Pseudodifferential operators on manifolds with a Lie structure at infinity

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).

Prime Lie algebras satisfying the standard Lie identity of degree 5

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X ∈ A. Theorem If K is a locally maximal compact set of zeroes of X ∈ A and the Poincaré-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish.

In the previous paper [1], we studied Pursell-Shanks type theorem for orbit spaces of G-manifolds. The purpose of this paper is to prove that this still holds for smooth orbifolds and for fibrations over smooth orbifolds with compact generic fibre. Let B and B be connected smooth orbifolds. Let S)(5) (resp. K(5)) be the Lie algebra of all smooth vector fields (resp. strata preserving smooth vector fields) on B with compact support.

Transitive local Lie algebras of vector fields can be easily constructed from dilations of $\mathbb{R}^n$ associating with coordinates positive weights (give me a sequence of $n$ positive integers and I will give you a transitive nilpotent Lie algebra of vector fields on $\mathbb{R}^n$). It is interesting that all transitive nilpotent local Lie algebra of vector fields can be obtained as subalgebras of nilpotent algebras of this kind. Starting with a graded nilpotent Lie algebra one constructs graded parts of its Tanaka prolongations inductively as `derivations of degree 0, 1, etc. Of course, vector fields of weight $k$ with respect to the dilation define automatically derivations of weight $k$, so the Tanaka prolongation is in this case never finite. Are they all such derivations given by vector fields or there are additional `strange'ones? We answer this question. Except for special cases, derivations of degree 0 are given by vector fields of degree 0 and the Tanaka prolongation recovers the whole algebra of polynomial vectors defined by the dilation. However, in some particular cases of dilations we can find `strange' derivations which we describe in detail

Projectable Lie algebras of vector fields in 3D

Defining relations for Lie algebras of vector fields

Israel Moiseevich Gelfand—one of the most original and versatile mathematicians, whose contributions had a profound effect on the study of theoretical and quantum physics—died in New Brunswick, New Jersey, on 5 October 2009.Gelfand was born on 2 September 1913 in the small town of Okny near Odessa in Russia. He was expelled from school in the ninth grade because he was the son of a “bourgeois element” —his father was a mill manager—so he learned mathematics on his own. In a 1989 interview with the Russian magazine Kvant, translated into English for Quantum in 1991, he recounted, “It’s my deeply held conviction that mathematical ability in most future professional mathematicians appears precisely at that time—at 13 to 16 years of age. … This early period formed my style of doing mathematics. The subject of my studies varied, of course, but the artistic form of mathematics that took root … became the basis of my taste in choosing problems that continue to attract me right up to the present time.”In 1930 Gelfand moved to Moscow. His next twist of fate was one of many paradoxes of Soviet life. Because of his status as a bourgeois element’s son, he could not enroll as an undergraduate student at a university. However, at 18 he taught at a technical college, and at 19 he entered the PhD program at Moscow State University without a diploma from college or even high school. The reason was simple: Although the Soviet Union used all possible resources to provide an education for the future elite that had “proletarian origins” and to keep others out of higher education, it did not bother to regulate graduate schools at that time.Gelfand’s thesis adviser was Andrei Kolmogorov, who taught that a true mathematician must be a philosopher of nature. In 1935 Gelfand defended his thesis, “Abstract Functions and Linear Operators,” and in 1940 he obtained a doctor of science degree. In 1933 he began teaching at Moscow State. He became a full professor in 1943 but lost the position temporarily in 1952 during Joseph Stalin’s anti-Semitic campaign. He worked at the Steklov Mathematical Institute starting in 1939, and in 1953 he moved to the Institute of Applied Mathematics. After World War II he took part in the Soviet version of the Manhattan Project.In 1989 Gelfand moved to the US. He spent a year at Harvard University and MIT before becoming a professor at Rutgers University, where he worked until his death.Gelfand became a worldwide celebrity among mathematicians after introducing in one of his early papers the idea of considering maximal ideals of Banach algebras as points of a topological space—an approach that had a great impact in modern mathematics. The paper clearly demonstrated Gelfand’s style: Turn hard theorems into easy ones.Gelfand produced a cascade of fundamental contributions in different areas, partly described in his celebrated six-volume monograph Generalized Functions. In 1978 he received the Wolf Prize “for his work in functional analysis, group representation, and for his seminal contributions to many areas of mathematics.” It was a recognition of his enormous influence on modern mathematics and physics. Surprisingly, the fundamental work on Gelfand-Tsetlin bases, a household term in representation theory, was at first ignored by mathematicians but heavily used by theoretical physicists. One of his favorite subjects was integral geometry, which he developed with Simon Gindikin and Mark Graev. It is now used to convert the data of magnetic resonance imaging and computerized axial tomography scans into three-dimensional images.Gelfand noticed that the index of an elliptic operator is a homotopy invariant of the leading symbol; that observation led to the celebrated Atiyah-Singer index theorem, which explained that for an elliptical operator on a compact manifold, the dimension of the space of solutions can be defined in terms of topological data. He wrote classical papers on differential equations and integrable systems with Boris Levitan and Leonid Dickey, and he originated with Dmitry Fuchs the study of the cohomology of Lie algebras of vector fields, which has numerous connections in physics. His work with Graev, Mikhail Kapranov, and Andrei Zelevinsky on hypergeometric functions is widely used in mirror symmetry. The legendary Gelfand Monday seminar he organized at Moscow State was a mathematical stock exchange, a breeding ground for young scientists, a one-man show, and much more. Gelfand interrupted every speaker with comments such as “Do not learn secondhand physics from mathematicians; talk to real physicists” and “You are trying to play violin just by listening to a symphony orchestra.” His motto was, “My seminar is for high-school students, decent undergraduates, bright graduates, and outstanding professors.”Out of Gelfand’s passion grew his famous correspondence school, also based at Moscow State, for mathematically gifted students. He wrote several textbooks—for his school and for universities—that each bear an imprint of his style and personality.Gelfand’s interests spread far beyond mathematics. He also published papers in biology, physiology, and other fields.The mathematics and physics communities will greatly miss Gelfand’s sharp-minded personality, his legendary intuition, and his unique ability to see the universe in a drop of water.Israel Moiseevich GelfandRUTGERS UNIVERSITYPPT|High resolution© 2010 American Institute of Physics.

We solve a long standing problem of the classification of all simple modules with finite-dimensional weight spaces over Lie algebra of vector fields on n-dimensional torus for any n. This generalizes the classical result of O.Mathieu on simple weight modules for the Virasoro algebra ( n = 1 ${n=1}$ ). Every such module is either of a highest weight type or is a quotient of a module of tensor fields on a torus, which was conjectured by Eswara Rao.

In this article, we exploit the theory of graded module categories with semi-infinite character developed by Soergel (Character formulas for tilting modules over Kac–Moody algebras, Represent. Theor. 2 (1998), 432–448) to study representations of infinite-dimensional Lie algebras of vector fields W(n), S(n) and H(n) (n \geq 2) , and obtain a description of indecomposable tilting modules. The character formulas for those tilting modules are determined.

with the bracket [ei, ej ] = (j − i)ei+j . The Lie algebra L is naturally graded, the degree of ei being i. The most natural modules over L are the so-called tensor field modules. A tensor field on the circle is of the form g(φ)(d/dφ). A vector field acts on this by infinitesimally changing the coordinate φ, where g(φ) is a section of some line bundle on the circle S with a flat connection. In the space of tensor fields we choose a basis fi, i ∈ Z, such that ei(fj) = (−λ(i + 1) + μ + j) · fi+j . Here λ, μ ∈ C are the invariants characterizing the module, i.e., the power of d/dφ and the logarithm of the monodromy of the flat connection. We denote such a module by Fλ,μ (see [3]). Denote by L1 the subalgebra of L with basis (e1, e2, e3, . . . ). It is easy to see that L1 is isomorphic to the Lie algebra of vector fields on the line, with polynomial coefficients, having a two-fold zero at the origin. The strategy of the cohomology computation for L1 with coefficients in the adjoint module is the following: we first compute the cohomology of L1 with coefficients in Fλ,μ, and then remark that the adjoint representation of L1 is a submodule of such an Fλ,μ. After this the spaces H(L1, L1) can easily be determined. The computations of H(L1, L1) and H (L1, L1) are contained in [3]. Deformations of L1 are studied in [5]. In this paper we shall describe a more general method for the computation of H∗(L1,Fλ,μ). It will be more convenient for us to deal with homology instead of cohomology. It is easy to see that H ∗ (L1,Fλ,μ) is dual to H∗(L1,F−1−λ,−μ). Then, using the fact that L∗1 is the factor of some Fλ,μ, we can compute Hi(L1, L ∗ 1). Notice that for almost every (λ, μ) the module Fλ,μ is an irreducible representation of L, and L1 is the maximal nilpotent subalgebra in L. That means that the problem of determining H∗(L1,Fλ,μ) is analogous to that of determining the cohomology of the maximal nilpotent subalgebra of a complex semisimple Lie

Over the (1,n)-dimensional real superspace, n>1, we classify K(n)-invariant binary differential operators acting on the superspaces of weighted densities, where K(n) is the Lie superalgebra of contact vector fields. This result allows us to compute the first differential cohomology of K(n) with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities—a superization of a result by Feigin and Fuchs [“Homology of the Lie algebras of vector fields on the line,” Funct. Anal. Appl. 14, 201 (1980)]. We explicitly give 1-cocycles spanning these cohomology spaces.

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples from physics, mathematics, and control theory.

We study the vector fields Vec (\mathbb A^n) on affine n -space \mathbb A^n , the subspace Vec ^c(\mathbb A^n) of vector fields with constant divergence, and the subspace Vec ^0(\mathbb A^n) of vector fields with divergence zero, and we show that their automorphisms, as Lie algebras, are induced by the automorphisms of \mathbb A^n : \mathrm {Aut}(\mathbb A^n) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}(\mathbb A^n)) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}^0(\mathbb A^n)) \xrightarrow{\sim} \mathrm {Aut_{Lie}}(\mathrm {Vec}^0(\mathbb A^n)). This generalizes results of the second author obtained in dimension 2, see [Reg13]. The case of Vec (\mathbb A^n) goes back to Kulikov [Kul92]. This generalization is crucial in the context of infinite-dimensional algebraic groups, because Vec ^c(\mathbb A^n) is canonically isomorphic to the Lie algebra of Aut (\mathbb A^n) , and Vec ^0(\mathbb A^n) is isomorphic to the Lie algebra of the closed subgroup SAut (\mathbb A^n) \subset Aut (\mathbb A^n) of automorphisms with Jacobian determinant equal to 1.

In this paper, we study the Lie algebra of vector fields ${\operatorname{Vec}}(\textrm{D}_p)$ of a smooth Danielewski surface $\textrm{D}_p$. We prove that the Lie subalgebra $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ of ${\operatorname{Vec}}(\textrm{D}_p)$ generated by locally nilpotent vector fields is simple. Moreover, if the two Lie algebras $\langle{\operatorname{LNV}}(\textrm{D}_p) \rangle$ and $\langle{\operatorname{LNV}}(\textrm{D}_q) \rangle$ of two Danielewski surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic, then the surfaces $\textrm{D}_p$ and $\textrm{D}_q$ are isomorphic. As an application we prove that the ind-groups ${\operatorname{Aut}}(\textrm{D}_p)$ and ${\operatorname{Aut}}(\textrm{D}_q)$ are isomorphic if and only if $\textrm{D}_p \simeq \textrm{D}_q$ as a variety. We also show that any automorphism of the ind-group ${\operatorname{Aut}}^\circ (\textrm{D}_p)$ is inner.