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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.
11
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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 …
- 356
8
votes
2
answers
373
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The minimum codimension of Lie subalgebra of $\chi^{\infty}(M)$
Assume that $M$ is an arbitrary manifold.
Is there a Lie subalgebra of $\chi^{\infty}(M)$, the space of smooth vector fields on $M$, whose codimension is equal to one?
If not, what is a counter …
- 356
7
votes
1
answer
848
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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 …
- 356
6
votes
1
answer
570
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A vector space associated with a vector field on a symplectic manifold
$\DeclareMathOperator\Div{Div}$Edit: The correct formulation of the vector space $S(X)$ which is defined in this question is the following:$$S(X)=\{Y\in \chi^{\infty}(M)\mid X.\omega(X,Y)=(1 …
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6
votes
1
answer
322
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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 …
- 356
5
votes
1
answer
110
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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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5
votes
0
answers
371
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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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4
votes
1
answer
224
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A Manifold for which $\chi^{\infty}(M)$ is rich
Is there a manifold $M$ for which $\chi^{\infty} (M)$, the lie algebra of smooth vector fields on $M$ contains all finite dimensional Lie algebras(Up to isomorphism)?
A weaker question:
Is there a …
- 356
4
votes
1
answer
284
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A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$
In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed …
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4
votes
0
answers
330
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The geometry of the holomorph of a Lie group
Every Lie group $G$ is naturally contained in its holomorph Hol($G$) = $G \rtimes $ Aut($G$)
Is Hol$(G)$ always a Lie group?
If the answer is yes our main questions:
1.For a left invariant metr …
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3
votes
1
answer
265
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Lie algebra of invariant polynomials or invariant smooth functions
Is there a symplectic structure on $M_{2n}(\mathbb{R})$, not necessarily with constant coefficients, such that the space of smooth invariant functions, those smooth functions $f:M_{2n}(\mathbb …
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3
votes
1
answer
667
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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 …
- 356
3
votes
1
answer
363
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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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3
votes
1
answer
392
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A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to...
Edit: According to the comment by @LSpice we realise the existing link to the main motivation of the question is not available. Then we search for the paper we found the following version:
https://www …
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2
votes
1
answer
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The centralizer of Lienard equation
Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined o …
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