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 ...

learn more… | top users | synonyms

0
votes
0answers
23 views

Show $\phi$ is a isomorphism as a lie algebra homomorphism

Show $\phi$ is a isomorphism as a lie algebra homomorphism $\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K ...
0
votes
1answer
25 views

Root system independent of chosen Cartan algebra

I have read on "Lectures on Lie groups and Lie algebras" (by Carter, Segal, Macdonald) that Cartan subalgebras are related by some automorphism of the Lie algebra and this is proved using a density ...
1
vote
0answers
32 views

Symmetric Polynomials in Geometry

I'm interested in symmetric polynomials. Could you name some nice examples in Differential Geometry where they are clearly useful? I would also be interested in algebraic examples if connected with ...
0
votes
1answer
17 views

Borel density theorem

I know the following version of Borel density theorem: If $G$ is a connected real Lie group such that every continuous homomorphism from $G$ to a compact group is trivial, and if $H$ is a closed ...
0
votes
0answers
29 views

How many subalgebras are there in $sl_3$?

The Lie algebra $sl_3$ is 8 dimensional and $B=\{h_1, h_2, e_1, e_2, [e_1, e_2], f_1, f_2, [f_1, f_2]\}$ is a basis of $sl_3$. For every $x \in B$, $\text{Span}\{x\}$ is a one-dimensional subalgebra ...
0
votes
1answer
83 views

Isomorphism between homotopy groups of Lie group, Grassmann manifold

It is asserted without proof in a book edited by Novikov and Rokhlin that $$\pi_{k - 1}(\text{GL}_n^+(\mathbb{R})) \cong \pi_k(\tilde{G}_n).$$ I know how to show that these two spaces are bijective. ...
0
votes
0answers
48 views

Do we have $\{g x g^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Fix $g \in G$. Do we have $\{gxg^{-1}: x \in \mathfrak{g}\} = \mathfrak{g}$? Thank you very much. Edit: I think that the answer is yes. We ...
1
vote
1answer
49 views

Strange parametrization of SO(3)

I have this parametrization of the sphere that seems quite a mess \begin{equation} R_{ij}=\cos r\left(\delta_{ij}-\hat{\theta}_{i}\hat{\theta}_{j}\right)+\hat{\theta}_{i}\hat{\theta}_{j}+\sin ...
5
votes
0answers
64 views

Pauli Matrices as Representations of Reflection Operators?

How does one derive, using, say, the Householder transformation $$ R(r) = (I - nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) + ...
0
votes
0answers
15 views

Lie group actions

I am looking for a nice reference to study the action of a Lie group $G$ on a smooth manifold $M$, $\psi : G\to\mathrm{Diff}(M)$: Linearization: in a neighborhood of a fixed point, what we can ...
1
vote
0answers
26 views

When a group of ismorphisms is a Lie group

What are the known cases where a group of isomorphisms of a smooth manifolds (diffeomorphisms that respect a given structure on the manifold) is a Lie group? such as: isometries of a compact ...
1
vote
0answers
21 views

Differential of a Lie group homomorphism

If $f : G\to H$ is a Lie group homomorphism, what can we say about its differential $d_ef : \mathcal{G}\to\mathcal{H}$? Is it a Lie algebra morphism or anti-morphism? ...
-1
votes
1answer
21 views

Dimension of the indefinite orthogonal group [closed]

How to find the dimension of the group $O_{p,q}(\mathbb R)= \{g \in GL_n(\mathbb R): g^TI_{p,q} \ g = I_{p,q}\}$, where $I_{p,q}= diag(1,..., 1,-1,...,-1)$ and $p+q=n$?
0
votes
0answers
21 views

Lie group of differential operators

I have the following three partial differential operators $$A=y \frac{\partial}{\partial y}$$ $$B=y^{-1}(z\frac{\partial}{\partial z}+y\frac{\partial}{\partial y}+c-1)$$ ...
1
vote
0answers
13 views

Basis vectors for “perturbed slicings” of a function, using SE(3)

Given a function $\Phi: \mathbb{R}^3\rightarrow\mathbb{R}$, we intruduce its planar cross section slices $\phi^{s}:\mathbb{R}^2\rightarrow\mathbb{R}$, using a rigid mapping $s \in SE(3)$ such that for ...
0
votes
0answers
12 views

evaluationmap is a submersion iff lie group acts transitively on connected components

Let $M$ be a manifold and $G$ a connected Liegroup acting smoothly on $M$. Take $x \in M$ and define by $ev_x \colon G \to M, \ g \mapsto g.x$ the evaluationmap at $x$. Is it true, that $G$ acts ...
0
votes
1answer
21 views

Unitary dual of $\{0\}$ and $\mathbb R$

How to prove that, the unitary dual of $\{0\}$ and $\mathbb R$ are the trivial identity representation $id$ and the representation $\chi_x (y) = e^{i x y}; y\in \mathbb R$, respectively. Thank you ...
-2
votes
2answers
63 views

prove $RP^3\cong SO(3)$ [closed]

Suppose $RP^3$ is the real 3-dimensional projective space,prove the rotation group $SO(3)$ is homeophoric to $RP^3$.
1
vote
1answer
37 views

prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$

$GL(n,R)$ is the general linear group ,$O(n)$ is the orthogonal group,how to prove $GL(n,R)\cong O(n)\times R_{+}^n\times R^{\frac{n(n-1)}{2}}$
2
votes
1answer
56 views

How to compute the center of $SU(2)$?

It is stated in our lecture notes without proof that the center of $SU(2)=\{\pm 1\}$. I understand how to find the center of $SO(3)$, which is $\{1\}$ and that is given in the notes, is that somehow ...
5
votes
1answer
48 views

The category of Lie algebra representations

A representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a homomorphism of Lie algebras $\mathfrak{g} \to \mathfrak{gl}(V)$. We define morphisms between representations as ...
0
votes
0answers
24 views

Jet-groupois, left translation and smoothness of a vector field

In this question I work in the smooth category. For basic definitions on lie groupids see "General Theory of Lie Groupoids and Lie Algebroids" by Kirill Mackenzie. Let us consider the Lie groupoid $G ...
6
votes
0answers
134 views

Duistermaat & Kolk's lost chapters on Lie groups

I am reading the book Lie Groups by Duistermaat and Kolk. It is written in the preface that "the text contains references to chapters belonging to a future volume". I could not find this second ...
3
votes
0answers
48 views

Two group structures on the cotangent bundle of a Lie group. Are they related?

Let $G$ be a compact connected Lie group, and consider its cotangent bundle $T^*G$. There are two ways of viewing this space. Using left translation, we can trivialize $T^*G\cong G\times{\frak ...
0
votes
1answer
42 views

Matrix representation of Heisenberg group

Equiped with the law $(a,b,c)\circ (a',b',c') = (a + a', b + b', c + c' + ab'),$ the matrix representation of the Heisenberg group $H^3$ is given by $$ \begin{pmatrix} 1 & a & c\\ 0 & ...
0
votes
0answers
25 views

Lie group action and Lie algebra action.

Let $G$ be a Lie group and $g$ its Lie algebra. Let $r \in g \otimes g$ and $b \in g$. Consider the adjoint action $g \times g \otimes g \to g \otimes g $ given by $(b, x, y) \mapsto b.(x \otimes y) = ...
3
votes
2answers
65 views

Why do we require that a simple Lie algebra be non-abelian?

We say that a Lie $k$-algebra is simple if it is a simple object in the category of Lie algebras, and also nonabelian. The only simple object which we do not consider to be a simple Lie algebra under ...
0
votes
1answer
39 views

What is the manifold underlying the Lie group $SU(p,q)$?

I've been trying to google around this topic without success, apologies in advance if I missed an obvious resource. I'm trying to understand what manifold (compact or not) underlies the complex Lie ...
0
votes
0answers
35 views

How does this product of matrices define a local diffeomorphism?

Let $$H_1 := \left\{\begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix} \mid x \in \mathbb{R}\right\},$$ $$H_2 := \left\{\begin{pmatrix} 1 & 0 \\ y & 1\end{pmatrix} \mid y \in ...
0
votes
1answer
24 views

Chevalley basis for $G_2$

I want to find the Chevalley basis for the exceptional group $G_2$. Could you point to literature where the computation is done in detail or show me how to do it?
1
vote
2answers
36 views

What can we say about $Aut(G)$ for an arbitrary Lie group $G$?

Let $G$ be a Lie group, $\mathfrak g$ be its Lie algebra, and $Aut(G)$ be the group of its smooth automorphism. Then, my questions are: (1) Is $Aut(G)$ again a smooth manifold? and particularly a Lie ...
3
votes
1answer
27 views

What the expression of a one-dimensional representation of $H$

Let $G= \{ g=(x,y,t); \quad x,y,t \in \mathbb R\}$ be the Heisenberg group and $H= \{ g=(x,y,t) \in G; \quad x=0\}= \{ h=(0,y,t); \quad y,t \in \mathbb R\}$ be a subgroup of $G$. I want to know why ...
1
vote
1answer
44 views

A proof that every compact Lie Group has torsion second homotopy group

I am trying to prove that every compact lie group has torsion second homotopy group but I get stuck. My argument is the following: Since $\Pi_1(G)$ is finite the universal cover G* of G is also a ...
1
vote
1answer
29 views

Are surjective Lie group homomorphisms which induce isomorphisms of Lie algebras covering maps?

Question: Suppose that $\varphi:G\to H$ is a surjective Lie group homomorphism whose differential $\varphi_*:{\frak g}\to{\frak h}$ is a Lie algebra isomorphism. Is $\varphi$ necessarily a smooth ...
0
votes
0answers
19 views

Splitting of short exact sequence. with the existence of non-deg. bilinearform

Let $\mathfrak{h}$ be a subalgebra of $\mathfrak{g}$,and $j \colon \mathfrak{h} \to \mathfrak{g}$ be the injective Liealgebra-Homomorphismus. Assuming now, that we have a non-degenerate bilinearform ...
2
votes
0answers
46 views

embeddings $SU(2) \to SU(N)$ and representations

How can we prove that group immersions $SU(2) \to SU(N)$ (up to conjugacy) are in 1-1 correspondence with (non-trivial) $N$-dimensional representations of $SU(2)$ (up to equivalence)? Feel free to ...
0
votes
1answer
21 views

Complexification of maximal compact subgroup of $GL(2,\mathbb{R})$.

Given the Lie group $G=GL(2,\mathbb{R})$, we have that $K=O(2)$ is a maximal compact subgroup of $G$. I am trying to describe the complexification $K_\mathbb{C}$ of $K$. The Lie algebra $k_0$ of $K$ ...
1
vote
0answers
18 views

Composition of analytic functions is analytic in Manifolds

My problem is in analytic manifolds.According to Cohn's book a function $f$ in a manifold $M$ is analytic at $p \in M$ if it can be expressed as a power series of $\sigma(p)=(x_{0})$. That means ...
1
vote
1answer
28 views

Different methods to compute a unitary representation

Given a nilpotent Lie group $G$ (for example the Heisenberg group), what is the most effective method to calculate their unitary representation: The orbit method due to Kirillov; or The induction ...
1
vote
1answer
24 views

Lie group homomorphism with injective Lie algebra homomorphism

Question: Suppose that $\varphi:G\to H$ is a Lie group homomorphism such that $G$ is simply connected and $\varphi_*:{\frak g}\to{\frak h}$ is injective. Is $\varphi$ injective? Since Lie group ...
10
votes
1answer
111 views

Flows and Lie brackets, $\beta$ not a priori smooth at $t = 0$

Let $X$ and $Y$ be smooth vector fields on $M$ generating flows $\phi_t$ and $\psi_t$ respectively. For $p \in M$ define$$\beta(t) := \psi_{-\sqrt{t}} \phi_{-\sqrt{t}} \psi_{\sqrt{t}} ...
3
votes
1answer
118 views

Identity for bracket operator in tangent space at identity

Let $G$ be a Lie group and $X,Y\in T_eG$. Show that $$[X,Y]=\left.\frac{\partial}{\partial s}\right\vert_{s=0}\left.\frac{\partial}{\partial t}\right\vert_{t=0}\exp(sX)\exp(tY)\exp(-sX)\exp(-tY).$$ ...
0
votes
0answers
39 views

Finite dimensional representation of $SL_{2}$

Let ($\pi , V$) be a finite dimensional representation of $SL_{2}$. Also, let $\alpha$ be highest weight vector. I want to show that for any $m>0$, then the following holds: ...
0
votes
0answers
29 views

Need someone explain to me how to invert exponential map from SO(3) to so(3)

I need someone to explain to me how to invert exponential map from $SO(3)$ to $\mathfrak{so}(3)$. \begin{equation} \begin{aligned} & R \in SO(3) \\ \text{From:}\\ \exp(\mathbf{M}) & = ...
0
votes
0answers
41 views

Proof of Howe-Moore Property for SL(n,R)

On page 210 of Howe and Tan's Non-Abelian Harmonic Analysis there is the following proposition and proof: Let $(\rho, V) $ be a unitary representation of $SL(n,\mathbb{R})$. The following are ...
0
votes
0answers
23 views

Identifying $\bar{D^3}$ with $S^3$ and then $\mathbb{R}P^3$ and $SO(3)$

After working it out, I found that we can identify the closed 2-disk, $\bar{D^2}$ with the 2-sphere, $S^2$. Let $\bar{D^2}$ have radius $\pi$. Then $\partial \bar{D^2}$ is identified with the south ...
1
vote
0answers
25 views

Product rule of exponential matrix differentiation

Let $X,Y$ be two $n\times n$ complex matrices. Consider the function $f(t)=e^{tX}e^{tY}$. Is it correct that $\frac{d}{dt}f(t)=Xe^{tX}e^{tY}+e^{tX}Ye^{tY}$? Otherwise, how to prove that $X +Y$ is ...
1
vote
1answer
22 views

Coadjoint representation $Ad^*$ of the Heisenberg group

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$ (z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right). $$ For $z=x+ i y \in \mathbb C$ ...
0
votes
0answers
18 views

The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
3
votes
1answer
55 views

Characters group and cocharacters group Hom duality

Let $T$ be an algebraic torus over $\mathbb{C}$. For brevity denote $C = \mathbb{C}^\times = \mathbb{G}_{m,\mathbb{C}}$ the multiplicative subgroup of $\mathbb{C}$. Define character group by $$ X^*(T) ...