A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the ...

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21 views

Adjoint action on Lie algebra su(2) ($A \in SU(2), X \in \mathfrak{su(2)} \Rightarrow AXA^{-1}\in \mathfrak{su(2)}$)

I am trying to understand ho $SU(2)/\{\pm I\} \cong SO(3)$ (see: how to show $SU(2)/\mathbb{Z}_2\cong SO(3)$) but i am not sure about the adjoint action. In especially, as I understand, the adjoint ...
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1answer
45 views

$su(2) $ and $ sl(2;R)$ are not isomorphic? [duplicate]

As real Lie algebras, both are three-dimensional. The basis of $su(2)$ is $$ \left( \begin{matrix} i & 0 \\ 0 & -i \end{matrix} \right), \left( \begin{matrix} 0 & 1 \\ -1 & 0 ...
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28 views

Lie functor produces an antihomomorphism in Lavendhomme's synthetic differential geometry text?

Classically the Lie functor maps a Lie group homomorphism to a Lie algebra homomorphism. But in Proposition 15 on page 249 in Basic Concepts of Synthetic Differential Geometry, Lavendhomme states that ...
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2answers
65 views

Lie algebra of the automorphism group of a Lie group?

Let $G$ be a Lie group and $\text{Aut}(G)$ the group of all Lie group automorphisms of $G$. If $\text{Aut}(G)$ can be interpreted to be a Lie group (for example, in the context of synthetic ...
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585 views

Can you give me an example of topological group which is not a Lie group.

I know the definitions of Lie group and topological group are different. Can you give me an example of topological group which is not a Lie group.
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0answers
35 views

Can the parameter $t$ in the exponential map $e^{tX}$ be complex?

From a Lie algebra to a Lie group, can the parameter $t$ in the exponential map $t\rightarrow e^{tX}$ be complex? If the Lie algebra is a complex one, this is legal, right?
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0answers
21 views

Lie transformation applied to Poisson brackets

Given the following Lie transformation: $$ \exp(\lbrace H, \cdot \rbrace):=\sum_{n=0}^{\infty} \frac{(\lbrace H, \cdot \rbrace)^n}{n!} $$ and apply it to a Poisson Bracket $\lbrace g_1, g_2 ...
2
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30 views

Action of $sl(2,\mathbb{C})$ on Dual of Polynomials does not Exponentiate

Let $V$ be the space of holomorphic polynomial functions in two complex variables $\xi,\eta$ and let $V^\ast$ be its dual space with subspace $W$ of linear functionals of the form $Df(1,0)$ where $D$ ...
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1answer
23 views

Dimension of centralizer of unitary matrix

Let $G=U(n)$ be the unitary group. We know that any unitary matrix is diagonalizable. Let $x$ be a unitary matrix. Then I've read the statement that any matrix $b$ commuting with $x$ is block-diagonal ...
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1answer
42 views

Map from unit quaternions to SO(3)?

On the wikipedia page for "Rotation Group SO(3)" I read that there is a 2:1 surjection from the unit quaternions, $q=w+xi+yj+zk$, to the rotatation matrix $$Q= \left( \begin{array}{ccc} 1-2y^2-2z^2 ...
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0answers
26 views

How does a group of transformations lead to a geometry?

I am reading Vinberg's algebra text, and on page 144 he says "Of course, not every transformation group leads to a geometry which is interesting and also important for some applications. All such ...
2
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43 views

Is the exponential map of complex spin group surjective?

The complex spin group $Spin(n,C)$ is defined as the double cover of $SO(n,C)$. If the the exponential map is surjective, it will give a parametrization of this Lie group. Is it true for this ...
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33 views

A sufficient condition for a subgroup being closed

Let $ G $ is a Lie group with Lie algebra $ g $ and $ H \subset G $ is a subgroup. If there exists $ h \in H $ and a closed neighborhood $ V $ of $ h $ such that $ H \cap V $ is closed in $ G $, then ...
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1answer
23 views

Closed orbits for reductive group actions

Let $G$ be a complex reductive group acting algebraically on a complex affine variety $X$. Is it true that an orbit $G.x$ is closed in $X$ if and only if the stabilizer $G(x)$ of $x$ is a reductive ...
4
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2answers
59 views

Does a lattice in $SL_n(\mathbb R)$ which is contained in $SL_n(\mathbb Z)$ have finite index in $SL_n(\mathbb Z)$?

A lattice $H$ in a locally compact group $G$ is a discrete subgroup such that the coset space $G/H$ admits a finite $G$-invariant measure. I have read several places that any lattice H in ...
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1answer
55 views

Is the group of symplectomorphisms of a symplectic manifold the symplectic group?

The group of symplectomorphisms of a symplectic manifold $M$ is a subgroup of the group of diffeomorphisms $GL(n)$, actually it is a subgroup of $SL(n)$. My question is whether this group of ...
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0answers
51 views

Proving $e^{-i\left( \frac{\theta}{2}n\cdot \sigma\right) }=\cos\left(\frac{\theta}{2}\right) - i n \cdot \sigma \sin\left(\frac{\theta}{2}\right) $

I believe I've almost proved that $e^{-i\left( \frac{\theta}{2} n \cdot \sigma\right) } = \cos\left(\frac{\theta}{2}\right) - i n \cdot \sigma \sin\left(\frac{\theta}{2}\right)$ such that $ n ...
2
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1answer
16 views

Can a point and a compact set in a Tychonoff space be separated by a continuous function into an arbitrary finite dimension Lie group?

Given a topological space $X$ which is Tychonoff (i.e., completely regular and Hausdorff), we know that given a compact set $K\subseteq X$ and a point $p \in X$ with $p\not\in K$, we can construct a ...
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1answer
60 views

A Lie group that has an immersion in $\mathrm{GL}(n,\Bbb R)$ but no embedding?

Question: Is there a Lie group $G$ that admits a smooth immersion $$i:G\longrightarrow\mathrm{GL}(n,\Bbb R)$$ for some $n\in\Bbb N$, but no smooth embedding ...
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0answers
18 views

If $G$ is a compact Lie group acting effectively on $X$ then it is a subspace of Homeo$(X)$?

Let $G$ be a compact Lie group acting effectively on a simply connected space $X$. Let Homeo$(X)$ be the group of all homeomorphisms of $X$ with itself given the compact open topology. Is the ...
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1answer
50 views

Closures of one-parameter subgroups of lie groups

I'm reading some basic facts of Lie groups. I meat with difficulties when I try to solve the following statement: "Prove the closure of a non-closed one-parameter subgroup of a Lie group is a torus." ...
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8answers
1k views

Why do we care about two subgroups being conjugate?

In classifications of the subgroups of a given group, results are often stated up to conjugacy. I would like to know why this is. More generally, I don't understand why "conjugacy" is an equivalence ...
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13 views

Flag varieties from the representation of a solveable Lie algebra

I've been reading Lie Algebras, and I've come across this problem: "Let $\mathfrak{g}$ be a solveable Lie Algebra over $\mathbb{R}$. $V$ a vector space over $\mathbb{R}$, and $\rho$ a representation ...
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1answer
30 views

Radical of a Lie algebra bracket itself

Let $\mathfrak{g}$ be a Lie Algebra over $k$, $\mathfrak{n}$ its radical. Why is $[\mathfrak{n},\mathfrak{g}]$ the smallest of its ideals $\mathfrak{a}$ such that $\mathfrak{g}/\mathfrak{a}$ is ...
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0answers
38 views

Nilpotent Lie Algebra with determinant 0 [duplicate]

If I have a nilpotent Lie Algebra $\mathfrak{g}$ and a representation $\rho(X)$ in a vector space $V$ such that $det \rho(X) = 0 $ for all $X \in \mathfrak{g}$, then how do I show that there is a ...
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0answers
14 views

Invariants of a PDE by Lie Symmetries

I have a little Problem in understanding how to derive invariants form Lie Symmetries (or theire infinitesimals). As one can show the heat equation $u_t=u_{xx}$ has the following symmetries ...
2
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0answers
15 views

Quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice

How can I show that a quotient of the abelian Lie group $(\mathbb{C}, +)$ by a full rank lattice has no faithful finite-dimensional linear representation as a complex Lie group? I was thinking of ...
1
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1answer
24 views

Every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple.

Can I deduce that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{R})$ is semisimple from the fact that every finite-dimensional representation of $\mathfrak{sl}_2(\mathbb{C})$ is ...
1
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0answers
28 views

Distributional derivative $Xu=0$ then $u$ is constant over the flow lines of $X$

Suppose we have a Lie group on $\mathbb R^n$, something like $(\mathbb R^n, \cdot)$, where with $\cdot$ we denote the group law on $\mathbb R^n$. Call $\mathfrak g$ the Lie algebra of $(\mathbb R^n, ...
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1answer
62 views

Let $\rho$ f.d. rep of a nilpotent Lie algebra such that $\rm{det} \rho(X) = 0$, $\forall X$. Then $\exists v \neq 0$: $\rho(X)v = 0, \forall X$.

Could you help me with this question? Let $\mathcal{g}$ be a nilpotent Lie algebra over $k$ and $\rho$ a representation of $\mathcal{g}$ in a finite-dimensional nonzero vector space $V$ over $k$. ...
0
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1answer
31 views

Gauss decomposition of an algebraic group.

Let $K$ be any field. Consider $GL_n(\mathbb{K})$ as an algebraic group. I know that it has a Gauss decomposition, i.e $GL_n(K)=I^- D I^+$, where $I^-$ and $I^+$ are the lower unipotent matrix and ...
3
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1answer
42 views

Lie group question: If $\gamma^{-1}\dot{\gamma}\in\mathfrak{g}$ everywhere, does $\gamma(t)\in G$?

Let $G$ be a Lie subgroup of $GL(n,\Bbb R)$ and $\mathfrak{g}\subseteq M(n,\Bbb R)$ its Lie algebra. Suppose that we have a smooth curve $$\gamma:\Bbb R\to G$$ with $\gamma(0)=I$. Then, it induces a ...
2
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0answers
25 views

complex reductive Lie group

I am reading A. L. Oniscik's paper Decompositions of Reductive Lie Groups, and the author cited a proposition that a complex reductive Lie group $G=ZS$ is locally isomorphic to the reductive ...
1
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0answers
50 views

Form for pseudo-unitary matrices of particular dimension

I know that the unimodular pseudo-unitary group is a Lie group defined by $$\text{SU}(p, q)= \{M \in \text{SL}_{p+q}(\mathbb{C}): MAM^{*} = A \} \text{,}$$ where $A= \begin{pmatrix} 1_p & 0 \\ 0 ...
6
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1answer
45 views

Universal Enveloping Algebra of $\mathfrak{gl}(n,\Bbb R)$

I am just learning about universal enveloping algebras, and I am wondering about the following. Question: Is the universal enveloping algebras of $\mathfrak{gl}(n,\Bbb R)$ just ...
5
votes
1answer
52 views

Is the exponential map to the indefinite special orthogonal groups $SO^+(p,q)$ surjective?

Is the exponential map to the identity component of the special indefinite orthogonal groups $$ \mathrm{exp} \colon \mathfrak{so}(p,q) \to SO^+(p,q)$$ surjective?
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0answers
18 views

One-parameter subgroup K in G with pi(K) non-trivial where pi:G->G/H and given G/H has no small subgroups

This is exercise 12.12.8 in Dieudonne's Treatise On Analysis, Volume 2 Let G be a locally compact metrizable group H a closed normal subgroup of G G/H is not discrete and has no small subgroups ...
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0answers
17 views

Help with a stage in Peter-Weyl proof: that “matrix entry” functions separate points

Edit The question as originally phrased was clumsy. What I really need is the simplest proof, or reference, anyone can rustle up of this: "for $G$ a compact Lie group, and $g$ and $h$ distinct ...
0
votes
0answers
45 views

Why is $[\mathfrak{q}, \mathfrak{g}]$ the smallest ideal $\mathfrak{a}$ of $\mathfrak{g}$ such that $\mathfrak{g}/\mathfrak{a}$ is reductive. [duplicate]

Does anyone know how I would go about answering this question? Any feedback is appreciated. I'm not too sure where to start. Thanks. Let $\mathfrak{g}$ be a Lie algebra over $k$ and $\mathfrak{q}$ ...
0
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1answer
43 views

Why simply connected solvable analytic groups have no nontrivial compact subgroups?

Why do simply connected solvable analytic groups have no nontrivial compact subgroups? I'll appreciate any help on this question.
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1answer
46 views

Principal bundles with compact simply connected structure group over 2-manifolds

I'm reading Thomas Friedrich's "Dirac Operators in Riemannian Geometry," where the following is stated (in the Remark on page 42 before section 2.2 begins, if anyone is following along with the ...
2
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0answers
26 views

Prove that a subgroup in a Lie group is homogeneous

Let $\mathbb E:=(\mathbb R^4, \cdot)$ be a Carnot group whose Lie algebra is given by $\mathfrak g=V_1\oplus V_2 \oplus V_3$, where $V_1=span\{X_1,X_2\},$ $V_2=span\{X_3\},$ $V_3=span\{X_4\}$, the ...
7
votes
1answer
90 views

Symmetric power of tautological representation of $U(n)$

Let $S^kV$ be the $k$-th symmetric power of tautological representation of $U(n)$ how to see that it's irreducible? I'm trying to do it using weight, but with no benefits..
2
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0answers
25 views

Classifying left invariant metrics on the 3-dimensional heisenberg group

Recently I read that all left invariant metrics on the Heisenberg group are equivalent up to scaling,however no reference was given for this result. I've made some attempt to prove this myself. In ...
0
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0answers
88 views

Left invariant Vector Field on $S^2$

How intuitively look like all left invariant vector fields on this manifold: the 2 dimensional unit sphere $S^2$ with the smooth structure inherited from $\mathbb R^3$? Why all left invariant vector ...
1
vote
1answer
32 views

Determining if a (Lie algebra) central extension is trivial.

Given a central extension for a given Lie algebra, is there any simple way to check that it is/isn't isomorphic to the trivial extension ("simple" meaning, not as tedious [and daunting, for an algebra ...
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0answers
38 views

Primitive of the matrix elements of irreducible representations of Lie groups

I am interested in the matrix coefficients $U_{ij}(g)$ of unitary irreducible representations of a Lie group $G$. In my case, these coefficients arise from the Peter-Weyl theorem. I would like to ...
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0answers
26 views

Tensor product of $Spin(2k)$ representations

I am trying to find the tensor product of spinor representations of $SO(2k)$. Labels are given as $$(n+I/2,I/2,\ldots,I/2,s)\otimes(I/2,\ldots,I/2).$$ Where $I$ and $n$ positive integers. How can ...
1
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1answer
31 views

What exactly is the group Omega(n,q) in MAGMA?

Let $n>2$ and $q$ be a prime power. In MAGMA I'm having a lot of trouble identifying the group Omega(n,q). I'm trying to use a source that asserts that it is the group of $n\times n$ orthogonal ...
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0answers
29 views

Map to a Lie group as exponential of a map to the Lie algebra

Let $U$ be an open subset of $\mathbb{R}^n$, $G$ a Lie group and $f:U\rightarrow G$ a smooth surjective map. Under which conditions there is a smooth function $\phi:U\rightarrow\mathfrak{g}$ such ...