New answers tagged lie-groups
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Show that $\lambda:G \to \mathbb{R}^\ast$ where $R_h^\ast \mu = \lambda (h) \mu$ is smooth
The multiplication map $m\colon G\times G\to G$, $m(g,h) = gh$, is smooth. This means that it induces a smooth bundle map $Dm\colon T(G\times G)\to TG$. In particular, $R_h(g)=m(g,h)$ and so for $X\in\...
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Homomorphisms $U(1) \to SU(2)$
The matrices in the image commute and thus are simultaneously diagonalizable, so indeed the same $h$ works for all.
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Generators for general / special linear groups
The reference for the 2-element generators mentioned by
brett stevens is
D. E. Taylor:
Pairs of Generators for Matrix Groups. I,
The Cayley Bulletin, 3, 1987, 76-85
https://arxiv.org/abs/2201.09155v1
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Definition of smooth action
Let $G=S^1=SO(2)$, the group of rotations around the origin in the Euclidean plane. Every rotation is either rational (has finite order) or is irrational (has infinite order).
Verify that $G$ is ...
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The natural representation of the real group $G=SO(2)$ on $V=\Bbb R^2$ is irreducible.
A unitary representation requires that the representation space is a vector space over the complex numbers, which $\Bbb{R}^2$ is not. Thus, you cannot use $(a)$. This is not a minor annoyance, it is ...
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Prove that $\exists g \in SO(n)$ such that $gv_i = w_i$ for $i = 1, 2$ where all these vectors are O.N.
$v_1, v_2$ can be completed to an orthonormal basis (of $\Bbb{R}^n$) $v_1, v_2, \ldots, v_n$. Now put the $v_i$ as column vectors in a matrix $V$. Then $V\in O(n)$. If $\det(V)=-1$ you can invert ...
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Is the fundamental group of a closed orientable hyperbolic $3$-manifold isomorphic to a non-discrete subgroup of $\mathrm{PSL}(2, \mathbb{C})$?
Here is a construction, but at this moment I do not have a proof of nondiscreteness (in general).
Let $\Gamma< SL(2, {\mathbb C})$ be a cocompact discrete subgroup (a uniform lattice): The ...
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Commutator of a connected semi simple Lie group
The commutator subgroup $ [G,G] $ is a normal subgroup of $ G $. And $ G/[G,G]=G^{ab} $ is abelian. So suppose that $ G $ is semisimple and connected. Then $ G^{ab} $ is a semisimple Lie group. So $ G^...
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G abelian Lie group, $(\pi, V)$ a finite-dim unitary rep. $\exists $ mutually orthog. 1-dim invariant linear subspaces s.t $V= \bigoplus_i V_i$
Let $\{\pi(g):g\in G\}\subset GL(V)$. These are unitary operatots that also commute ($\pi(g_1)\pi(g_2)=\pi(g_1g_2)=\pi(g_2g_1)=\pi(g_2)\pi(g_1))$.
Moreover, as they are unitary, they are also ...
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details in the proof: G a commutative Lie group and $(\pi, V)$ a finite-dim unitary representation of G. Then $\pi$ is irreducible iff dim $V = 1$
A representation assumes that the map $\pi:G\times V\to V$ is continuous. So the composition $V\to \{g\}\times V\subseteq G\times V \xrightarrow{\pi} V$ is continuous. The topologies taken are the ...
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Why is $ V \longrightarrow V: v \mapsto \pi(g)(v)$ an endomorphism of $G$-modules?
Lets write down explicitly:
$$f:V\to V$$
$$f(v)=\pi(g)(v)$$
Or simply $f=\pi(g)$. It is linear because $\pi(g)$ is. It is a bijection because $\pi(g^{-1})$ is inverse of $\pi(g)$.
So the only question ...
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How do projective representations map elements? Are they mulivalued?
I have rewritten my answer.
Let $V$ be a vector space on a field $k$. Let us recall that if $V$ is a vector space, $\mathbb{P}(V)$ is the set of equivalence classes of vectors up to the multiplication ...
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Accumulation point in a topological group of orthogonal matrices over R
Let $G$ be a compact Hausdorff group (in your case the group of orthogonal matrices) and let $g\in G$. Of course $\overline{\langle g\rangle}$ is a compact subgroup of $G$.
If $|g|<\infty$ then $\...
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A question related to the action of a group on the real projective space
The action $G \times \mathbb{P} \to \mathbb{P}$ is what we might expect: $(g, \mathbb{R}v) \mapsto \mathbb{R}gv$, but we have to verify that it's well-defined. For this consider any nonzero multiple $...
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dual representation map as a linear map
Every linear map $f:V\to W$ induces a linear map $W^*\to V^*$. This linear map is denoted by any of the following symbols: $f^*,f^t,f’$, and is called the dual map or adjoint map or transpose map or ...
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How can I show this property about lie algebras and lie subgroups?
Continuing off of your approach, it suffices to show that $\overline{\mathfrak h}$ is commutative. Consider any $X,Y \in \overline{\mathfrak h}$. $e^{sX}$ and $e^{tY}$ commute for all $s,t \in \Bbb R$ ...
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Extensions of simple Lie groups with a topological splitting
Yes, indeed, if $H$ is abelian then the Lie group central extensions of $G$ by $H$ are classified by the 2ned continuous cohomology $H^2_c(G; H)$. A good survey can be found in
Stasheff, James D., ...
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Is there a way to convert an element of $\mathfrak{so}(3)/\mathfrak{so}(2)$ into a "geometric" tangent vector on $S^2$?
Consider $\Bbb S^n\subseteq \Bbb R^{n+1}$, write $\Bbb R^{n+1} = \Bbb R^n \times \Bbb R$, and fix the north pole $p = (0,1)$. Note that $T_p(\Bbb S^n) = \Bbb R^n\times \{0\}$. Then ${\rm SO}(n+1)$ ...
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What are the invariant polynomials of representations of the Lorentz group $SO^+(3,1)$ and $SL(2,\mathbb{C})$?
The case $SL(2)$ is the well known problem of the classical invariant theory. You may find lists of invariants for example here
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Set of positive definite Hermitian matrices as quotient
In general, when a group acts on a space $\Omega$, for any $x\in\Omega$, the fibers of the map $G\to\mathrm{Orb}(x)$ given by $g\mapsto gx$ are the cosets of $\mathrm{Stab}(x)$. Thus, $g\mathrm{Stab}(...
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Understanding the Geometry Spawned from Quotient Spaces $GL^+(4,R)/SO(3,1)$, $GL^+(4,R)/Spin(3,1)$, and $GL^+(4,R)/Spin^c(3,1)$
The group $Spin(3,1)$ is nothing but $SL(2, {\mathbb C})$. The latter has a natural embedding in $GL(4, {\mathbb R})$. The subgroup
$SL(2, {\mathbb C})< GL(4, {\mathbb R})$ preserves two things:
...
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Reference for universal cover of $\mathrm{SL}_2(\mathbb{R})$
I doubt this will answer all the parts of your question, but the following paper on Geometries of 3-manifolds by Peter Scott has a nice description of $\tilde SL_2(\mathbb{R})$.
Algebraically, it can ...
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Complexification of $G-$representation where $\mathrm{End}_G(V)\cong\mathbb{C}.$
So I believe that I have an answer. We know that $$\mathrm{End}_G(V\otimes_{\mathbb{R}}\mathbb{C})\cong \mathrm{End}_GV\otimes_{\mathbb{R}}\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}\cong \...
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Manifold of $SU(2)$ and it's relation to physical rotations
I think I figured it out!
Indeed, any element $U\in SU(2)$ can be represented as
\begin{equation}
U=e^{i\frac{1}{2}\vec{\sigma}\cdot \hat{n}\theta}
\end{equation}
where $\hat{n}$ is a 3-dimensional ...
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Isomorphism between $\mathbb{G}_m\times SL(n)$ and $GL(n)$
Yes, this holds quite generally.
Let us assume that $G$ is an algebraic group i.e., a finite type group scheme over some field $k$. There is a natural map
$$p\colon G^\mathrm{der}\times Z(G)\to G,\...
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The order in which diffeomorphisms should be applied
Introducing extra notation for the intermediate steps of the computation and worrying about the point evaluations at $z\in S^1$ separately might make things clearer.
Let $g:=\pi(\xi_2)f$. Applying the ...
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Lie algebras are to Lie groups what Lie modules are to …?
This is not the common definition of a Lie module. Usually, a Lie module (aka Lie algebra representation) is defined as a module on which a Lie algebra acts (in a suitable way). A Lie algebra can be ...
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Understanding Representation theory
We know the group $G=\mathrm{SL}(2,\mathbb{R})$ acts on $\mathbb{R}^2$. Thus, it acts on the real projective line $\mathbb{RP}^1$. With the congruence $\big[\begin{smallmatrix} x \\ y \end{smallmatrix}...
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Isomorphic connected Lie subgroup
It seems to me that your approach is too complicated. By construction, the homomorphism $\Phi:K\to H$ has bijective derivative. But surjectivity of the derivative and connectedness of $H$ imply that $\...
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Triangle inequality for geodesic distance on groups
Saying "geodesic distance" is not sufficient to characterize the distance function on a Lie group with, say, a bi-invariant metric (note that the latter may not be unique), because the ...
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Irreps of crystallographic groups
I'll stick to the notation from the ncatlab article linked in the post. $S$ is the crystallographic group, $N \cong \mathbb{Z}^n$ is the normal subgroup of translations, and $G = S/N$ is the point ...
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Explicit expression for simple roots of the root systems $A_2$, $B_2$ and $G_2$ in 2D
To get this off the unanswered list:
An explicit realization of a root system of type $A_n$ in $n$-dimensional Euclidean space (that is, with coroots / reflections defined to match the standard ...
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Prove that every finite subgroup of $SL_2(\mathbb{C})$ is conjugate to a finite subgroup of $SU_2$
There is such a Moore's theorem from 1898 (Eliakim Hastings Moore: 1862-1932). Every finite group $G\leq GL_n(\mathbb{C})$ is conjugate to a subgroup of the unitary group $U_n$. In your case $n=2$.
...
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$G$-equivariant map between Lie group representations
It depends on the context, but the answer to your first question should be yes if you are thinking of complex semi-simple Lie groups. Indeed, this should be true in any setting where your ...
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Complexification of $G-$representation where $\mathrm{End}_G(V)\cong\mathbb{C}.$
The key idea is that the center of a compact Lie group is trivial, meaning that the only possible irreducible real representations of G are trivial.
Let's analyze each case:
EndG(V)≅C: Based on the ...
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How to define a Lie group that is a Riemannian manifold and has an anticommutative group operation?
The OP clarified in a comment that by "anticommutative" he means "noncommutative". If so, the simplest example of a noncommutative Lie group is the 3-sphere $S^3$. A natural ...
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Product of spherical tensors
I denote the $(2j+1)$-dimensional representation by $S^{2j}\mathbb{C}^2$ (physicists, I believe, just write the dimension $\mathbf{2j+1}$ in bold).
Let $M_j=S^{2j}\mathbb{C}^2 \otimes S^{2j}\mathbb{C}^...
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Weyl Group of Parabolic subgroups
In this case the Weyl group is trivial. For $SL(n,\Bbb R)$, if we set $A_o$ equal to the usual diagonal split component of $B$ then the Weyl group of $A_o$ is $S_n$ acting by permuting the diagonal ...
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Uniform open subgroups of $p$-adic Lie group
No: there is no guarantee that the subgroup generated by $H_1$ and $H_2$ is still torsion-free. For a simple example, let $\Gamma=\mathbb{Z}_p\times\mathbb{Z}/(p)$, let $H_1$ be the closed subgroup ...
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Equivariant map to homogeneous space is a fiber bundle
$p$ is a fiber bundle without assuming the $G$ action on $X$ has closed orbits. In fact we can write down a local trivialization of $p$ in terms of a local trivialization of $G \to G/H$.
Consider a ...
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Finding the dimension of the symplectic group
According to Wikipedia, we have the diffeomorphism
$$Sp(2n,\Bbb R)\cong U(n)\times\Bbb R^{n(n+1)}$$
of manifolds. You may see also this bounty question.
So,
$$\dim Sp(2n,\Bbb R)=\dim U(n)+n(n+1)=n^2+n^...
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How can I write $M_2(\mathbb{C})$ as a direct sum?
$$M_2(\mathbb C)=\bigg\{\begin{pmatrix}a&b\\0&0\end{pmatrix}:a,b\in\mathbb C\bigg\}\oplus\bigg\{\begin{pmatrix}0&0\\a&b\end{pmatrix}:a,b\in\mathbb C\bigg\}$$
is a decomposition into $G$...
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