), which become trivial once the action is restricted to (2) and such that the Vect(ℝ)-action on them is expressed in terms of differential operators, are classified by the elements of the weight basis of , where denotes the differential cohomology (i.e., we consider only cochains that are given by differential operators) and where D λ,μ = Homdiff(F λ, F μ) is the space of differential operators acting on weighted densities. The main result of this paper is computation of this cohomology. In addition to relative cohomology, we exhibit 2-cocycles spanning and (2).

Annihilators of tensor density modules

We describe the two-sided ideals in the universal enveloping algebras of the Lie algebras of vector fields on the line and the circle which annihilate the tensor density modules. Both of these Lie algebras contain the projective subalgebra, a copy of sl 2 . The restrictions of the tensor density modules to this subalgebra are duals of Verma modules (of sl 2 ) for Vec ( R ) and principal series modules (of sl 2 ) for Vec ( S 1 ) . Thus our results are related to the well-known theorem of Duflo describing the annihilating ideals of Verma modules of reductive Lie algebras. We find that, in general, the annihilator of a tensor density module of Vec ( R ) or Vec ( S 1 ) is generated by the Duflo generator of its annihilator over sl 2 (the Casimir operator minus a scalar) together with one other generator, a cubic element of U ( Vec ( R ) ) not contained in U ( sl 2 ) .

Reducible Ordinary Differential Equations

The class of reducible differential equations under consideration here includes the class of symmetric systems, and examples show that the inclusion is proper. We first discuss reducibility, as well as the stronger concept of complete reducibility, from the viewpoint of Lie algebras of vector fields and their invariants, and find Lie algebra conditions for reducibility which generalize the conditions in the symmetric case. Completely reducible equations are shown to correspond to a special class of abelian Lie algebras. Then we consider the inverse problem of determining all vector fields which are reducible by some given map. We find conditions imposed on the vector fields by the map, and present an algorithmic access for a given polynomial or local analytic map to \({\mathbb R}.\) Next, reducibility of polynomial systems is discussed, with applications to local reducibility near a stationary point. We find necessary conditions for reducibility, including restrictions for possible reduction maps to a one-dimensional equation.

How to realize a Lie algebra by vector fields

We describe an algorithm for embedding a finite-dimensional Lie algebra (superalgebra) into a Lie algebra (superalgebra) of vector fields that is suitable for a ground field of any characteristic and also a way to select the Cartan, complete, and partial prolongations of the Lie algebra of vector fields using differential equations. We illustrate the algorithm with the example of Cartan’s interpretation of the exceptional simple Lie algebra $$\mathfrak{g}$$ (2) as the Lie algebra preserving a certain nonintegrable distribution and also several other examples.

Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

We compute the homology of the Lie algebra W1 of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W1-valued currents on the line.

A category of modules for the full toroidal Lie algebra

Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ġ and considers Fourier polynomial maps from an N + 1-dimensional torus into ġ. Setting tk = e ixk , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t0 , t ± 1 , . . . , t ± N ], and the Lie algebra of the ġ-valued maps from a torus with the multi-loop algebra C[t±0 , t ± 1 , . . . , t ± N ] ⊗ ġ. When N = 0, this yields the usual loop algebra. Just as for the affine algebras, the next step is to build the universal central extension (R⊗ ġ)⊕K of R⊗ ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R⊗ ġ)⊕K. The resulting algebra,

Sobolev spaces on Lie manifolds and regularity for polyhedral domains

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping properties of pseudodifferential operators. A tubular neighborhood theorem for Lie submanifolds allows us also to extend to regular open subsets of Lie manifolds the classical results on traces of functions in suitable Sobolev spaces. Our main application is a regularity result on polyhedral domains \mathbb P \subset \mathbb R^3 using the weighted Sobolev spaces \mathcal K^m_a(\mathbb P) . In particular, we show that there is no loss of \mathcal K^m_a –regularity for solutions of strongly elliptic systems with smooth coefficients. For the proof, we identify \mathcal K^m_a(\mathbb P) with the Sobolev spaces on \mathbb P associated to the metric r_{\mathbb P}^{-2} g_E , where g_E is the Euclidean metric and r_{\mathbb P}(x) is a smoothing of the Euclidean distance from x to the set of singular points of \mathbb P . A suitable compactification of the interior of \mathbb P then becomes a regular open subset of a Lie manifold. We also obtain the well-posedness of a non-standard boundary value problem on a smooth, bounded domain with boundary \mathcal O \subset \mathbb R^n using weighted Sobolev spaces, where the weight is the distance to the boundary.