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 (group-...

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25 views

Every orthogonal matrix represents a rotation around an axis

Is it true that every element of the group $O(n)$ represents a rotation around some axis? I'd like this to be true in order to decompose any matrix $R \in O(n)$ as a block matrix in $O(n-1)$ and a 1 ...
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25 views

SU(N) tensor product decomposition

Let's consider the group SU(N). The adjoint representation is $\textbf{Adj}= $ $\textbf{N}^2\textbf{-1}$. The following decomposition holds generally ( have a look at this ref ) $$ \textbf{Adj}\otimes\...
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62 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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36 views

The Virasoro-Bott group and the KdV equations

The Euler Arnold equation expresses equations (usually from mathematical physics) as geodesic equations on a Lie group. For the famous $KdV$ equations these equations are given on the Virasoro-Bott ...
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2answers
38 views

A semisimple Lie group has no character; Am I right?

Let $G$ be a compact connected Lie group with semisimple Lie algebra ${\frak g}$. With the following reasoning, I show that there is no non-trivial Lie group homomorphism $$\chi:G\to S^1.$$ Is that ...
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20 views

Exponentiating an ``affine subalgebra''

Consider the Lie algebra $u(N)$. If I exponentiate an $x \in u(N)$, I will obtain an element of the $U(N)$ group. My understanding is that $exp(u(N))= \{exp(x)| x \in u(N)\}$ is, in fact, the group $U(...
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1answer
37 views

Left-invariant vector fields on the circle $S^1$

I'm trying to find the left-invariant vector fields on the circle $S^1$. If I understand correctly, $S^1$ is given the group structure of the multiplicative group of complex numbers on the unit ...
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1answer
26 views

Coset spaces of a lie group and fiber bundles

I am reading Steenrod. He writes: Another example of a bundle is a Lie group $B$ operating as a transitive group of transformations on manifold $X$. The projection is defined by selecting a point $...
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29 views

Topologies of partially exponentiated lie algebras, especially in regard to $SU(2)$

Consider the fundamental respresentation of $\mathfrak{su}(2)$ given in terms of the Pauli matrices as $\mathfrak{su}(2) = \langle \frac{i\sigma_1}{2},\frac{i\sigma_2}{2},\frac{i\sigma_2}{2}\rangle_{\...
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36 views

Harmonicity on semisimple groups

Let $G$ be a semisimple real Lie group, $U(\mathfrak{g})$ its universal enveloping algebra, let $\Omega$ be the Casimir element in $U(\mathfrak{g})$ and let $f$ be a smooth (or analytic) real-valued ...
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1answer
21 views

What's its use of the nonsingular 2-step nilpotent Lie algebras

What's its use of the nonsingular 2-step nilpotent Lie algebras which form an a class of 2-step nilpotent Lie algebras ? Recall: A $2$-step nilpotent Lie algebra $N$ is non-singular if $ad X : N \to ...
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21 views

how to show that the representation of $SL(2, \mathbb{C})$ is holomorphic

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to GL(...
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1answer
35 views

Showing that a very well-known representation is really a representation

Fix an integer $n\geq 0$, and let $V_n$ be the complex vector space of polynomials in two variables $z_1$ and $z_2$ homogeneous of degree $n$. Define a representation $$\phi_n:SL(2,\mathbb{C})\to GL(...
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0answers
20 views

Problems with Matrix of irreducible representation of SO(3)

I'm working out the irreducible representation of $SO(3)$. Let's call $R_{\theta}$ as a general rotation and $Y_{m}^{l}\left(\eta,\varphi\right)$ the spherical harmonics. I now like to have the matrix ...
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1answer
45 views

Classification Problems in Lie algebra

My field of interest is Lie group analysis of PDEs, currently I am struggling with techniques for classification of Lie algebra into mutually conjugate classes of 1- and 2-dimensional sub-algebras ...
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17 views

Centralizer of Maximal Toral Subalgebra of $L$

I am trying to understand the proof of the following result from Humphreys Lie Algebra book: Let $L$ be a semisimple complex Lie Algebra.Let $H$ be Maximal toral subalgebra of $H$,Let $C_L(H)$ ...
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1answer
80 views

Relations between center (fundamental group) and (co)root and weight lattices for Lie groups

I would like to find some explanation or reference for the following facts, provided they are correct, and clarify some of the assumptions. Denote by $G$ a (perhaps semisimple compact connected) Lie ...
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26 views

Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition $$V_l=...
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1answer
24 views

Can we discribe a Lie group action from some local property?

Let G be a Lie group,and it acts on a smooth manifold M.Then can we get that the action is transitive from some local property of the Lie group action.More precisely,Can we get the action is ...
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1answer
47 views

Quotient by a discrete subgroup of a Lie group

I was reading Fulton Harris' Representation theory, A first course, where I came across the following: Let $H$ be a Lie group and $T$ be a discrete subgroup of its center $Z(H)$. Then there exists ...
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1answer
19 views

Lie group structures inside of Clifford algebras

I am reading a text by Jean Gallier on Clifford algebras, Pin and Spin groups. I have a problem with one little innocent-looking paragraph establishing Pin and Spin as Lie groups on the page 37. I don'...
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1answer
25 views

$S^1$ acting on $SO(n+1)/SO(n-1)$ by translations

I'm ready right now in a paper, that $S^1$ acts on $SO(n+1)/SO(n-1)$ by right translations. I thought that a Liegroup $G$ acting by right translations, means that we have a right action $\varphi \...
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18 views

If the continuous action of a non-compact Banach-Lie group on a Banach space preserves the zero element, then it is non-proper.

I am studying the differential geometry of Banach-Lie groups, specifically, the differential geometry of the orbits of an action of a Banach-Lie group on a Banach space, and I ended up "proving" the ...
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25 views

Relationship between Casimir and index of a representation of a Lie algebra.

In several QFT textbooks (namely, those of Peskin and Shroeder and of Schwartz) there is presented an identity for representations of Lie algebras, $$ d(R) C_2(R) = T(R) d(G),$$ where $d(R)$ is the ...
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17 views

Is this covering group $G'$ of $G$ unique?

Let $G$ be a Lie group (not necessarily connected) acting effectively/faithfully on a connected, locally path connected, semi-locally simply connected space $X$ (not necessarily with fixed points). ...
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0answers
16 views

In what condition, the universal covering group of some Lie group is a matrix Lie group.

As we know the universal covering group of $GL(n,\mathbb{R})$, $SL(n,\mathbb{R})$ is a Lie group which cannot be faithfully represented by a finite dimensional matrix. Therefore what's the sufficient ...
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39 views

Why Bi invariant metric on noncompact lie group doesn't exist??

In the book "Lectures on Differential Geometry" by Sternberg page 233 "Given a representation,p, of a Lie group G (in particular the adjoint representation) on a vector space F, if p(x) is compact ...
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24 views

Show that $\left.\dfrac{d}{d s}\right|_{s=0}X_\eta (m)(f\circ \varphi_{\exp(s\xi )})=(X_\eta f)_{*,m}(X_\xi (m))$

Let $G$ be a Lie group, $M$ be a manifold, $\varphi:G\times M \rightarrow M$ smooth action of $G$ over $M$ where $\varphi(g,m)=g\cdot m$. For each $\xi \in \mathfrak{g}=T_e G$ we difine the ...
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1answer
87 views

Do I understand the Chevalley Restriction Theorem correctly?

Let $G$ be a complex semisimple Lie group with Lie algebra $\frak g$, and let $\frak h$ be a Cartan subalgebra with Weyl group $W$. The Chevalley Restriction Theorem states that the restriction map $\...
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30 views

what's the universal covering group of general linear group $GL(n,\mathbb{R})$ and $GL(n,\mathbb{C})$

Because the general linear group is not simply connected, they must be able to be covered by some simply connected Lie group. But I cannot imagine which Lie group with same dimension can cover $GL(n,\...
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0answers
18 views

Generators in adjoint representation are structure constants

Given that $g T_a g^{-1} = D^b_a T_b$ one can show that the generators in the adjoint representation of a group $G$ are the structure constants of the lie algebra satisfied by the $T_a$. Write $g$ ...
0
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1answer
26 views

Compute the associated induced Lie algebra action $\text{d}\pi$

Let $G=\mathrm{SL}_2(\mathbb{C})$ and consider the action of $G$ on the space of smooth functions on column vectors $\mathbb{C^2}$ given by $\big(\pi(g)\phi\big)(v)=\phi\left({g^\top}\,v\right)$ for ...
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1answer
35 views

Let $G$ be a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogeneous space $\frac{G}{H}$ are connected, then $G$ is connected.

Let $G$ ge a Lie group and $H$ be a closed subgroup of $G$. Show that if $H$ and homogenuous space $\frac{G}{H}$ are connected, then $G$ is connected. Remark: For this proof I use the following ...
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2answers
67 views

Calculating the differential of the inverse of matrix exp?

Let $A(t)$ and $B(t)$ be two matrix-valued smooth function satisfying the equation, $B(t) = e^{A(t)}$. I need to express $\frac{dA(t)}{dt}$ in terms of $B(t)$. I know that there is a formula of Wilcox,...
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2answers
51 views

Differential of the multiplication and inverse maps on a Lie group

I'm trying to solve the following two problems: Let $G$ be a Lie group with multiplication map $\mu\colon G\times G\to G$, and let $\ell_a\colon G\to G$ and $r_b\colon G\to G$ be left and right ...
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1answer
34 views

Why does $(X_a,Y_b)(f\circ \mu(a,b)) = X_a(f\circ R_b(a)) + Y_b(f\circ L_a(b))$?

On the first page of these notes: if $\mu\colon G\times G\to G$ is the multiplication map on a Lie group $G$, then given a point $(a,b)\in G\times G$ and letting $R_b$ and $L_a$ denote right-...
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1answer
25 views

Proving that Lie groups are locally connected

I'm trying to show that if $G$ is a Lie group, then it is locally connected, i.e., for any point $p$ in an open subset $U$ of $G$, there is a connected neighborhood $V$ of $p$ such that $V\subset U$. ...
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0answers
29 views

References for actions of infinite-dimensional Banach-Lie groups on infinite-dimensional Banach manifolds

I am starting to study infinite-dimensional manifolds, specifically, Banach manifolds. I found some interesting introductory texts in which the mathematical background is developed with some detail. ...
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1answer
30 views

Is there a harmonic analysis theory for the group of strictly increasing continuous invertibe functions from $\mathbb{R} \mapsto \mathbb{R}$

I have very little to add to the title, would appreciate pointers to the literature. A follow up question I have is: If one considers differentiable strictly increasing functions, do they lend ...
4
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1answer
38 views

Differentiability of an action of the group of invertible elements of a $C^{*}$-algebra $\mathcal{A}$ on the dual of $\mathcal{A}$

I am studying the actions of Banach-Lie groups on Banach manifolds, and I am not able to concretely evaluate the differentiability properties of a specific action. Let $\mathcal{A}$ be a unital $C^{*}...
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0answers
41 views

Why can we find a basis for the elements of the Lie algebra?

I am a physicist and we do Lie algebras pretty informally, so I hope my question makes any sense to a mathematician. There is one thing that I don't quite understand, which is why we can find a basis ...
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0answers
24 views

How to construct free subgroup of SO(3) using Baire's Theorem?

I'm looking to demonstrate that there exists a free group on two generators $ G\subseteq SO(3) $ (this is for homework, so hints are preferred to complete answers I've turned it in now, so full ...
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1answer
51 views

Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
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0answers
38 views

Compact Lie group $G$ with Lie algebra $\frak g$ satisfying $gZg^{-1}=-Z$ for $Z\in\frak g$ and $g\in G$

Let $G$ be a compact Lie group with Lie algebra $\frak g$. Are there known conditions on $G$ guaranteeing the following property: $$ \hbox{For each $Z\in\frak g$ there exists an element $g\in G$ ...
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1answer
16 views

The Lie algebra of the generalized unitary group $\{g \in GL_n(\mathbb{C}) : gS\bar{g}^t=S\}$ is $\{XS+S\overline{X}{}^t=0\}$

Let $ S \in M_n(\mathbb{C}) $ be a square matrix and let $ X$ be in the Lie algebra $\mathbb{\mu(S)} $ of the generalized unitary group, $$U(S):=\{g \in GL_n(\mathbb{C}); gS\bar{g}^t=S\} .$$ ...
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1answer
29 views

Adjoint matrix in $\mathbb{so_3}$

$\mathbb{so_3}$ has the following basis: $X_1=\begin{bmatrix} 0 & & \\ & &1 \\ & -1 & \end{bmatrix}$, s: $X_2=\begin{bmatrix} & & 1\\ & 0& \\ -1 & & ...
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1answer
32 views

Show that $ (\pi(g)\phi)(v)=\phi({^t}gv) $ defines a representation

Let $ G=SL_2(\mathbb{C}) $ and consider the action of $ G $ on the space of smooth functions on column vectors $ v \in \mathbb{C^2} $ given by: $ (\pi(g)\phi)(v)=\phi({^t}gv) $ Question 1: Show that ...
1
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1answer
37 views

Compute $ad_X$, $ad_Y$, and $ad_Z$ relative to a basis

For a lie algebra $\mathbb{g} $ we can define the adjoint representation as: $ ad: \mathbb{g} \rightarrow End(\mathbb{g}) $ as the map such that $ad_x(y)=[x, y] $ for all $\in \mathbb{g} $ I am ...
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0answers
22 views

Lie algebra of $SL_2(\mathbb{R})$ and show $\exp(X)=I+X$ where $I \in SL_2(\mathbb{R}) $ and $X \in sl_2(\mathbb{R})$

I am doing an undergraduate course on Representation Theory and am trying to solve these consecutive questions. The first two I am ok with (I just included them for context), but I could do with some ...
3
votes
0answers
24 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: \...