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
votes
1answer
153 views

$\mathfrak{sl}(2,\mathbb C)$ real v. complex

I'm a Lie theory novice, so please bear with me. My understanding is that the Lie algebra $\mathfrak g$ of a matrix Lie group $G$ is the pair $(V, [\cdot, \cdot ])$ where $V$ is the real vector space ...
4
votes
1answer
109 views

Some questions about $S^n$

I have some questions about the $n$-sphere: I know that for $n=0,1,3$, $S^n$ forms a Lie group and I also know why it's true, but why is it not the case for other $n$? I have the same question for ...
5
votes
1answer
426 views

Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
3
votes
0answers
142 views

Infinitesimal generators of actions

Is there a method to obtain an action of an infinite dimensional Lie group starting with its infinitesimal generator ? I'm interested about actions of G on itself . And I was wondering if I can ...
3
votes
1answer
147 views

Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) + U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
1
vote
0answers
276 views

jacobian involving SO(3) exponential map: $\log(R \exp(m))$

I would like to compute the 3 × 3 Jacobian of $$ \log(R \exp(m)) $$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...
9
votes
2answers
1k views

On surjectivity of exponential map for Lie groups

A recent question made me realize I didn't know any proof that exponential of a Lie algebra $\mathfrak g$ of a compact connected Lie group $G$ is surjective. After a bit of thinking I've come up with ...
2
votes
1answer
120 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
351 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
3
votes
0answers
44 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
2
votes
1answer
670 views

Showing that left invariant vector fields commute with right invariant vector fields

I'm trying to prove that if $G$ is a Lie group, $X$ is a left-invariant vector field on $G$, and $Y$ is a right-invariant vector field on $G$, then $[X,Y] =0$. When I imagine what it means to be ...
4
votes
0answers
30 views

Terminology: how to call the compact version of an affine space?

How should I call briefly the space $M$, which is obtained when "forgetting" the origin (i.e. the identity) of an $n$-dimensional torus $T$ (i.e. $T$ is a compact $n$-dimensional abelian Lie group)? ...
9
votes
0answers
116 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
1
vote
1answer
90 views

Connection between spaces of cosets and homotopy cofibers

Suppose we have an inclusion of a closed sub-Lie group $H\to G$ and take the space of left cosets of $H$, $G/H$. How is this related to the homotopy cofiber of the inclusion of topological spaces ...
6
votes
1answer
226 views

Connectedness of the orthogonal subgroup $O^+_+(k,l)$

Let $O(k,l)$ be the orthogonal group associated to the quadratic form $q$ on $\mathbb{R}^{k+l}$ with signature $(k,l)$. Let $O^+_+(k,l)$ be the connected component of the identity, i.e. the connected ...
4
votes
1answer
225 views

Computing Lie algebra homomorphism from Lie group homomorphism

I'm pretty much stuck on the following question (taken from the book Lie groups and introduction to linear groups by Rossman W.): I've found some clues, but I think I lack proper understanding of ...
0
votes
0answers
42 views

What is the relationship between various groups

Does the projective linear group of a given dimension share a representation with a group of automorphisms?
3
votes
2answers
146 views

On a theorem on Lie derivatives

I am a little confused proving this theorem (for $p$-forms on $M^n$, $M^n$ is a smooth manifold): $L_X\cdot i_Y-i_Y\cdot L_X=i_{[X,Y]}$ where $X,Y$ are smooth vector fields. Now it is clear that both ...
3
votes
1answer
111 views

Question about special orthogonal Lie group construction

Working through homework and I run into this problem: Suppose the Lie group $SO^{+}(2,2)$ is presented as the group of all transformations in its associated space. How do you determine whether a ...
6
votes
1answer
440 views

The Gram-Schmidt process is a deformation retraction

Consider the Gram-Schmidt process $r : GL(n) \rightarrow O(n)$ that sends invertible matrices to orthogonal matrices. I need to show this is a deformation retraction and, by restrictions of $r$, ...
5
votes
2answers
168 views

Classification of irreducible representations via Casimirs

Physicists almost always label irreducible representations via Casimirs (e.g., characterizing the irreducible representations of $SO(3)$ by spin). I've been looking far and wide to see the general ...
0
votes
1answer
108 views

Lie group reps induced by Lie algebra reps

Let $G$ be a Lie group and $\mathfrak g$ its Lie algebra. Suppose that $\rho_\mathfrak{g}$ is a representation of $g$ on a vector space $V$. Is it true that the mapping $\rho$ from the identity ...
1
vote
1answer
37 views

U(m) as a subgroup of SO(2m)

We know $U(m)$ is one the subgroups of $SO(2m)$ acting transitively on the sphere $S^{2m-1}$ (one of the groups in the Borel's list). What is the explicit formula of this embedding (or it's action)?
2
votes
1answer
124 views

Exponent of a matrix and commutative property

A question given at lie group theory course: $P$ and $Q$ are $n \times n$ matrices, and some $k \in \mathbb{R}$. Assume that $$\displaystyle\forall_{x,y \in \mathbb{R}} e^{xP}e^{yQ} = ...
3
votes
1answer
69 views

Existence of a Lie subgroup

Let $G=SU(k)\times T^1$, $S$ a subgroup of the center of $SU(k)$ ($Z(SU(k)\cong \mathbb{Z}_k$) and $\eta$ a homomorphism from $S$ into $T^1$. Suppose $(S, \eta)$ denotes the subgroup of $G$ contains ...
1
vote
0answers
40 views

Symmetric algebra and complex polynomial on Lie algebra

Let $G$ a Lie group and $\mathfrak{G}$ its Lie algebra. How can I identify the symmetric algebra on $\mathfrak{G}$ ($S(\mathfrak{G})$) with the algebra of complex polynomials on $\mathfrak{G}$, that I ...
4
votes
0answers
202 views

Representations of non-semisimple Lie algebras

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$, and suppose $\mathfrak{g}$ is semisimple. An integral weight for $G$ is an element $\lambda \in \mathfrak{t}^*$ with ...
1
vote
1answer
38 views

maximal tori and principal $N(T)$-bundles.

Let $U(n)$ be the unitary group and $T^{n}= S^{1} \times \cdots \times S^{1}$ a maximal torus in $U(n)$. Let $N(T^{n})$ be the normalizer in $U(n)$ of $T^{n}$. How can i prove that $U(n) \rightarrow ...
4
votes
1answer
111 views

$U(n)/U(n-1)$ as homogeneous space

How can I prove that the quotient $U(n)/U(n-1) \simeq S^{2n+1}$ (where $U(n)$ is the unitary group). Is il correlated with the teory of homogeneous spaces?
3
votes
2answers
149 views

why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
2
votes
1answer
291 views

a neighbourhood of identity U generates G where g is a connected lie group

Let G be a connected Lie group and U any neighbourhood of the identity element. How to prove that U generates G.
1
vote
0answers
59 views

Invariants of representation theory of Lie groups

How to compute the determinant of a representation of an element of the special linear group? How do I argue that it doesn't change? (@Marek: @rschwieb: Yes well, given one represenation (with ...
9
votes
4answers
192 views

Examples of familiar, easy-to-visualize manifolds that admit Lie group structures

I have a trouble learning Lie groups --- I have no canonical example to imagine while thinking of a Lie group. When I imagine a manifold it is usually some kind of a $2$D blanket or a circle/curve or ...
2
votes
1answer
145 views

Weyl group, permutation group

Let $U(n)$ be the unitary group and $T$ its maximal torus (group of diagonal matrix) and $N(T)$ the normalizer of $T$ in $G$. Why $N(T)/T$ is the permutation group $S_{n}$?
1
vote
1answer
104 views

Defining Lie groups without the notion of a manifold

I like to introduce (matrix) Lie groups without the notion of manifolds. (And I am happy to scarify the "few" Lie groups which are not matrix groups in favor of a simpler definition.) I was thinking ...
1
vote
1answer
207 views

Dimension of Lie algebra according to root system

I was wondering how is it possible to find the dimension of a semi-simple lie algebra $L$ if its corresponding root system is (lets make it simple) of type $B_2$. We can find the number of roots and ...
3
votes
2answers
119 views

Nilpotent Lie Group that is not simply connect nor product of Lie Groups?

I have been trying to find for days an non-abelian nilpotent Lie Group that is not simply connect nor product of Lie Groups, but haven't been able to succeed. Is there an example of this, or hints to ...
1
vote
1answer
64 views

Differential action on a complex manifold

Let $M$ be a complex manifold of dimension $n$. Furthermore assume that we have a action of a Lie-Group $G$ on $M$ i.e. $G \times M \rightarrow M$, which is differential, meaning that for every $g \in ...
3
votes
1answer
132 views

Invariant inner products on infite-dimensional representations

Let $G$ be a compact group and let $V$ be it's continuous representation. It is well known that if $V$ is finite-dimensional, then there is an $G$-invariant inner product on $V$. I haven't found a ...
2
votes
2answers
211 views

When does the $\mathfrak g$-invariance of the symplectic form imply $G$-invariance?

Let $G$ be a Lie group with Lie algebra $\mathfrak g$, and let $M$ be a smooth manifold. Suppose $G$ acts on $M$, $G \to \text{Diff}(M)$. This naturally induces an action $\mathfrak g \times M \to M$ ...
4
votes
0answers
42 views

Is it possible to further simplify the product of three exponentials $e^A e^B e^C$ when $[A,C]=kB$ (k is a scalar)

The background is calculation of the little group elements of Poincare group for massless particles. I start with a bunch of exponentials of operators, and the end goal is to crunch them into the ...
3
votes
1answer
60 views

Nilpotence of Lie Algebra

I am trying to show that if $L$ is a Lie algebra and $L/Z(L)$ is nilpotent than $L$ is also nilpotent. Can someone please help me? I tried to first show by induction: $(L/Z(L))^k=L^k/Z(L)$. Is it ...
15
votes
1answer
306 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
2
votes
1answer
360 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
4
votes
2answers
524 views

$SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
2
votes
2answers
336 views

Lie group and SO3 visualisation

Maybe I'm asking a very vague question but I'd like to know if there are some visualisation tools available already that explain lie algebra exponential map or logarithm? I'd like to be able to ...
2
votes
1answer
88 views

Schur's first lemma for finitely generated continuous groups of $SU(d)$

Suppose that a finite set $S$ of $d\times d$ special unitary matrices densely generates a representation $\rho$ of a continuous subgroup of $G$ of $SU(d)$. That is, for every $\epsilon>0$ and ...
2
votes
0answers
68 views

How to find the induced Lie algebra homomorphism

Consider the quaternions $H=\{1+bi+cj+dk, a,b,c,d \in \mathbb{R}\}$ and the norm $\|h\|=\sqrt{h^*h}$, which is a Lie group homomorphism between $H^*$ and $\mathbb R^*$. How can I find the Lie algebra ...
0
votes
2answers
274 views

Is there a general expression for the adjoint representation of $U(N)$ or $u(N)$?

At least for low values of $N$ like $2$ or $3$ and such I would like to know if there are explicit matrices known giving the representation of $u(N)$ or $U(N)$ in the adjoint? (..a related query: ...
3
votes
0answers
99 views

How to prove that a lie group is simply connected

I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...