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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32 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 : ...
2
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38 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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47 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
47 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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1answer
55 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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11 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 ...
2
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1answer
32 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
36 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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28 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
40 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), ...
3
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1answer
25 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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27 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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2answers
67 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
24 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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2answers
102 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
49 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 ...
4
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2answers
90 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
14 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
31 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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0answers
24 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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0answers
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
62 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
56 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 ...
3
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1answer
43 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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39 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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0answers
41 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 ...
2
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2answers
70 views

Do these vector fields span an integrable distribution?

Let $X_i$, $1 \le 1 \le n$, be smooth vector fields on the open set $U \subset \mathbb{R}^n$, which are linearly independent at each point. Set $[X_i, X_j] = \sum_{k=1}^n c_{ij}^kX_k$. Suppose that ...
2
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0answers
37 views

$GL(V)$ representations and Schur modules.

Let $W$ be a fine dimensional complex vector space of dimension $n$ and $L_{\lambda}W$ the Schur module associate to the partition $\lambda=(\lambda_1, \cdots,\lambda_{n-1})$, where $\sum_i ...
4
votes
0answers
98 views

Identifying the cotangent bundle of the flag variety

Suppose $G$ is a Lie group (or I guess a linear algebraic group), $P \subset G$ a Lie subgroup with Lie algebras $\mathfrak{g}$ and $\mathfrak{p}$ respectively. In Chriss and Ginzburg's book ...
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28 views

Moment map and Hamiltonian

Take the manifold $M$ to be $M=\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ (hence $x\in M$ is given by $x=(p,q)$ with $p$ and $q$ three dimensional vectors) and take the possion bracket on $M$ given ...
2
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0answers
39 views

Exercise 11, chapter 2 in Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

I am reading the book: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall. I am stuck at the following exercise: exercise 11, chapter 2 . Can you help me? ...
21
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191 views

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc… & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...
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1answer
61 views

Could one modify S^1=U(1) to move the Earth under group multiplication?

The easiest way (to my humble understanding) to think about the group $\Bbb S^1$ is to consider the set of all complex numbers $z=a+bi$, for which $a^2 + b^2=1$ and use multiplicative operation to ...
4
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41 views

Homotopic properties of the Spin group from geometric algebra

There are two possible ways to define the $\mathrm{Spin}(n)$ group of Euclidean $n$-space from $\mathrm{Pin}(n)$. First is that $\mathrm{Spin}(n)$ is the identity component of $\mathrm{Pin}(n)$. ...
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29 views

Inclusion of subring in Ideal

Let $K$ be a commutative ring with mutpilicative identity and $m \ge 3$. Let $L(m,K)$ be a subring of Lie ring of matrices with coefficients from $K$ and traces = $0$: $ \{ (a_{ij}) \in M_m (K) | ...
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1answer
40 views

Embedding so(n) in su(n)

Is there any way of embedding $\mathfrak{so}(n)$ into $\mathfrak{su}(n)$ for any $n$ other than picking the antisymmetric matrices of $\mathfrak{su}(n)$? I know that for small $n$ one can use ...
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0answers
27 views

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

Let $K$ be a commutative ring and $m \ge 3$. 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}) \in M_m(K) | ...
2
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3answers
48 views

Showing that order of $SL_2(Z_3)$ is 24

I am having some trouble proving that order of $SL_2(Z_3)$ is 24, First I know that the number of elements in $M_2(Z_3)$ is 81 because we have four entries and for each entry we have 3 different ...
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2answers
61 views

How do you prove Euler's angle formula?

Euler's rotation theorem states that any rotation in $\mathbb{R}^3$ can be described by $3$ parameters. Theorem Any rotation of the $xyz$-space is the composition of a rotation around the $z$ ...
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1answer
25 views

Lie Groups of bigger cardinality

A Lie Group is to be a group that is also a manifold, and of course a manifold is a second countable Hausdorff space. Now the maximum cardinality for a second countable (Hausdorff)space is ...
2
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1answer
27 views

matrix Lie group embedding as a manifold

Given a Lie group of matrices, and suppose for simplicity that it is globally generated through exponential map from its Lie algebra on a element. Is there a canonical way to embed it into ...
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1answer
38 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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0answers
41 views

Clifford algebra and Spin group of 4-dimensional Euclidean space

I’m seeking for a straightforward construction of well-known $\mathrm{Spin}(4) = \mathrm{Spin}(3)\times\mathrm{Spin}(3)$ isomorphism using geometric algebra-based definition of “Spin”, without ...
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0answers
41 views

Induced Lie Algebra Representation, Left invariant vector fields and more…

The following is an excerpt from a proof in John Lee's Introduction to Smooth Manifolds I am struggling to understand. I would appreciate if someone was able to help me with whatever it is I am ...
1
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2answers
60 views

Young tableaux of $8\otimes 8$ in $SU(3)$

In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$: I am unsure of all the cancellations. Let us number the canceled tableaus increasing ...
3
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1answer
120 views

Guided study of Lie theory

I'm willing to become a mathematical physicist and, as such, it is mandatory that I become better acquainted with some essential mathematical theories, Lie groups and algebras being one of them. ...
15
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1answer
135 views

When do two matrices have the same exponential?

Let $A$ and $B$ be $n\times n$ hermitean matrices. When do we have $e^{iA}=e^{iB}$? Can we somehow classify those pairs of matrices that have the same exponential? Here are some observations that I ...
2
votes
1answer
62 views

kernel of the exponential map is isomorphic to the singular homology group

Let $G$ be an algebraic torus or an abelian variety over the complex numbers. Then $G(\mathbb{C})$ is a complex Lie group. Is it true that we have the following exact sequence ? $ 0 \to ...
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28 views

How to prove this statement for a Lie algebra?

Let $\mathfrak{L}$ be a semi-simple Lie algebra. Let $X^A$ be the elements of this algebra with $A=1, \ldots, N$. The bracket is given by $$[X^A, X^B]=if_{\,\,\,\,\, C}^{AB}X^C$$ where ...
2
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43 views

Lie algebras of GL(n,R) and differentials

This question comes from a proof in John Lee's Introduction to Smooth Manifolds, page 194. I am questioning a line in the proof of the following proposition: The composition of the maps ...