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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7
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1answer
40 views

Can the Lie group structure be recovered from the geometry of an invariant metric?

Is there a manifold $M$ with two non isomorphic Lie group structures $G_{1}$ and $G_{2}$, and two left invariant metrics $g_{1}$ and $g_{2}$, respectively such that $(M,g_{1})$ is isometric to ...
1
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0answers
31 views

Can one speak of a threefold (or other) symmetry of SU(3) and the Gell-Mann matrices?

A torus has a rotation symmetry along the axis, a sphere has "spherical" symmetry under rigid motions; doesn't SU(3) also have a symmetry? The Gell-Mann matrices ( see ...
3
votes
0answers
26 views

How do roots act on weights?

In Lie theory it's possible to compute things very explicit using tensor methods. For example, we can use an explicit matrix for each generator $T^a$ and compute the "action" of this generator on an ...
0
votes
0answers
43 views

Difference between infinitesimal motion and finite motion

I was reading an article about back ground of Killing's work by Thomas Hawkins from Historia mathematica 1980.In it Hawkin's says that,Killing was trying to generalise all types of space ...
1
vote
0answers
38 views

Problem of Arnold's book and covering spaces

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point. Especially, I am talking about the part that ...
1
vote
1answer
54 views

Show Lie bracket left invariant

I want to prove from the definition that the Lie bracket $[X,Y]$ of two left-invariant vector fields $X,Y: G \rightarrow TG$ where $G$ is a Lie group is again left-invariant. Left-invariance ...
0
votes
1answer
35 views

Derivative of a group action

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ where $g: I \rightarrow G$ and $d: I \rightarrow M$ are ...
2
votes
1answer
29 views

Lie algebra and Lie group cohomology, reference request

Who can give me a good reference (better if introductory/motivated) about Lie group cohomology, Lie algebra cohomology, and the link between the two? Thanks.
0
votes
0answers
19 views

Differential of square map in Lie group away from identity

I've looked everywhere for this specific example but couldn't find it. Probably simple but I only need it for a small application and my Lie theory is very rusty. Let $G$ be an arbitrary Lie group ...
0
votes
1answer
36 views

equivalence of Lie group and Lie algebra intertwiner

I encountered this problem while working on my research. Let $G$ be a Lie group, and consider an intertwiner of the complex representations (possibly infinite-dimensional) $$ \pi:G\rightarrow ...
0
votes
0answers
29 views

Derivative group action [duplicate]

Let $\phi : G \times M \rightarrow M$ be a group action on a smooth manifold $M$ and Lie group $G$. Then we define $$f(t):=\phi(g(t),d(t)).$$ Now I'd say: $$f'(t) = D\phi(g(t),d(t))(g'(t),d'(t)).$$ ...
0
votes
1answer
26 views

$SL(n,\mathbb R)$ diffeomorphic to $SO(n) \times \mathbb R^{n(n+1)/2-1}$?

Question : How to show $SL(n,\mathbb R)$ diffeomorphic to $ SO(n) \times \mathbb R^{n(n+1)/2-1}$? Also, how to show $SL(n,\mathbb C)$ diffeomorphic to $ SU(n) \times \mathbb R^{n^2-1}$? I have ...
0
votes
0answers
29 views

Is the product of group representations commutative?

Consider, for example, the product of $E_6$ representations $$ 78 \cdot \overline{351}_s \cdot 78 \cdot 351_s, $$ where the $s$ denotes symmetric. Is this equal to $$ 78 \cdot 78 \cdot ...
2
votes
1answer
32 views

References and suggestions about the elementary theory of Lie groups and Lie algebras

I am looking for suggestions on how to approach the field of Lie groups and Lie algebras. I am acquainted with both the elementary algebraic concepts, having studied from Bourbaki's "Algebra I-III", ...
-3
votes
0answers
11 views

A connected Lie group G decompose into a one-parameter subgroup and a invariant Lie subgroup [closed]

show that $G=G_1C_1$,where $G$ is a $n$ dimensions connected Lie group ,and $G_1$ is the $n-1$ dimensions invariant Lie subgroup of $G$ ,$C_1$ is a one-parameter subgroup.
1
vote
0answers
22 views

Vector field from group action

Let $\Phi: G \times \mathbb{R}^4 \rightarrow \mathbb{R}^4$ be a group action where $G = \mathbb{R}/(2 \pi \mathbb{Z}).$ Then $$\Phi(\theta,(x_1,x_2,p_1,p_2)) = ( R(\theta) (x_1,x_2)^T, R(\theta) ...
2
votes
0answers
26 views

signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...
1
vote
0answers
19 views

How do I know the length of a Weyl group element from its Weyl orbit result

How do I know the length of a Weyl group element from its Weyl orbit result? For example, I know that $[2,2]$ under $s_1s_2s_1$ transforms into $[-2,-2]$,but given the result, how can I tell ...
1
vote
1answer
19 views

Finding the tangent space of $Z(U(n))$

Recall that the center of the matrix unitary group for a given $n\in \mathbb N$ is\begin{equation*}Z(U(n))=\{\omega I:|\omega|=1\}\end{equation*} I'm trying to find the tangent space at the identity ...
1
vote
0answers
13 views

G-orbits have equal dimension on a neighborhood, so there exists a cross-section

I'm working right now with the book of Guillemin/Sternberg (Symplectic techniques in Physics) and there is one statement I can't prove right now. Namely: They assume, if we have a symplectic manifold ...
-1
votes
0answers
45 views

The subgroup that is a Lie group

Suppose that $G$ is a subgroup of Lie group $GL(k,\mathbb C)$, $H$ is a Lie subgroup of $GL(k,\mathbb C)$. Let $\dfrac{G}{H}$ be a countable quotient group.($H\lhd G$) I guess that with above ...
1
vote
0answers
26 views

Is $\mathfrak{s}\mathfrak{u}(d) \otimes \mathfrak{s}\mathfrak{u}(d)=\mathfrak{s}\mathfrak{u}(d^2)$?

Given a basis $a_i$ for the lie algebra $\mathfrak{s}\mathfrak{u}(d)$, does the set of elements $a_i \otimes a_j$ form a basis for $\mathfrak{s}\mathfrak{u}(d^2)$?
2
votes
1answer
23 views

The space of orientations of a 3-d object ($SO(3)$, $RP^3$, $S^3$, $S^2 \times S^1$, etc)

It seems like I am missing something really basic here. I am thinking of the following 2 representations of the orientations of a 3d object (excluding reflections). Take a sphere at the origin for ...
0
votes
0answers
29 views

What are the generators of the coset of SU(2)/U(1)?

I need to understand the coset, $SU(2)/U(1)$ for the fundamental representation. How would I go about doing so? From my understanding of coset it means that any transformation of $SU(2)$ mod a $U(1) ...
2
votes
0answers
13 views

Curvature of $K$-invariant connection (principal bundles)

Here is a proposition from Kobayashi & Nomizu's Foundations of Differential Geometry. I don't understand how they obtain the final line of the proof. They write: \begin{align} ...
3
votes
1answer
69 views

Symplectic group action

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
2
votes
0answers
22 views

Quotient space of equivariant vector bundle.

Let $p: P \to B$ be a principal $G$-bundle, and $\pi : E \to P$ a vector bundle with action of $G$ on $E$ such that $G$ acts by vector bundle isomorphisms and $\pi$ is equivariant. Is it always the ...
1
vote
1answer
15 views

Lie algebra associated to Lie group of algebra automorphisms

I'm working through Fulton and Harris's Representation Theory, and I'm stuck on Exercise 8.27. I'm trying to show that if $A$ is an algebra and $G$ is the Lie group of algebra automorphisms ...
2
votes
1answer
47 views

1 parameter subgroups and Lie groups

I was just reading some lectures notes (that are not online available unfortunatley) on Lie groups and found that sometimes the author just says if he wants to prove something for all Lie group ...
2
votes
0answers
31 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
1
vote
1answer
42 views

Nonlinear representation of SU(2) and SU(4)

Consider a nonlinear representation of a group acting on the space of real vector $\phi_i$ in the form: \begin{equation} \phi_i\rightarrow \sum_jM_{ij}\phi_j+\delta\phi_i \end{equation} where $M$ is ...
1
vote
0answers
48 views

How To Formalize the Fact that $(g, h)\mapsto dL_g|_h$ is smooth where $g, h\in G$ a Lie Group

Let $G$ be a Lie group. I am wondering if there is a way to say that the map $(g, h)\mapsto dL_g|_h$ defined on $G\times G$ is a smooth map (Here $L_g$ is the left translation map from $G$ to $G$ ...
4
votes
1answer
38 views

Questions about fundamental representations of $SL_3/U$.

Consider the group $SL_3$. Let $U$ be the subgroup of $SL_3$ consisting of all upper triangular unipotent matrices. Then the algebra $\mathbb{C}[SL_3/U]$ is generated by $a_{11}, a_{21}, a_{31}, ...
-2
votes
0answers
45 views

Lie Group Structure on the Spheres [duplicate]

Show that the spheres admitting Lie group structure are $S^0, S^1, S^3, S^7$ and give their Lie structures directly.
1
vote
2answers
39 views

A Lie Group Homomorphism $f:G\to H$ Induces a Functor from Principal $G$-Bundles to $H$-Bundles

I am trying to understand Qiaochu Yuan's answer to this question. The first line of the answer reads: A Lie group homomorphism $f:G\to H$ induces a functor from the category of principal ...
1
vote
0answers
28 views

Defining relations of SU(2) and SU(4)

Defining relations are relations among the generators of a group, $\{g_a\}$, such that $\prod g_a^{n_a}=e$, where $e$ is the identity element. The multiplication rule of a group can always be ...
1
vote
0answers
40 views

Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
2
votes
0answers
19 views

Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$ \int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU $$ where the integration is with respect to the Haar measure on the orthogonal group, ...
5
votes
1answer
63 views

Is an injective morphism from a Lie group to itself surjective?

I have a question about Lie groups. Let $G$ be a finite dimensional (real or complex) Lie group and $f:G \rightarrow G$ an homomorphism of Lie groups. Edit : in the view of some counter-examples let's ...
2
votes
0answers
20 views

Closure of a Manifold is a Manifold with Corners?

Is there a general theorem that shows that if you have a manifold $S$ then its closure $\overline{S}$ is a manifold with corners? I am dealing with a specific set $S$ (I would rather not say which ...
1
vote
0answers
28 views

Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
0
votes
0answers
18 views

The definition of Fell topology

Let $G$ be a Lie group, $\pi$ is a representation, then with some conditions, we have the following branching law $\pi|_N=\int^{\oplus}m_\pi(\mu)\mu\mathrm{d}\mu$ where $m$ is the multiplicity ...
1
vote
1answer
27 views

Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
1
vote
1answer
27 views

The fundamental vector fields of a principal bundle are vertical.

Let $p:P\to M$ be a principal $G$-bundle. To each $A$ in the Lie algebra of $G$ corresponds a fundamental vector field $A^*$ on $M$ defined by $$A^*_u=\frac{d}{dt}|_{t=0} u(exp(tA))$$ How can we see ...
0
votes
0answers
28 views

Action on an irreducible lattice of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ [closed]

Let $G_1 = G_2 = \mathrm{SL}_2(\mathbb{R})$. Let $\mathrm{SL}_2(\mathbb{Z}[\sqrt{2}]) = \Gamma < G_1 \times G_2$ be the usual example of an irreducible lattice. Is the action of $G_1$ on $(G_1 ...
2
votes
1answer
46 views

Left and right action?!

The adjoint of the adjoint representation $Ad^* : G \times \mathfrak{g}^* \rightarrow \mathfrak{g}^*, (g,x) \mapsto Ad^*_{g}(x)$ is a group action on the dual space of the Lie algebra. Now, we said ...
1
vote
3answers
89 views

Proving smoothness of left-invariant metric on a Lie Group

Assume $G$ is a Lie group. The standard construction of a left invariant metric on $G$ goes as follows: Take an arbitrary inner product $\langle,\rangle_e$ on $T_eG$ and define $\langle u , ...
1
vote
0answers
22 views

Lie algebra operations from lie group

According to wikipedia, if $G$ is a closed subgroup of $GL(n, \mathbb{R})$ then the Lie algebra of $G$ can be thought of informally as the matrices $m$ of $M(n, \mathbb{R})$ such that $1 + εm$ is in ...
0
votes
0answers
32 views

Is SU(2) a subgroup of the exceptional lie group $G_2$?

I am not an expert in Lie groups and I have spent ages looking at textbooks; I assume that because I haven't found this statement explicitly it must either be untrue or obvious ;) The only thing I ...
1
vote
1answer
23 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...