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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A question about a proof in Lang's $SL_2(\mathbb{R})$

The following is a lemma in Lang's book $SL_2(\mathbb{R})$. It's the last line of the proof that I don't understand. Let $G=SL_2(\mathbb{R})$ , $E$ a Banach space, and let $\pi$ be an irreducible ...
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26 views

İf x is diagonalizable then ad(x) is also diagonalizable

I start to study lie algebras from K. Erdmann, Mark J. Wildon-Introduction to Lie Algebras and i try to solve question below but actually i can't see .How can i start ? Give me hint please Let ...
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14 views

Isomorphism of Principal Bundles with structure groupoid

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Suppose that $\pi:P\rightarrow B$ is a $\mathcal{G}$-principal bundle and let $(h_s,g_s):(P,B)\rightarrow (P,B); s\in [0,1]$, be a homotopy ...
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1answer
18 views

Nested commutators that don't vanish

So I've been reading up on Lie Groups and Lie Algebras and the Baker-Campbell-Hausdorff formula. I understand how the formula works and that most of the time the nested commutators vanish at a certain ...
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79 views

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $?

$S^2=SO(3)/SO(2)$. Does this mean that $S^2 = SU(2)/U(1) $ since $SO(3) \approx SU(2)$ and $SO(2) \approx U(1)$? Is there some more generic rule on how to relate $S^{n-1} = SO(n)/SO(n-1)$ to the ...
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26 views

Why for simple roots in Lie algebras the master formula reduces to one integer?

The master formula for two generic weights (roots) is $$ 2 \frac{\vec{a} \cdot \vec{b} }{\vec{a} \cdot \vec{a} }=q-p $$ but if we require that the roots are simple then this reduces to $$ 2 ...
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7 views

Inverting the the decomposition of tensor product representation into irreps

Suppose I have two unitary representations $U_V, U_W$ of a group $G$ on finite-dimensional vector spaces $V$ and $W$. I know that the tensor product representation $U_V\otimes U_W$ need not be ...
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1answer
25 views

Clarification on notation of “left invariant fields” (Lie groups)

In these notes in Definition 1.4 we learn that A vector field $X$ on a Lie group $G$ is called left invariant if $d(L_g)_h(X(h))=X(g(h))$ for all $g,h \in G$, or for short $(L_g)_*(X)=X$. where ...
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2answers
56 views

Normal Subgroups of $SU(n)$

I was wondering if there is any classification for normal subgroups of $SU(n)$? In particular, I think that the answer is no for $n = 2$ by looking at the covering map onto $SO(3)$, but I was curious ...
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30 views

What is the differential of left translation?

Let $G$ be a Lie group, $g\in G$ and $L_g$ be left translation by $g$. I want to compute the differential $dL_g|_0$ of $L_g$ at $0$. Attempt: Let $v\in T_0G$ be a tangent vector at $0$. Let ...
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1answer
29 views

Diffeomorphism in Lie Group

$G$ is a Lie group and consider $L_{g}: G \rightarrow G$ ($L_g(h)=gh$). What i need to show that $L_{g}$ is diffeomorphism. Is it something obvious? Can someone explain it to me?
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5 views

Proof that ideal in Lie ring can be represented as sum of 2 Lie subrings.

Let $K$ be a commutative ring and $m≥3$. Let $L(m,K)$ be a Lie subring of matrices with coefficients from ring $K$ that contains matrices with null traces, $L(m,K)={(a_{ij})∈M_m(K)|\sum\limits_{i = ...
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3answers
245 views

Lie group. How is manifold defined?

If I have the Lie group $SL(2,\mathbb{R})$. Then how is the manifold structure on this algebraic group defined, could anybody explain this to me? I mean this is the group of matrices that determinant ...
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1answer
23 views

Quaternions and Lie Groups

It's obvious that Quaternions, (denote by $H$, without $0$) form a non-commutative group under multiplication ( it's even non commutative division algebra ). It seems that it's also obvious that ...
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1answer
37 views

Are unipotent algebraic groups connected?

Is a unipotent algebraic group over a field of characteristic zero always connected?. As far as I know, every unipotent algebraic group over field of characteristic zero is isomorphic to a closed ...
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10 views

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$?

How to prove $e^{π(ω)} = cos(ω)I_3 + (1 − cos(ω))ωˆ ⊗ ωˆ + sin(ω)π(ωˆ )$, where $ω = |ω|, ωˆ =ω/|ω|.$ My Attempt: In my understanding, $\pi (w)$ is an element of the lie algebra, which is a ...
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21 views

Why is left-invariant vector fields needed to construct a Lie algebra from a Lie group?

Since the set of all vector fields $V$ on a Lie group $G$ forms a vector space, one can impose algebraic structure (a Lie algebra) by defining the bracket $[\cdot,\cdot]$ between these vector fields. ...
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1answer
38 views

Diagonalizing elements of compact lie groups

Chapter 5 of Sepanski's Compact Lie Groups starts with this paragraph: "Since a compact Lie group $G$ can be thought of as a Lie subgroup of $U(n)$, it is possible to diagonalize each $g\in G$ using ...
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43 views

Any continuous group homomorphism from $\mathbb{R}$ to $GL(n,\mathbb{C})$

Any continuous group homomorphism $\phi$ from $\mathbb{R}$ to $GL(n,\mathbb{C})$ is of the form $\phi (t)=exp(tX)$ for some $X\in M(n,\mathbb{C})$. Can anyone give hints for the proof of this fact? I ...
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1answer
27 views

Exterior derivatives involving representations

I have two questions regarding the exterior derivative of vector valued forms when representations are involved: Suppose $V$ is a vector space, $M$ a smooth manifold and $\omega$ is a $V$ valued ...
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20 views

The negative of a vector field and its flow

I have a relatively short question about vector fields. Let $G$ be a Lie Group, and $X$ a smooth vector field on it. If its flow is $\left\{\phi_t\right\}$ what is the corresponding flow for $-X$? ...
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63 views

Principal bundles, connection forms and fundamental vector fields

Suppose $\pi:P\rightarrow M$ is a principal bundle, $\omega\in \Omega^1(P;\mathfrak{g})$ is the connection one form and $\sigma(\cdot)$ is the fundamental vector field associated to some vector field ...
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2answers
35 views

Universal covering space of X x classifying space of \pi_1(X)

I am trying to learn about classifying spaces for a Lie group $G$. The question I have is the following: Suppose $X$ is a manifold and $G=\pi_1(X)$ is its fundamental group, is it true that ...
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1answer
33 views

Proof of $G\rightarrow G/H$ is a Principal H bundle

Let $G$ be a Lie group and let $H$ be a closed subgroup (not necessarily normal). Then $G$ is a principal $H$-bundle over the (left) coset space $G/H$. I could proof that the fibers are all ...
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1answer
28 views

Question on banach space over an extension of $\Bbb{Q}_p$

Let $G$ be a compact locally $\Bbb{Q}_p$ analytic group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Let $M$ be a $O[G]$ module. I was reading an article which says : ...
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32 views

A mixed state integrated over Haar measure in QFT

I think that we can write a state in Quantum Mechanics like: $$ \int dx\lambda _x \vert x\rangle\langle x\vert $$ and when we talk about QFT, we would replace integral with functional integrals: $$ ...
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41 views

Special linear lie algebra relative with Angular Momentum

Let have the special linear algebra $\operatorname{Sl}(2,\mathbb F)$ ,which is the set of $ 2 \times 2$ matrix with trace zero. I have to prove that the lie algebra $ g=\operatorname{Span}\{ ...
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1answer
36 views

Role of Group actions in Differential Geometry

This is a rather soft question, my hope is to bring some order into the stuff I would like to learn about differential geometry -- here it is: I was told over and over again that Geometry has to do ...
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38 views

Associated bundles: isomorphism between spaces of differential forms.

I think this will be an easy question for numerous people. Let $\pi:P\rightarrow M$ be a principal bundle and $\rho:G\rightarrow GL(V)$ a representation. The space of $k$ forms on $M$ with values in ...
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7 views

When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
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1answer
30 views

When is $SO^0(n,1)$ isomorphic to a complex Lie group?

The group $SO^0(3,1)$ is isomorphic to a complex Lie group, namely $PSL_2(\mathbb{C})$. Are there further examplex when $SO^0(n,1)$ isomorphic to a complex Lie group? An obvious necessary condition is ...
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1answer
32 views

How to find subalgebras of standard lie algebras

As I understand it, the symplectic Lie group $Sp(2n,\mathbb{R})$ of $2n \times 2n$ symplectic matrices is generated by the matrices in ...
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27 views

Tangent map of the special linear group

Let $G=SL(2,\mathbb{R})$ be the special linear group of $2\times 2$ linear matrices withe real entries, $g=\left( \begin{array}{cc} 2 & 1 \\ 1 & 1\end{array}\right)$, and $L_g$ be the left ...
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1answer
35 views

Is $U(2)=SU(2) \times U(1)$?

In the many textbook of standard model, i encounter the relation \begin{align} SU(2)_L \times U(1)_L = U(2)_L \end{align} Here $L$ means the left-handness, (It is a physical meaning(representation), ...
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1answer
20 views

Different definitions for semisimple Lie group

I am confused about two definitions for the notion of a semisimple Lie group i found. Lets say for simplicity i am only interested in matrix groups. In this case, do the following two object-classes ...
3
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24 views

Center of the dual of a Lie algebra

Let $\mathfrak{g}$ be a Lie algebra. Let $C \subset \mathfrak{g}^*$ be the subspace of linear forms that vanish on Lie brackets: $$C = \{\alpha \in \mathfrak{g}^*, ~ [\mathfrak{g}, \mathfrak{g}] ...
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0answers
23 views

Gauge fields and restrictions of the connection one form

I am working through some lecture notes on principal bundles and am stuck on the proof of a certain proposition. In the following, $\pi:P\rightarrow M$ is a principal bundle, $\omega$ is the ...
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1answer
23 views

Computing the matrix of rotation

Let $\gamma_P$ denote the conjugation by $P\in SU_2$. Let $P=(\cos\theta)I+(\sin\theta)A$ where $A$ is on the equator. I want to know how I compute $\gamma_P$. I know it's defined by conjugation ...
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1answer
27 views

Quotient spaces $SO(3)/SO(2)$ and $SO(3)/O(2)$

I have a question similar to this one, but that question is not answered. The question is to show that $SO(3)/SO(2)$ is isomorphic to the 2-sphere: $$ SO(3)/SO(2)\cong S^2 $$ How does one establish ...
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2answers
46 views

Confusion about Lie groups in Fulton & Harris

Near the beginning of chapter 8 (titled Lie groups and Lie algebras) authors motivate the definition of Lie algebra. I'm confused by two things in just one sentence: ($G$ is a Lie group) The ...
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2answers
74 views

Lie Groups/Exponential map identity

I have come across this identity a few times and I have absolutely no idea why it holds. $g^{-1}\exp(tX)g=\exp(t(\text{ad}_{g^{-1}}X))$ Would any one be able to explain exactly why this holds or ...
2
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1answer
13 views

Finite subgroups which are normal in a matrix Lie group

I have the following question: Let $G$ be a closed subgroup of $GL(n,\mathbb{C})$. Denote $Z(G)$ by the center of $G$. ${\bf Question}:$ Is it true that every finite normal subgroup of $G$ are ...
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1answer
27 views

Finitely Generated Matrix Group Decompositions

If I take a finite collection of n x n invertible matrices and generate a group G under matrix multiplication, is it the case that there always exists a maximal normal solvable group R from which I ...
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17 views

The commutator of a Lie algebra element with a Lie group element

Is there a way to evaluate the trace of generators of the Lie algebra and group elements? For example take $SO(N)$, with $\lbrace T^a\rbrace$ the set of generators, normalized such that ...
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15 views

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$?

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$? $H^{1}(K)$ is the first de Rham cohomology group of $K$.
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2answers
61 views

Bijective isometry which fixes origin from $\mathbb{R}^{n} \longrightarrow \mathbb{R}^{n}$ is linear

I was going through Hall's book about Lie groups.While presenting Euler groups $E(n)$ and on the way to prove that they form a matrix Lie group hee made a proposition that Every one one onto distance ...
4
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1answer
49 views

Does a left-invariant vector field on a complex Lie group preserve holomorphic functions?

Let $G$ be a (finite-dimensional) complex Lie group, and suppose $f : G \to \mathbb{C}$ is holomorphic. Let $X$ be a left-invariant vector field on $G$. Must $Xf$ be holomorphic? I think I have a ...
2
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1answer
35 views

Representations of a product of Lie groups

Let $G=G_1\times G_2$ be a product of two compact Lie groups. Is every finite dimensional irreducible representation of $G$ a tensor product of irreducible representations of $G_1$ and $G_2$? This ...
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37 views

Sections of inverse image sheaf of sheaf of sections of vector bundle

Let $\eta \colon Y \to Z$ be holomorphic map between complex manifolds, $E$ holomorphic vector bundle over $Z$, $\eta ^\ast E$ its pullback over $Y$, ${\cal O}_E$ sheaf of sections of $E$, and ...
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37 views

$SO(3)$ has a subgroup $U(1) \times U(1)$?

I am wondering - and asking you - whether there is a subgroup $U(1) \times U(1)$ of the Lie group $SO(3)$. Equivalently, I can reformulate it from a geometrical point of view: does there exist a torus ...