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

1
vote
0answers
127 views

decomposition of right Haar measure on homogeneous space

For simplicity, let $G_n=GL(n,\mathbb{R})$, $N_n$ be the upper trianguler unipotent subgroup, $P_{n-i}$ be the standard parabolic subgroup associated to partition $n=(n-i,i)$, and finally let $K=O(n)$ ...
7
votes
1answer
723 views

Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
0
votes
0answers
378 views

References for basic level Differentiable Manifolds and Lie Groups

I an undergraduate math student with a decent background in abstract algebra. I am looking forward to studying Lie groups this summer...I want some you to suggest good references for the following ...
1
vote
0answers
100 views

Asymptotic invariants of infinite groups

I am reading a Gromov's book " Metric Structures for Riemannian and Non-Riemannian Spaces ". Consider the following concept : $$ distort(X)\doteq sup \frac{length\ dist|_X}{dist|_X} $$ That is for ...
1
vote
2answers
81 views

Relationship between $PGL_2$ and $PSL_2$

I read somewhere that $PGL_2(\mathbb{C})=SL_2(\mathbb{C})/N$ where $N$ is the normal subgroup consisting of $\pm \left( \begin{matrix} 1 & 0\\ 0 & 1 \end{matrix}\right)$. It is unclear to me ...
8
votes
1answer
309 views

Why is the Lie derivative linear in the vector field?

This might seem a very basic question, but I can't manage to find a proper proof in the books I have on my desk (or simply cannot see that it's "just that"). So be sure of what we talk about, let $G$ ...
4
votes
1answer
55 views

Cohomology of Lie Groups finite dimensional?

Let $G$ be a connected Lie group, without any further assumptions. Is it true, that its rational cohomology ring $$H^\bullet(G,\mathbb Q)$$ is finite dimensional? Is $G$ homotopy equivalent to a ...
2
votes
1answer
46 views

Understanding Hilbert–Smith conjecture.

By a faithful action of a topological group G on a topological manifold M, we mean a continuous injection $G\to \operatorname{Homeo}(M)$ (where $\operatorname{Homeo}(M)$ has the compact open ...
0
votes
1answer
65 views

Range of the exponential map

I need to build a matrix $A\in Gl^+_n(\mathbb{R})$ ($det(A)>0$) that is not an exponent, i.e. there is no $B\in Mat_n(\mathbb{R})$ such that $A=exp(B)$. Could you give me a hint to note some ...
3
votes
2answers
41 views

Representations of $SO_3(\mathbb{R})$ from $SU_2(\mathbb{C})$

Define $V_n$ as the linear space of all homogeneous polynomials of degree $n$ in two variables $x$ and $y$. Define also the representation $\rho_n$ of $SL_2(\Bbb{C})$ on $V_n$ by: ...
1
vote
1answer
31 views

$SU_2(\mathbb{C})$ and the characters

i can prove that the irreducible characters $\chi_n$ of $SU_2(\mathbb{C})$ are equal to: $$\chi_n(e^{i\phi})=\frac{\sin((n+1)\phi)}{\sin(\phi)}$$ If i want to give the dimension of the representation ...
8
votes
1answer
286 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
9
votes
1answer
420 views

Learning Roadmap for Borel - Weil - Bott Theorem

Next semester I may study a course where the ultimate goal is to get to the Borel - Weil - Bott (BWB) Theorem, if not at least try to understand it in the case that we have $G = \text{SL}_n$. I have ...
1
vote
1answer
61 views

if $X$ is a vector field how can I find $Y$ such that $[X,Y]=0$?

Suppose I am given a holomorphic vector field $X$ over a complex manifold $M$. To simplify this we can suppose that $X$ is a holomorphic vector field in $\mathbb{C}^n$ for $n=2$ or $n=3$. How can I ...
2
votes
2answers
111 views

Question about representation of free products of groups.

Does anyone have an idea of books or papers that treats representation theory of free products of groups? What properties of factors of of a free product suggest a possible representation? ...
5
votes
0answers
139 views

Can one reformulate tensor methods and young tableaux to account for spinor representations on $\operatorname{SO}(n)$?

Standard tensor methods and Young tableaux methods don't give you the spinor reps of $\operatorname{SO}(n)$. Is this because spinor representation are projective representations? If so, where does ...
0
votes
0answers
26 views

Closed formula for the coefficients of a series obtained from an expansion.

The Heisenberg algebra is generated by $h_i, i\in \mathbb{Z}\backslash\{0\}$ and the central element $c$. We expand the function $$\exp (\sum_{n=1}^{\infty}h_{-n}\frac{z^n}{n}) = 1 + ...
2
votes
0answers
34 views

Classification of 6-D Nilmanifolds

I am reading the G.Cavalcanti and M.Gualtieri's Generalized Complex Structures on Nilmanifolds. In the introduction it is said that there are 34 nilpotent lie algebra isomorphism classes. There are ...
3
votes
3answers
219 views

Definition of Lie Algebra

I have just start studying Lie Algebra. I want to know the motivation for the conditions on the Lie Bracket in the Definition of Lie Algebra. Please explain me or tell me some references. Thanks.
5
votes
2answers
258 views

path-connected subgroup of Lie group is Lie group

Theorem (Yamabe): Let $G$ be a Lie group, and let $H$ be an arc-wise connected subgroup of $G$.Then $H$ is a Lie subgroup of $G$. I am reading this theory form an appendix in a book called (bilinear ...
6
votes
1answer
517 views

Prove that the universal covering space of a Lie group is unique

This is a problem in Lee's book, introduction to smooth manifold. I am trying to show that the universal covering space of a connected Lie group is unique. Suppose $G$ is a Lie group, and ...
12
votes
0answers
190 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
3
votes
0answers
74 views

Weights set spans

Definition Let $T$ be a torus and $R: G \to GL(V)$ a representation. $R(T)$ is a collection of commuting matrices and therefore can be simultaniously diagonalized. For a character $\lambda \in ...
3
votes
1answer
93 views

How to show $SL_{n}(\mathbb{R})=\bigsqcup_{w\in W}LwU$ where L (or U) are lower(or upper) triangular matrix?

I'd like to ask a homework problem that causes me many troubles for days. The problem is like below : Let W denote the subgroup of permutation matrices in $SL_{n}(\mathbb{R})$. Show the following ...
2
votes
2answers
141 views

Are there nonlinear operators that have the group property?

To be clear: What I am actually talking about is a nonlinear operator on a finitely generated vector space V with dimension $d(V)\;\in \mathbb{N}>1$. I can think of several nonlinear operators on ...
1
vote
0answers
255 views

Lie algebra of Euclidean group

From the book "Spinning Tops" by Audin, she claims that $$\mathfrak{so}(3)[\epsilon]/\epsilon^2$$ with coefficientwise Lie bracket is a Lie algebra of a Lie group that is $TSO(3)$ (group action not ...
14
votes
1answer
242 views

Why is the dimension of $SL(2,\mathbb{H})$ equal to $15$?

Let me ask a very basic question which is inspired by reading M. Atiyah's "Geometry and physics of knots". Could you explain me (or give a reference to) the definition of the special linear group ...
0
votes
1answer
178 views

Invariant subspaces of tensor product of SU(2)

Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$. I know that irreducible representations of $\varphi_2 \otimes \varphi_3 = \varphi_5 \oplus ...
2
votes
0answers
68 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
2
votes
0answers
66 views

A technical problem on constructing smooth vectors in a representation

Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$, and $\pi:G\to\mathrm{GL}(H)$ a continuous representation of $G$ in a Banach space $H$. Let $\pi^1:\mathcal{C}_c(G)\to\mathrm{End}(H)$ be ...
3
votes
1answer
305 views

Irreducible representation of tensor product

Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$. What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?
5
votes
0answers
587 views

Discrete subgroups of SU(n) and SO(n).

Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful. I would like to know whether there is a complete understanding of discrete ...
2
votes
1answer
125 views

Show that the SU(2) group is a Lie group

How can I prove that the SU(2) is a Lie group with the Pauli matrices as generators?
4
votes
1answer
96 views

Why is Lie derivative smooth?

Let $G$ be a linear Lie group, say $\mathrm{SL}(2,\mathbb{R})$. Suppose $X\in\mathfrak{g}$ and $f:G\to\mathbb{R}$ is smooth. The Lie derivative of $f$ with respect to $X$ is the function ...
8
votes
1answer
91 views

Are there more embeddings $U(2) \hookrightarrow SO(4)$?

It is easy to prove that $SO(4)$ acts transitively and freely on $S^2$ with fiber $U(2)$. Therefore, we can identify each point of $S^2$ with a particular embedding $U(2) \hookrightarrow SO(4)$. My ...
1
vote
1answer
65 views

Lie Algebra over Quaternions

We define the skew-symmetric hermitian inner product $$\phi(x,y)=\bar{x}^tjy$$ over $\mathbb{H}^2$ and are asked to calulate the Lie algebra of the group $G\le GL(2,\mathbb{H})$ of automorphisms of ...
1
vote
0answers
31 views

Lie map Question

I am trying to prove the below but I am having problems starting and I think that I am misunderstanding something? If we have that G is a connected linear group and $p:G\rightarrow GL(V)$ and ,V a ...
7
votes
1answer
184 views

How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?

Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
3
votes
1answer
79 views

Centralizers of connected linear group and its Lie algebra

If we have that $G$ is a connected linear group and $H<G$, where $H$ is also connected, with $\mathfrak{h}$ the lie algebra of $H$ and we define the centralizers of the elements in the following ...
7
votes
2answers
556 views

Calculating the lie algebra of $SO(2,1)$

I am trying to calculate the Lie algebra of the group $SO(2,1)$ where this is defined as: $SO(2,1)=\{X\in Mat_3(\mathbb{R})|X^t\eta X=\eta, \det(X)=1\}$ where $\eta$ is the matrix defined as: $$\left ...
4
votes
1answer
538 views

The dimension of $SU(n)$

$SU(n)$ denotes the special unitary group. I know its dimension should be $n^2-1$. However, I am trying to prove it and get a wrong result. I have no idea what is wrong with my proof. Therefore, I am ...
3
votes
2answers
147 views

Symplectic Form Preserved by Orthogonal Transformation

I'm trying to prove that the symplectic form $$\omega = d(\cos\theta) \wedge d\phi$$ is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply ...
2
votes
0answers
63 views

Action of a Lie group on a coset of its subgroup

I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) ...
3
votes
0answers
75 views

Maximal compact subgroup of $GL_n(\mathbb C_p)$

It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of $GL_n(\mathbb ...
4
votes
1answer
38 views

Proving that the Flag Variety $Fl(n;m_1,m_2)$ is connected.

I wish to prove that the flag variety $Fl(n;m_1,m_2) = \{ W_1 \subset W_2 \subset V | dimW_i = m_i \}$, for $0 \le m_1 \le m_2 \le n$ where V is an n-dimensional vector space over $\mathbb{C}$ and ...
1
vote
0answers
28 views

How to write down the maximal subgroups of $GL(9, \mathbb{C})$

I am wondering about the maximal subgroups of the group $GL(n^2, \mathbb{C})$. My motivation for wondering about these groups is a project (in its most general form) I am working on where I am trying ...
1
vote
0answers
110 views

are closed orbits of Lie group action embedded?

Consider a smooth action $G\curvearrowright M$ of a Lie group on a manifold. Suppose that an orbit $G\cdot p$ is closed. Is the orbit an embedded submanifold. In general we know that the orbits are ...
1
vote
1answer
39 views

Lie subalgebra, Lie subgroup and membership

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $H$ be a connected Lie subgroup with Lie algebra $\mathfrak{h}$. We have that $X \in \mathfrak{h} $ iff $exp(tX) \in H \ \ \ \forall t ...
3
votes
3answers
129 views

Show that $\exp: \mathfrak{sl}(n,\mathbb R)\to \operatorname{SL}(n,\mathbb R)$ is not surjective

It is well known that for $n=2$, this holds. The polar decomposition provides the topology of $\operatorname{SL}(n,\mathbb R)$ as the product of symmetric matrices and orthogonal matrices, which can ...
4
votes
0answers
503 views

The Symplectic group is connected

Let $K = \mathbb{R}, \mathbb{C}$ be a field and consider the skew-symmetric matrix $$ J = \left( \begin{matrix} 0 & I_n \\ -I_n & 0 \end{matrix} \right) $$ where $I_n$ is the unit matrix of ...