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Results tagged with lie-algebras
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user 36688
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
5
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A (possible) Lie algebra extension of the Lie algebra of a foliation
Motivation: The aim of this post is to extend the Lie algebra of a foliation to a bigger Lie algebra. We assume that a manifold $M$ is foliated by compat leaves. The Lie algebra of the foliation is th …
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Rotation number for homeomorphisms of a Lie group other than $S^1$
Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$.
For what kind of Lie group $G$ the standard process of definition of rotation number …
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1
answer
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A Lie group whose Lie algebra is equal to (the Lie algebra? of )all functions with fibrewise...
Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket …
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When is the exterior derivation $d$ a Lie algebra morphism?
In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We …
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An extension of symplectomorphism group
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$.
We consider the …
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When a finite codimensional subalgebra contains a finite codimension ideal?
What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property:
Every finite codimensional subalgebra $B$ of $A$ con …
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Fredholm elements of a Lie algebra
An element $a$ of a Lie algebra $L$ is called a Fredholm element if the adjoint operator $\mathrm{ad}_a:L \to L$ is a Fredholm linear map. That is: its kernel is a finite-dimensional space and its ran …
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A linear map on $\chi^{\infty}(\mathbb{R}^2)$ arising from the Cauchy integral formula
The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\mat …
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The idealizer of the space of vector fields with vanishing divergence
The space of smooth vector fields on a manifold is counted as a Lie algebra with the usual Lie bracket structure.
Is there a Riemannian manifold of dimension at least $2$ which satisfies either of th …
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1
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Alternative Lie bracket on $\chi^{\infty}(M)$ where $M$ is a Riemannian manifold with a symp...
Let $(M,\omega, g)$ be a Riemannian symplectic manifold. The Poisson bracket of two functions $f,g$ is denoted by $\{f,g\}$. The Hamiltonian vector field and gradient vector field …
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A Lie algebra associated to a symplectic manifold
Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ with the volume form $\omega^n.$
In this question we associate a Lie algebra $L(M,\omega)$ to $(M,\omega)$. Then we are interested …
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A vector space associated with a vector field on a symplectic manifold
For $n>1$, and the standard symplectic structure $\omega=\sum dx_i\wedge dy_i$ of $\mathbb{R}^{2n}=\{(x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n)\}$ and for the vector field $X=\partial/\partial_{x …
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1
answer
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A particular embedding of a Lie group in Euclidean space
I apologize in advance if my question is elementary.
Before I present my question I mention my motivation:
Motivation:
A Lie group is a manifold. At the same time it is a Riemannian manifold equippe …
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answer
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The Hessian of invariant functions on a Lie group
Assume that $G$ is a Lie group with Lie algebra $\mathfrak{g}$. We fix an invariant Riemannian metric on $G$ and fix its corresponding $LC$ connection.
Consider the natural right action of …
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answer
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Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n = \dim M$
For a connected $n$-manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a point $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\}$. Of course …
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