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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Difference between the pairings $\text{Tr}(xy)$ and $\text{Tr}(x^t y)$

Let $\mathfrak{g}$ be the tangent space to $GL_n(\mathbb{C})$ at the identity. What is the difference between the two maps? Any subtle geometric or algebraic difference between the two pairings $$ ...
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35 views

Ideals in the unitary group

What would be examples of one-dimensional ideals in the lie algebra of the unitary group? Moreover, how would one show that it is in the tangent space of the center of the unitary group and that the ...
2
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1answer
44 views

Lie Group and Universal Covering Groups

Is every Lie group realized as the quotient of its universal covering group by a discrete group of isometries? Basically, a Lie group analog for the uniformization theorem. It seems reasonable but I'm ...
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23 views

Induced automorphism on a tangent bundle

I had a pretty simple question but was having trouble finding the answer anywhere. If I have an orthogonal matrix $A: \mathbb{R}^n \to \mathbb{R}^n$, it should induce an automorphism on the tangent ...
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166 views

$SO(n)$ is connected

The question really is that simple: Prove that the manifold $SO(n) \subset GL(n, \mathbb{R})$ is connected. it is very easy to see that the elements of $SO(n)$ are in one-to-one correspondence with ...
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28 views

Given tangent space of submanifold of Lie group, is it possible to recover the submanifold?

I have computed the tangent space of certain submanifolds (the unstable manifolds) of a Lie group at a particular point. I know that the exponential map lets us move between the Lie algebra and the ...
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33 views

Action of a Lie group, a map of constant rank

Consider some Lie group $G$, smooth manifold $X$ and some action of $G$, i.e. a group homomorphism $\mathcal{A}: G\longrightarrow \mathrm{Diffeo}(X)$ such that the map $(g,x)\mapsto ...
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68 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
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2answers
60 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
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30 views

Reciprocity for branching rules of $\mathrm{GL}_n(\mathbb C)$

[Separated from another question] If I have information about the restriction of representations of the general linear group, can I make any statements about the induction (by Frobenius reciprocity)? ...
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33 views

How do non-semisimple modules over $\mathbb C\mathrm{GL}_n(\mathbb C)$ look like?

[Separated from another question] Can you give an example of a non-semisimple module over $\mathbb C\mathrm{GL}_n(\mathbb C)$? (Preferably one without direct summands, i.e. an indecomposable module) ...
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40 views

Orthogonal invariants of an irredubile GL-representation

Let $n\in 2\mathbb Z$ be an even number. Let $G=\operatorname{GL}_n(\mathbb{C})$ and $V_\lambda$ the irreducible complex $G$-module corresponding to the partition ...
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29 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over ...
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23 views

Question regarding surjectivity of lie algebra

Consider the general linear group $\mathrm{GL}(2,\mathbb{C})$. How would I prove that the mapping from lie algebra of this group into $\mathrm{GL}(2,\mathbb{C})$ is surjective? Is there some way I can ...
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1answer
20 views

Space of tangents of a matrix group G?

Given a smooth path A(t) through the identity in any matrix group G, how would one prove that the smooth path through any g in G, is of the form gA(t)? It is clear that gA(t) is differentiable and ...
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47 views

Irreducible representations of $SO(5)$

I am looking for irreducible representations of the group $SO(5)$ that can be described by a tensor of at most rank two. My own considerations have brought me to the conclusion that there is a ...
1
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1answer
28 views

Any explicit pictures of root datum

Consider the root datum $(X^*, \Delta,X_*, \Delta^{\vee})$ of a reductive algebraic group, where $X^*$ is the lattice of characters of a maximal torus, $X_*$ the dual lattice (given by the 1-parameter ...
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1answer
48 views

simple Lie groups

A Lie group is a group which is a smooth manifold such that the multiplication and inversion are smooth. When does a Lie group become simple? What is the difference between simple and semi-simple Lie ...
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82 views

Lie algebra $\implies$ Lie group?

Lie's third theorem says that every finite-dimensional Lie algebra g over the real numbers is associated to a Lie group G. So say I have an $r-$ parameter group of symmetries whose tangents at the ...
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66 views

Correspondence of representation theory between $\mathrm{GL}_n(\mathbb C)$ and $\mathrm U_n(\mathbb C)$

If I know something about the representation theory of the general linear group $\mathrm{GL}_n(\mathbb C)$, what can I say about the representation theory of the unitary group $\mathrm U_n(\mathbb ...
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1answer
47 views

Classifying point stabilizers for the groups associated with 3D model geometries.

For those who have the book, this question is regarding p181 in Thurston's "Three Dimensional Geometry and Topology" (although I will do my best to summarize it). Basically, there's an entire ...
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22 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
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Non-isomorphic Group Structures on a Topological Group

Which Topological Groups Have a Unique Group Structure (up to isomorphism)? I know that there are many non-isomorphic finite groups of same order, so there are many group structures possible for ...
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0answers
26 views

First integrals and invariants of one-parameter groups

Please, how do I go about showing that the first integrals of the following n-th order differential equation: $$ \frac{d^n u}{dx^n} = H(x, u^{n-1})~~ $$ on $M\subset X \times U ...
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3answers
141 views

Lie group, Lie algebra and surjectivity

Let $G$ be a connected Lie group. If the Lie algebra $\mathfrak{g}$ is commutative, is the exponential mapping surjective? If not, do we at least have that $G$ is abelian? Any counter-examples as I ...
3
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1answer
45 views

Unique Square Root Neighbourhood in Topological Group

For a Lie Group $\mathfrak{G}$ and any neighbourhood $\mathcal{V}\subset\mathfrak{G}$ of the identity $\mathrm{id}\in\mathfrak{G}$, $\exists$ neighbourhood $\mathcal{U}\subset\mathcal{V}$ of ...
3
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1answer
169 views

Baker–Campbell–Hausdorff formula for [exp(x),exp(y)]

Can someone provide a explicit (the first priority with leading orders, then the secondary consider as complete as possible, or) expansion like Baker–Campbell–Hausdorff formula for the commutator: ...
2
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2answers
88 views

Proving a submanifold of $SL_2(\mathbb{R})$

I already showed that $SL_2(\mathbb{R})$ is a 3-dimensional manifold. Now I want to show that the subspace $E$ of symmetric matrices whose eigenvalues are positive in $ SL_2( \mathbb{R})$ is a ...
6
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1answer
93 views

How to obtain a diffeomorphism between $\mathrm{SL}(2,\mathbb{R})$ and $ (\mathbb{C}\!\smallsetminus\!\{0\})\times\mathbb R$?

Could someone please give me a tip on how to show that the map $\mathrm{SL}(2,\mathbb{R}) \to (\mathbb{C}\setminus\{0\})\times\mathbb{R}$ \begin{equation}\begin{pmatrix} a & b\\ c & d ...
2
votes
2answers
65 views

Understanding that $GL_n(\mathbb{R})$ has two connected components

I am trying to understand the proof of the theorem: $GL_n(\mathbb{R})$ has two components. The proof says that The group of matrices with positive and negative determinant, ...
1
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1answer
30 views

Question on lifting the Weyl group into the group of inner automorphisms of $\mathfrak{g}$

I'm looking for some clarification of a statement that I found in Kac and Peterson's paper (112 realizations of the basic representation of the loop group of $E_8$). Let $\mathfrak{g}$ be a simple ...
2
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1answer
46 views

Computing Lie algebra of a subgroup

I will like to know how does one compute the Lie algebra of an abstractly given subgroup of a Lie group? Specifically, let $G = \mathrm{SO} ( n + 1, 1 )$ and consider the flow $$ g_t = ...
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63 views

Proof of Horn theorem with moment map

Please look at this problem: Let $\mathcal{H}$ be the space of $(n,n)$ hermitian matrix. $\phi:\begin{align*} &\mathcal{H} \to \mathfrak{u}(n):=Lie(U(n)) \\&A \mapsto iA \end{align*}$ ...
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32 views

Integration on associated vector bundle

Let $G$ be a compact lie group and $\mathfrak{g}$ be its Lie algebra then we can construct the integral on $G\times \mathfrak{g}$ by $$\int_G\int_{\mathfrak{g}}f(x,Y)dxdY$$ Where $x\in G$ and $Y\in ...
2
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0answers
35 views

Real-valued Irreducible Representations of Lie Groups

I'm interested in the real-valued irreducible representations of a number of Lie groups. For concreteness I'll use the group $M(2)$ of Euclidean motions, which can be parameterized as follows: $$ g(t, ...
2
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1answer
196 views

How do I represent such a transformation?

Let's say I have a 2d rectangle defined by $ [0,x_0] \times [0,y_0]$. Now lets say I cut out the middle rectangle $[\frac{1}{3} x_0, \frac{2}{3} x_0] \times [\frac{1}{3} y_0, \frac{2}{3} y_0]$. Now ...
2
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0answers
29 views

Existence of maximal tori in infinite-dimensional Lie groups

Reading about Lie groups and maximal tori I came up with a lemma that states that any Lie group $G$ has maximal tori. The proof goes like this: firstly, it is proven that if $H \subset G$ is a proper ...
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1answer
21 views

Is the radical of a Lie algebra preserved by any of its derivations?

Let $\mathfrak{g}$ be a finite dimensional complex Lie algebra. A derivation $D: \mathfrak{g} \rightarrow \mathfrak{g}$ is $\mathbb{C}$-linear map $\mathfrak{g} \rightarrow \mathfrak{g}$ such that for ...
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1answer
58 views

How to find Casimir Operators and their degree.

Consider the quite general problem of computing all Casimir Operators of a given Lie Algebra $\mathfrak{g}$. How does one proceed, in general? And how is possible to compute the degree of a given ...
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1answer
41 views

Question related to tangent space of $U(n)$ at a matrix $g\in U(n)$

I was working on a homework problem that involved showing that the map $f:U(n)\rightarrow S^1,g\mapsto det(g)$ is a submersion (which is given here) And the following question emerged: Given $g\in ...
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26 views

Why $\widehat{G^{\mathbb{C}}}$ can be identified with the space of highest weights

Let $G$ be a compact connected Lie group and $G^{\mathbb{C}}$ be the complexification of Lie group $G$ and we denote $\widehat{G^{\mathbb{C}}}$ the set of isomorphism classes of irreducible rational ...
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1answer
61 views

Is there a Relationship between Quantum Groups and Lie Groups?

I know that the Lie Group is all about continuous transformation groups. I know that the quantum group denotes various kinds of noncommutative algebra with additional structure. Transformation group ...
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24 views

How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
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1answer
64 views

Tangent space of $\mathfrak{ so}(3)$ Lie algebra

Very basic question and the terminology makes it difficult to find a reference. I just know the basics of differential geometry but my question is simple. Is the tangent space at the point ...
2
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2answers
101 views

Unitary group and unit circle

Let $U(n)$ denote the group of complex unitary matrices, let $S^1$ be the unit circle in the complex plane. Then the map $$f:U(n)\to S^1,\quad f(A)=det(A)$$ is a group homomorphism and a submersion. ...
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22 views

Translation Group

We know that the Poincare group, $P(3,1)$ has $10$ generators. $6$ of them are the generators of the Lorentz group, $O(3,1)$ and $4$ of them are the generators of the four dimensional translation ...
2
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1answer
38 views

$SO(n)$ is a deformation retract of $SO(n,\mathbb{C})$

Does anyone know how to prove the fact that $SO(n)$ is a deformation retract of $SO(n,\mathbb{C})$? Here $SO(n,\mathbb{C}):= \{ A\in M_{n\times n}(\mathbb{C}) | A \cdot A^T = Id \} $ and ...
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1answer
38 views

What are the classical compact connected lie groups?

I think they are $SO(n),SU(n),Sp(n)$, and $Spin(n)$ but I'm not sure. Any help would be appreciated.
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1answer
68 views

A question about differential forms on Lie groups

Let $G$ is a Lie group and $\mathfrak{g}$ is its Lie algebra and $\mathfrak{g}^{\mathbb{C}}$ be its complexification. Also assume that $\mathfrak{h}\subset \mathfrak{g}^{\mathbb{C}}$ be its ...
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33 views

Maximal tori in lie groups?

How would one prove that a maximal torus in a lie group is a maximal abelian subgroup? For example, in the specific cases of SO(n) or SU(n). I know that the maximal torus of SO(2n) and SO(2n+1) is Tn ...