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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4
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2answers
81 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
votes
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 ...
0
votes
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 ...
0
votes
0answers
18 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 ...
0
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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 $K$.
5
votes
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
votes
1answer
52 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
votes
1answer
38 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 ...
0
votes
0answers
38 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 ...
1
vote
0answers
38 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 ...
1
vote
2answers
64 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
votes
0answers
34 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
96 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 ...
0
votes
0answers
24 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
votes
0answers
30 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? ...
19
votes
0answers
172 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 ...
0
votes
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
votes
0answers
40 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 $\mathrm{Pin}(n)$. ...
2
votes
0answers
28 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) | ...
0
votes
1answer
39 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 ...
1
vote
0answers
26 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
votes
3answers
46 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 ...
1
vote
2answers
56 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 $z$ ...
0
votes
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
votes
1answer
21 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 ...
-1
votes
1answer
36 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 $$ ...
1
vote
0answers
38 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 ...
1
vote
0answers
34 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
vote
1answer
48 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
votes
1answer
112 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. ...
10
votes
0answers
55 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 ...
1
vote
1answer
58 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 ...
0
votes
0answers
27 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
votes
0answers
39 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 ...
1
vote
0answers
39 views

Maximal tori in Lie vs algebraic groups

If $G$ is a Lie group, we define a maximal [Lie] torus in $G$ to be a maximal connected compact abelian Lie subgroup of $G$. These guys correspond to Cartan subalgebras of $\mathfrak{g}=Lie(G)$. If ...
0
votes
0answers
26 views

Left invariant vector field question

This question is an exercise from Jeff Lee's Manifolds and Differential Geometry Let $\tilde{X}$ be the left invariant vector field corresponding to $X\in L(V,V)$ (That is, $\tilde{X}_g=(dL_g)_e X$). ...
1
vote
0answers
25 views

Lie Groups map question

This is a question about Exercise 5.59 in Jeff Lee's Manifolds and Differential Geometry. He writes For a fixed $A\in GL(V)$, the map $L_A:GL(V)\rightarrow GL(V)$ given by $A\mapsto A\circ B$ has ...
1
vote
1answer
21 views

Proving that on a Lie group $G$ the space of left-invariant vector fields is isomorphic to $T_e G$

Let $G$ be a Lie Group. Given a vector $v\in T_e G$, we define the left-invariant vector field $L^v$ on $G$ by $$L^v(g)=L^v|_g=(dL_g)_e v.$$ I want to show that $v\mapsto L^v$ is a linear ...
1
vote
0answers
39 views

Is a matrix group with a continuum of generators always a Lie group?

Suppose that you have a set of matrices $S \subseteq SL(2,\mathbb{C})$, and furthermore you know that $S$ contains a continuum of matrices. (In particular, assume that there is some one-parameter ...
2
votes
2answers
64 views

how to find an integral curve in Lie group?

Given a Lie group $G$, $e$ is its identity element and $g$ is one element of $G$. I want do find a curve $\gamma(t)$ that satisfies these conditions: 1) passes $g$ and $e$, that is ...
2
votes
0answers
19 views

The product of Weyl groups is the Weyl group of the product

Let $G$ and $H$ be compact, connected Lie groups. Write $W(-)$ for the Weyl group. Then it is not hard to see that $$W(G \times H) \cong W(G) \times W(H).$$ Where can I find a published statement of ...
2
votes
1answer
57 views

Tangent space and differential forms of quotient groups

I have difficulty understanding the following argument because it seems that many details are swept under the rug and I am looking for a detailed rigorous exposition on this (I'll try to make clear ...
2
votes
1answer
66 views

how to extend a vector at $e$ of a Lie group to a left invariant vector field?

I am reading some books about Lie group and Lie algebra. Denote the set of all the left invariant vector fields as $\mathfrak{X}_L$, and the tangent space at $e$ of $G$ as $T_eG$. They say that the ...
0
votes
0answers
34 views

Chevalley group

Let $L=\mathfrak sl_6$ be the special linear Lie algebra over $\mathbb C$ and let $S=\{\alpha_1, \alpha_2, \alpha_3,\alpha_4,\alpha_5\}$ be a set of simple roots. Then the set of positive roots are ...
0
votes
0answers
45 views

Fundamental representation of $O(3)$

I want to check if the fundamental representation of $O(3)$ is irreducible on $\mathbb{R}^3$ and $\mathbb{C}^3$. I want to use isomorphism properties. I know this relation exists $$ ...
4
votes
0answers
55 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
1
vote
1answer
24 views

Tangent space of closed subgroup of Lie group upon action

I am trying to show the below statement which I very strongly feel should be "obviously correct", but I think I am missing the easy way to see this. Let $G$ be a Lie gorup and $H$ a closed subgroup. ...
4
votes
0answers
46 views

Geodesics on Homogeneous Spaces

Consider a homogeneous space $G/\text{Stab}_p \cong M$ where $G$ is a compact Lie group active transitively on $M$ (a compact manifold). If $F$ is a Finsler Metric on $G$ which pushes forward ...
0
votes
0answers
16 views

Reference on the Crystallographic restriction theorem and some related results

Firstly I precise that I am working on $\mathbb C$ the plan of complex numbers I have some result for which I look for references (mathematical books or articles) where the reader can find their ...
2
votes
1answer
37 views

Lie Subgroup Example - Explanation?

I'm currently working through Jeff Lee's 'Manifolds and Differential Geometry'. He defines a Lie Subgroup, $H$, to be an abstract subgroup of a Lie Group $G$, such that the inclusion map ...