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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6 views

The generators of $SO(n)$ are antisymmetric, which means there are no diagonal generators and therefore rank zero for the Lie algebra?

Okay, this may be a silly question but I can't figure it out myself right now. By definition $O \in SO(n)$ fulfils $O^T O=1$ and $\det(O)=1$. For the generators of the group $ T_a \in so(n)$, this ...
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1answer
43 views

Tangent space of quotient space

Let $\pi : M \rightarrow M/G$ be the canonical projection, where $M$ is a manifold and $M/G$ is a quotient manifold. Now, what can we say about $d \pi (p) : T_pM \rightarrow T_p(M/G)$? From my ...
4
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0answers
31 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
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1answer
26 views

Lie group and stabilizer quotient

Let $G$ be a Lie group and $$G_x:=\{g \in G; Ad^*_g(x)=x\}$$ the stabilizer, where $Ad_g^*$ is the adjoint of the adjoint representation. Now, I was wondering why $G/G_x$ has a manifold structure. ...
2
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0answers
41 views

Dual space isomorphism and the dual representation

Let $V$ be a complex finite-dimensional vector space. Then there always exists an isomorphism $V \simeq V^*$, where $V^*$ is the dual space. The isomorphism can be fixed by choosing a non-degenerate ...
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1answer
15 views

Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
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1answer
47 views

Orbits form a manifold?

A prominent example are the coadjoint orbits $O_x = \{Ad_u^*(x);u \in G\}$ where $x \in \mathfrak{g}$ and $G$ a Lie group with Adjoint map $Ad.$ Could anybody give me an easy argument why $O_x$ is a ...
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34 views

Exponential map only for matrix Lie algebras?

Recently, I stumbled over some proofs in Lie algebra theory and noticed that they often use the notion of an exponential map $e^{t \zeta}$ for $\zeta \in \mathfrak{g}$ such that $e^{t \zeta} \in G$ ...
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1answer
15 views

Tangent vectors to coadjoint orbits

Let $O_x:=\{Ad^*_g (x); g \in G\}$ be the orbit of $x \in \mathfrak{g}^*$ and $Ad$ the adjoint map. Now take $\xi \in \mathfrak{g}$ then $g(t):=Ad^*_{e^{t \xi}}(x)$ defines a map $g: I \rightarrow ...
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1answer
23 views

Lie algebra of affine linear maps

Let $G$ be the Lie group of affine transformations, $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ We can represent these maps as matrices $$\begin{pmatrix} A & b \\ 0 & 1 ...
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1answer
48 views

topic between algebra and geometry [on hold]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
2
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33 views

Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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13 views

Extending orthogonal representation of $SU_2$ to $U_2$

Let $\phi: SU_2 \rightarrow SO_3(\mathbb{R})$ be the orthogonal representation of $SU_2$, obtained by letting $SU_2$ act on the three-dimensional vector space of trace-zero skew-hermitian matrices. ...
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17 views

Lie algebra affine transformations [duplicate]

Let $G$ be the Lie group of affine transformations $$\{x \mapsto Ax+b,A \in GL(n), b \in \mathbb{R}^n\}.$$ Then we can represent these maps as matrices $\begin{pmatrix} A & b \\ 0 & 1 ...
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14 views

Adjoint and coadjoint orbits

I just read that for the Lie algebras $\mathfrak{gl}(N),\mathfrak{sl}(N),\mathfrak{so}(N),\mathfrak{sp}(2N)$ the adjoint and coadjoint orbits coincide. Now, the adjoint orbits are $O_{\xi} = ...
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1answer
45 views

Killing form - strange definition

I was just reading about Killing forms. In my opinion, the definition of these forms is quite strange. I mean why would one define $B(X,Y) = \mathrm{tr} (\mathrm{ad} (X) \circ \mathrm{ad} (Y))$? I ...
3
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1answer
27 views

Adjoint representation is Lie algebra homomorphism

Let $T_g:=L_g R_{g^{-1}}: G \rightarrow G$ be the standard automorphism of a Lie algebra, then $Ad_g:=DT_g(e): \mathfrak{g} \rightarrow \mathfrak{g} $is called the adjoint representation. Now, I want ...
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0answers
8 views

Stabilizer subgroup in adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
3
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1answer
33 views

Level set as the orbit of the action of a Lie Group?

I'm wondering the following. Given a smooth function $f:\mathbb R^n\rightarrow \mathbb R^m$ with $m<n$ and level sets $\mathcal O(y)=\{x\in\mathbb R^n| f(x)=y \}$. What are the conditions on $f$ ...
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19 views

Why $exp(0_{T_eG})=e$, where $exp$ is the exponential map of a Lie group?

I wonder if this fact is true: I consider the exponential map of a Lie group $G$. $$exp: \mathfrak{g} \rightarrow G.$$ Is it true that $exp(0_{T_eG})=e$, where $e$ is the identity element of $G$? ...
2
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3answers
46 views

Group actions on manifolds - exponential map

Let $M$ be a smooth manifold. Suppose $K$ is a Lie group (with Lie algebra $\mathfrak{k}$) acting EDIT: TRANSITIVELY on $M$ from the left and $G$ is a Lie group (with Lie algebra $\mathfrak{g}$) ...
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0answers
11 views

Transitive group actions on Principal bundles

I have a question in regards to page 107 of Kobayashi & Nomizu's Foundations of Differential Geometry Setup: Let $\pi:P\rightarrow M$ be a principal bundle with structure group $G$ whose Lie ...
4
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29 views

Understanding $G_2$ inside Spin(7)?

This is a rather embarrassing question, so please let me know of any duplication and I will happily remove it. I am seeking to understand the $\mathbb Q$-split form of the algebraic group $G_2$, and ...
2
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2answers
40 views

Connected Lie group is second countable?

I know this is true from various sources, unfortunately none of them give the full proof. I already have a start: Let $G$ be connected Lie Group. Choose $K$ to be any compact neighbourhood of the ...
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1answer
27 views

Determine the exponential map of the direct product of two Lie groups.

I know that the direct product of two Lie groups $G$ and $H$ is a Lie group. Knowing the exponential map of $G$ and $H$, I would like to find an expression for the exponential map of $G \times H$. ...
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0answers
15 views

Branching rules without previous knowledge of the projection matrix?

Given a representation $R$ of some group $G$ one can find in many books and papers (e.g. page 96ff here) the decomposition under certain subgroups: $$ R= R_1 + R_2 + \ldots$$ This is often called a ...
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0answers
14 views

Inner Automorphism of Lie algebras in Terms of Roots and Weights?

An automorphism is a homomorphism of a group $G$ onto itself. For Lie groups this induces a Lie algebra $g$ automorphism, i.e. a map of the Lie alegbra onto itself that preserves the Lie bracket. An ...
4
votes
1answer
86 views

How to see that SL(2,C) is simply connected?

I started reading about Lie groups and right now I'm trying understand why $SL(2,\mathbb{C})$ is simply connected. I have shown that $SU(2)$, being diffeomorphic to $S^3$, is simply connected. So my ...
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1answer
33 views

Do we have $\mathbb{C}[SL_n] = \oplus_{\lambda, \text{ht}(\lambda)\leq n} V_{\lambda} $?

The coordinate algebra $\mathbb{C}[SL_n]=\mathbb{C}[x_{ij}: i, j \in \{1, \ldots, n\}]/(\det(x_{ij}) - 1)$ is a representation of $SL_n$: $(g'.f)(g)=f(g'^T g)$. Let $V_{\lambda} = \langle e_T : T ...
1
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1answer
40 views

How to understand that minors are matrix elements in fundamental representations of $SL_n$?

In the video, Lecture 3 of June 14, 49:00-53:00, it is said that "minors are matrix elements in fundamental representations of $SL_n$". What are fundamental representations of $SL_n$? How to ...
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0answers
66 views

Orbit of a Weight Vector?

Given some element $\phi$ of a representation $R$ of a group $G$, the orbit $G(\phi)$ of $\phi$ is defined as the set $g \phi \ \forall \ g \in G$. We can write every element of a given ...
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27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
1
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1answer
27 views

Automorphism group of a lie algebra as a lie subgroup of $GL(\frak g)$

Let $G$ be a lie group with lie algebra $\frak{g}$. Let $Aut(\frak g)$ be the automorphism group of $\frak{g}$. Its clear to me that $Aut(\frak{g})$ $\subset GL(\frak{g})$ since any automorphism of ...
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1answer
34 views

Algebraic groups with no small subgroups

I have read in many textbooks proofs that any Lie group has no small subgroups, that is, there is an open neighborhood of the unity element containing no nontrivial subgroups. In particular, ...
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1answer
45 views

How to understand the definition of Killing form?

Define the matrix commutator $\text{ad}_X$ as $$\text{ad}_XY=[X,Y]=XY-YX$$ where $X,Y\in\mathfrak{g}$ and $\mathfrak{g}$ is the Lie algebra associated to Lie group $G$. Then on Lie group $G$, the ...
1
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2answers
67 views

Solving $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$

Given an arbitrary $A \in \mathfrak{su}(n)$, is it always possible to solve for $U,V \in SU(n)$ and $\lambda \in \mathbb{C}$ such that $\lambda U^{\dagger}V -\bar{\lambda} V^{\dagger}U = A$? Update: ...
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0answers
22 views

Elements of SO(n) is block-diagonalizable

I am not able to show that elements of SO(n) are conjugate to a block-diagonal matrix with 2x2 blocs that are rotation matrices, and a 1x1 bloc 1 if n is odd. Can someone help me please?
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1answer
45 views

Stabilizer subgroup of adjoint action

Given $b \in \mathfrak{su}(n)$, how can I find the stabilizer $\text{stab}(b)$ for the adjoint action of $SU(n)$ on $\mathfrak{su}(n)$ given by $Ad_U(b) = UbU^{\dagger}$ without using coordinates? The ...
1
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2answers
54 views

Representation of $sl(2,R)$.

I am interested in the unique (up to isomorphism) $5$-dimensional representation of the Lie algebra $sl(2,R)$. I understand that one can choose the module $V_4 = ...
2
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0answers
24 views

List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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1answer
27 views

Transitive Lie group actions and surjectivity of maps

I am reading a paper at the moment and I have come across two statements which I want to understand. Here is the setup: Suppose that $G$ is a Lie group which acts on a manifold $E$ differentiably and ...
1
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0answers
19 views

Ref. Request — Non-Transitive Lie Group Actions, Applications to Orbifolds/Groupoids

I'm working on a problem where I have a (highly) non-transitive Lie group action on a manifold, and I am trying to deduce the geometric structure of the quotient space. I've been looking at some ...
1
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1answer
27 views

Explicit Representation of the SU(N) Simple Roots in with redundant coefficents?

Commonly the simple roots for $SU(n)$ groups are given as $n$ dimensional vectors, although root-space is $n-1$ dimensional. The $SU(n)$ Wikipedia article explains: Here, we use n redundant ...
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15 views

Questions about distributions on $l$-spaces.

I am reading the paper. I have some about distributions on $l$-spaces. On page 7, Section 1.7. Let $X$ be an $l$-space. Locally constant complex-valued functions on $X$ with compact support are ...
1
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2answers
62 views

Prove that General Linear Group is a topological subgroup.

First of all for $\mathbb{R}$ in my book it is written that: "$GL(n,\mathbb{R})$ is an open subset of euclidian $n^2$-space and that is the topology is given. Matrix multiplication is given by ...
9
votes
1answer
124 views

gradient flow on $SU(n)$

Define the following cost functions $f_1, f_2 :SU(n) \rightarrow \mathbb{R}$ by $f_1(U) = Re \left( \text{Tr}\left(G^{\dagger} U \right) \right)$ and $f_2(U) = \left| \left( \text{Tr}\left(G^{\dagger} ...
2
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1answer
41 views

Does $GL(n,\mathbb{C})$ inject into $GL^+(2n, \mathbb{R})$ for all $n$?

I know that $GL(n, \mathbb{C}) \hookrightarrow GL^+(2n,\mathbb{R})$ for n = 1 and 2. I just did these with pen and paper (and mathematica). But is this statement true in general? What is a proof of ...
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0answers
9 views

Weights system corresponding to reflected Dynkin diagram?

Given a set of weights corresponding to the $SO(10) Dynkin diagram How can I transform these weights into weights that correspond to the Dynkin diagram ?
2
votes
1answer
29 views

Calculating the Lie algebra representation of the regular representation on subspace of functions on $\mathbb R$.

Let $G = \mathbb R$ and let $\pi$ be the regular representation of $G$ on $L^2(\mathbb R)$, that is, $\pi(g)(f)(x) = f(x-g)$ for $g \in G$. Let $V = \{f \in \mathcal C_c^\infty | supp f \subseteq ...
5
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1answer
67 views

Invariant subspaces of Lie group vs invariant subspaces of Lie algebra

I am starting to study infinite-dimensional representations of Lie groups and I am wondering about the following: Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$ and with a ...