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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12
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184 views

Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
11
votes
0answers
161 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 ...
9
votes
0answers
108 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 ...
8
votes
0answers
335 views

Campbell-Baker-Hausdorff formula for $\log(\exp(X+Y)\exp(X-Y))$

Given $X,Y\in \mathfrak g\mathfrak l_{\mathbb R}(n)$, and the CBH formula for $\log(\exp X\exp Y)$ (wiki), is it possible to derive the general term in the series of $\log(\exp(X+Y)\exp(X-Y))$ that ...
8
votes
0answers
78 views

Theorem about the subgroup of a Lie group fixed by an involution

When trying to do Lie-theoretic calculations on Lie groups (finding the Bruhat decomposition, etc.) I've often come across expositions that seem to be implicitly using a result something like the ...
8
votes
0answers
158 views

Weyl character formula for locally compact Lie groups.

I was just wondering if there exists such a formula. Specifically I need to calculate characters of irreducible representations of GSp$(4,\mathbb{C})$. I know how to do it for the compact Lie group ...
7
votes
0answers
124 views

Translating a passage of a paper by L. Bérard Bergery

I am currently studying the following paper on Einstein manifolds: L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, Inst. Elie Cartan, Univ. Nancy №6, 1-60 (1983). I have ...
7
votes
0answers
424 views

How do the infinitesimal generators generate the whole compact Lie group?

Suppose that $G$ is a connected compact Lie group, $g(a)\in G$ is its element, and $a=(a_1,a_2,...,a_r)$ is the parameter of the group element $g(a)$. Then the infinitesimal generator of the group can ...
7
votes
0answers
375 views

Klein's Erlangen program taken seriously

Felix Klein suggested in his Erlangen program a way to classify geometries based on group theory. According to Wikipedia, we have the following definition: A Klein geometry is a pair (G, H) where ...
7
votes
0answers
91 views

Can the infinite von Dyck groups be subgroups of $SU(n)$?

I know by constructing some particular cases that I can find unitary matrices $X$, $Y$ and $Z$ such that $X^m = Y^n = Z^p = XYZ = 1$ with $$ \frac{1}{m} + \frac{1}{n}+\frac{1}{p} < 1 $$ ...
6
votes
0answers
87 views

Are there common inequivalent definitions of Cartan subalgebra of a real Lie algebra?

I'm confused about seemingly different notions of a Cartan subalgebra of a real semisimple Lie algebra, and I'm wondering if there are common inequivalent definitions. In the book Lie Groups: Beyond ...
6
votes
0answers
81 views

A very difficult problem about the existence of following $SU(2)$ matrices?

Let $G_i$ be a sequence of $2\times2$ $SU(2)$ matrices, where $i=1,2,...,n$; and $P$ represents a permutation of $\left \{ 1,2,...,n \right \}$. The question is: Does there exist a sequence of ...
6
votes
0answers
47 views

Two definitions of real flag manifolds: do they coincide?

Let $G$ be a real semisimple Lie group with finite center. Definition 1 A real flag manifold is a homogeneous space $G/Q$ where $Q$ is a parabolic subgroup of $G$. Definition 2 Let $K$ be a ...
6
votes
0answers
133 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
6
votes
0answers
138 views

ODE system and Lie symmetries

The ODE system (see below), where $F$ is a given function together the algebraic condition (SYM) imply that $$\boxed{y(t)=k-x(t)} \tag{*}$$ for some $k$ constant. The result that $y$ is a translation ...
6
votes
0answers
259 views

Root Systems of Lie Groups.

Let $G$ be a compact Lie group assumed to be a subgroup of $U(n)$. Also, let $T$ be a maximal torus of G. Then there exists a basis $\{v_1, \ldots ,v_d\}$ of the Lie algebra of $G$, $\mathfrak{g}$, ...
6
votes
0answers
147 views

Finding the universal cover of a matrix group

Consider the group $G = \left\{\begin{pmatrix} a & b\\ 0 & 1\end{pmatrix}: a \in \mathbb{C}^{\times}, b \in \mathbb{C}\right\} \subset GL(2, \mathbb{C})$. How does one find the universal cover ...
6
votes
0answers
397 views

Alternating and special orthogonal groups which are simple

I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if ...
5
votes
0answers
54 views

Is every complex Lie algebra a complexification?

I'm wondering if every finite-dimensional complex Lie algebra can be written as a complexification of a real Lie algebra. At the vector space level, clearly every $\mathbb{C}^n$ is a complexification ...
5
votes
0answers
51 views

Convolution of matrix coefficients is also a matrix coefficients

I have a question about the convolution of matrix coefficients as follows: Let $G$ be a compact Lie group. A Map $f:G\rightarrow \mathbb{C}$ is called a matrix coefficient if there is a finite ...
5
votes
0answers
67 views

Exponential of a polynomial of the differential operator

Given that $$\exp(aD)f(x)=f(x+a)$$ where $\exp(D)$ is the exponential of the differential operator $D$, is there a similar closed-form, general expression for $\exp(g(D))f(x)$, where $g(D)$ is a ...
5
votes
0answers
77 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
5
votes
0answers
41 views

Covering spaces of Lie groups

In a paper of Tom Bridgeland's, he describes an action by the universal over $G:=\tilde{GL^+}(2,\mathbb{R})$ using a description of $G$ I find unintuitive. Namely, he indexes write the fiber over ...
5
votes
0answers
99 views

— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...
5
votes
0answers
181 views

Is $G$ a lie group if left multiplication is smooth and multiplication is smooth near $e$?

Suppose $G$ is a smooth manifold and also a topological group. Also suppose that left multiplication $L_g : G \rightarrow G$ is smooth for any $g \in G$. Finally suppose that the multiplication map is ...
5
votes
0answers
40 views

Does every element of $G_2 = \mathrm{Aut}(\mathbb{O})$ stabilize a quaternion subalgebra?

Has anyone heard of this result before? Let $\mathbb{O}$ denote the octonions and let $G_2$ denote its automorphism group (i.e. the 14 dimensional subgroup of $SO(7)$). Then any element of $G_2$ ...
5
votes
0answers
129 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 ...
5
votes
0answers
467 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 ...
5
votes
0answers
95 views

Quantum Adjoint Action of the Coordinate Algebra on the Enveloping Algebra

As is well known, any Lie group $G$ has a canonical action on its Lie algebra $\frak{g}$, namely the adjoint action $Ad$. Firstly, let me ask, does this extend to an action of $G$ on its enveloping ...
5
votes
0answers
150 views

Generating function for characters of representations

One example of such a generating function that I know how to derive is for $SU(2)$, $\frac{1}{(1-tx)(1-\frac{t}{x})}$. The coefficient of $t^n$ in the above function is the character in the $n+1$ ...
5
votes
0answers
111 views

Pullback of a 3-form to SU2

I have a left invariant 3-form, $\sigma$ on an simply connected Lie group, $G$ whose value at the identity is $\sigma=\langle[x,y],z\rangle$, where $\langle\cdot,\cdot\rangle$ denotes an invariant ...
5
votes
0answers
274 views

Short exact sequences of topological groups and Lie groups

could someone please clarify the definitions of extensions of topological groups and Lie groups. For topological groups, what I see in most papers is as follows: An extension of topological groups $0 ...
4
votes
0answers
78 views

Embedded Lie subgroups are closed.

This is Exercise 2.1 from Kirillov's Lie theory book. Let $G$ be a Lie group and $H$ a closed Lie subgroup. Show that the closure $\overline{H}$ of $H$ in $G$ is closed in $G$. Show ...
4
votes
0answers
61 views

Exact sequences of SU(N) and SO(N)

We know that the spin group $Spin(N)$ has a short exact sequence of Lie groups. $$1 \to Z_2 \to Spin(N) \to SO(N) \to 1$$ I wonder whether there are some examples for SU(N) and SO(N) group, such ...
4
votes
0answers
77 views

When do invariant measures arise from smooth differntial forms?

It is well known that the Haar measure of a Lie group $ G $ arises from a invariant differential form density $ |\omega| $ (of top dimension). Also, we know that if we have a closed subgroup $ H \leq ...
4
votes
0answers
81 views

Making the definition of dual root unambiguous

In 5.4 of his book Lectures on Invariant Theory, Igor Dolgachev introduces the dual of a root by requiring that $\check\alpha(t) f_\alpha(x) \check\alpha^{-1}(t)= f_\alpha(x)$ ...
4
votes
0answers
102 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
4
votes
0answers
53 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
4
votes
0answers
46 views

On compact, simply connected Lie group and its subgroup

Let $G$ be a compact, simply connected Lie group, and $H$ is its Lie subgroup that is also compact and simply connected, and has the same dimension with $G$, then should $H=G$? Note that these imply ...
4
votes
0answers
165 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
4
votes
0answers
138 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
4
votes
0answers
373 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 ...
4
votes
0answers
41 views

Relationship between representations of $\mathfrak{sl}_{2n}\mathbb{C}$ and $\mathfrak{sp}_{2n}\mathbb{C}$

If $V=\mathbb{C}^{2n}$ denotes the standard representation of $\mathfrak{sl}_{2n}\mathbb{C}$, what can we say about $\wedge^kV$ in terms of the standard representation $W$ of ...
4
votes
0answers
149 views

Irreducible representations of $\mathfrak{sl}_3\mathbb{C}$

I am working through the exercises in Fulton and Harris's Representation Theory, and am stuck on two on page 189. Let $\text{Sym}^2V$ denote the second symmetric power of the standard 3-dimensional ...
4
votes
0answers
28 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)? ...
4
votes
0answers
179 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 ...
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 ...
4
votes
0answers
424 views

Weyl Character formula applied to Sp$(4,\mathbb{C})\cap$ U$(4)$.

I posted a question a short while ago on this but got no response. I have worked on this more and so now have a more specific question. To start with we work with the $\mathbb{Q}$ version of ...
4
votes
0answers
263 views

Bi-invariant form on compact connected Lie group

Let $G$ be a Lie group and $\omega$ be a left invariant $k$-form, how to prove that $r^*_a \omega$ is left invariant? What I do: $(l^*_g (r^*_a \omega))_x (v_1, \ldots,v_k)=(r^*_a \omega))_{gx} ...
4
votes
0answers
133 views

Non-closed subgroups of Lie groups

The following are my impressions from playing with a line of irrational slope $\gamma$ in the standard torus $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$, $\mathbb{R}\hookrightarrow ...