scholarly journals Zero sets of Lie algebras of analytic vector fields on real and complex two-dimensional manifolds

2017 ◽  
Vol 39 (4) ◽  
pp. 954-979 ◽  
Author(s):  
MORRIS W. HIRSCH ◽  
F.-J. TURIEL
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Fixed Point Theorems ◽  
Zero Set ◽  
Analytic Vector ◽  
Ground Field ◽  
Zero Sets ◽  
Finite Dimensional ◽  
Differentiable Vector ◽  
Analytic Vector Field

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

scholarly journals On Lie algebras of vector fields with invariant submanifolds

1974 ◽  
Vol 55 ◽  
pp. 91-110 ◽  
Author(s):  
Akira Koriyama
Keyword(s):  
Lie Algebras ◽  
Compact Support ◽  
Vector Fields ◽  
Finite Dimensional ◽  
Invariant Submanifolds ◽  
Infinitesimal Automorphisms

It is known (Pursell and Shanks [9]) that an isomorphism between Lie algebras of infinitesimal automorphisms of C∞ structures with compact support on manifolds M and M′ yields an isomorphism between C∞ structures of M and M’.Omori [5] proved that this is still true for some other structures on manifolds. More precisely, let M and M′ be Hausdorff and finite dimensional manifolds without boundary. Let α be one of the following structures:

scholarly journals COMPUTATION OF COHOMOLOGY OF LIE SUPERALGEBRAS OF VECTOR FIELDS

2000 ◽  
Vol 11 (02) ◽  
pp. 397-413 ◽  
Author(s):  
V. V. KORNYAK
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Mathematical Physics ◽  
Lie Superalgebras ◽  
Graded Algebras ◽  
Topological Information ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Cohomology Of Lie Algebras ◽  
Mathematics And Physics

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. It carries most fundamental ("topological") information about algebra under consideration. At present, because of the need for very tedious algebraic computation, the explicitly computed cohomology for different classes of Lie (super)algebras is known only in a few cases. That is why application of computer algebra methods is important for this problem. We describe here an algorithm and its C implementation for computing the cohomology of Lie algebras and superalgebras. The program can proceed finite-dimensional algebras and infinite-dimensional graded algebras with finite-dimensional homogeneous components. Among the last algebras, Lie algebras and superalgebras of formal vector fields are most important. We present some results of computation of cohomology for Lie superalgebras of Buttin vector fields and related algebras. These algebras being super-analogs of Poisson and Hamiltonian algebras have found many applications to modern supersymmetric models of theoretical and mathematical physics.

A Weight Theory for Unitary Representations

1966 ◽  
Vol 18 ◽  
pp. 159-168 ◽  
Author(s):  
Thomas Sherman
Keyword(s):  
Lie Algebras ◽  
Subject Matter ◽  
Unitary Group ◽  
Group Representation ◽  
Unitary Representations ◽  
Real Numbers ◽  
Ground Field ◽  
Simple Lie Algebras ◽  
Finite Dimensional ◽  
The Subject

Over a field of characteristic 0 certain of the simple Lie algebras have a root theory, namely those called “split” in Jacobson's book (3). We shall assume some familiarity with the subject matter of this book. Then the finite-dimensional representations of these Lie algebras have a weight theory. Our purpose here is to present a kind of weight theory for the representations of these Lie algebras when their ground field is the real numbers, and when the representation comes from a unitary group representation.

scholarly journals Lie algebras of conservation laws of variational partial differential equations

2018 ◽  
Vol 18 (2) ◽  
pp. 207-228
Author(s):  
Emanuele Fiorani ◽  
Sandra Germani ◽  
Andrea Spiro
Keyword(s):  
Lie Algebras ◽  
Vector Fields ◽  
Differential Operators ◽  
Equivalence Classes ◽  
Conserved Quantities ◽  
Lagrange Equations ◽  
Bijective Correspondence ◽  
Independent Variables ◽  
Geometric Conditions ◽  
Finite Dimensional

Abstract We establish a version of Noether’s first Theorem according to which the (equivalence classes of) conserved quantities of given Euler–Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action from which the Euler–Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such a bijective correspondence Φ͠ between equivalence classes comes from an explicit (non-bijective) linear map Φ from vector fields into conserved differential operators, and not from a map into divergences of conserved operators as it occurs in other proofs of Noether’s Theorem. Where possible, claims are given a coordinate-free formulation and all proofs rely just on basic differential geometric properties of finite-dimensional manifolds.

scholarly journals On the zero set of G-equivariant maps

2009 ◽  
Vol 147 (3) ◽  
pp. 735-755
Author(s):  
P-L. BUONO ◽  
M. HELMER ◽  
J. S. W. LAMB
Keyword(s):  
Bifurcation Theory ◽  
Vector Fields ◽  
Isotropy Subgroup ◽  
Zero Set ◽  
Group Representations ◽  
Equivariant Maps ◽  
Zero Sets ◽  
The Difference ◽  
A Finite Group ◽  
Stratified Set

AbstractLet G be a finite group acting on vector spaces V and W and consider a smooth G-equivariant mapping f: V → W. This paper addresses the question of the zero set of f near a zero x with isotropy subgroup G. It is known from results of Bierstone and Field on G-transversality theory that the zero set in a neighbourhood of x is a stratified set. The purpose of this paper is to partially determine the structure of the stratified set near x using only information from the representations V and W. We define an index s(Σ) for isotropy subgroups Σ of G which is the difference of the dimension of the fixed point subspace of Σ in V and W. Our main result states that if V contains a subspace G-isomorphic to W, then for every maximal isotropy subgroup Σ satisfying s(Σ) > s(G), the zero set of f near x contains a smooth manifold of zeros with isotropy subgroup Σ of dimension s(Σ). We also present partial results in the case of group representations V and W which do not satisfy the conditions of our main theorem. The paper contains many examples and raises several questions concerning the computation of zero sets of equivariant maps. These results have application to the bifurcation theory of G-reversible equivariant vector fields.

scholarly journals MODULES FOR A SHEAF OF LIE ALGEBRAS ON LOOP MANIFOLDS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250079
Author(s):  
YULY BILLIG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Semidirect Product ◽  
Central Extension ◽  
Vector Fields ◽  
Vertex Algebra ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Rham Complex ◽  
De Rham

We consider a semidirect product of the sheaf of vector fields on a manifold ℂ* × X with a central extension of the sheaf of Lie algebras of maps from ℂ* × X into a finite-dimensional simple Lie algebra, viewed as sheaves on X. Using vertex algebra methods we construct sheaves of modules for this sheaf of Lie algebras. Our results extend the work of Malikov–Schechtman–Vaintrob on the chiral de Rham complex.

scholarly journals Classification of Simple Lie Superalgebras in Characteristic 2

2021 ◽  
Author(s):  
Sofiane Bouarroudj ◽  
Alexei Lebedev ◽  
Dimitry Leites ◽  
Irina Shchepochkina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Fields ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Simple Lie Algebra ◽  
Hamiltonian Vector Fields ◽  
Finite Dimensional ◽  
Characteristic 2

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

scholarly journals The Wielandt Subalgebra of a Lie Algebra

2003 ◽  
Vol 74 (3) ◽  
pp. 313-330 ◽  
Author(s):  
Donald W. Barnes ◽  
Daniel Groves
Keyword(s):  
Group Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Ground Field ◽  
Derived Length ◽  
Finite Dimensional ◽  
Definition Of

AbstractFollowing the analogy with group theory, we define the Wielandt subalgebra of a finite-dimensional Lie algebra to be the intersection of the normalisers of the subnormal subalgebras. In a non-zero algebra, this is a non-zero ideal if the ground field has characteristic 0 or if the derived algebra is nilpotent, allowing the definition of the Wielandt series. For a Lie algebra with nilpotent derived algebra, we obtain a bound for the derived length in terms of the Wielandt length and show this bound to be best possible. We also characterise the Lie algebras with nilpotent derived algebra and Wielandt length 2.

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