Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
3,059 questions
2
votes
0
answers
263
views
Type I unimodular groups in physics
Many of the physical symmetry groups are type I and unimodular. The unitary representations of type I second countable groups in separable Hilbert spaces can be given in a direct integral form which ...
- 21
-2
votes
1
answer
236
views
How to construct/characterize "Thermal" sections ?
There were errors in the way I framed the question last time. So doing a major revision this time.
Consider $SU(2)$ as a homogeneous space $SU(2)\times SU(2)/SU(2)$ and sections of this principle ...
- 3,541
1
vote
1
answer
163
views
Explicit Coquasi-Triangular Quantised Coordinate Algebra of a Complex Semi-Simple Lie Group?
Let $SL_q(N)$ be usual quantised coordinate algebra of the special linear group. As is well-known, this is co-quasi-triangular algebra with coquasi-triangular structure given by
$$
R(u^i_j \otimes u^...
- 1,462
0
votes
0
answers
187
views
reductive lie group without $R$-characters compact?
Let $G$ be the real points of an affine algebraic group
defined over $R$. If there is no non-trivial characters
$G\to R^*$, does it imply $G$ is a compact lie group?
I guess the paper of Borel and ...
- 1
2
votes
1
answer
197
views
Polytopes related to the conjugation action of a Lie group on multiple copies of itself?
Let G be a finite dimensional real Lie group. As I understand it, the quotient space of G acting on itself by conjugation is a well studied polytope which can be identified with the fundamental alcove ...
- 27.5k
1
vote
0
answers
66
views
topologies on globalizations
I am reading notes by David Vogan on Unitary representations and Complex analysis (pdf / dvi).
The setting is as follows (page 23): Let $X$ be a $(\mathfrak{g},K)$-module and let $X(\mu)$ denote its ...
- 8,597
1
vote
0
answers
117
views
How to Prove that nilpotent Lie groups satisfy the Leptin condition?
A topological group $G$ is said to satisfy the Leptin condition if for every compact subset $K\subseteq G$ and for every $\epsilon>0$ there exists a compact subset $L$ such that
$\mu(LK)$ < $ (...
- 2,479
0
votes
0
answers
61
views
Representation and Laplacian on the Heisenberg group
Let $\pi_{\lambda}$ be the the Schrödinger representations of the Heisenberg group $H=\mathbb C\times \mathbb R$. For $\varphi\in L^2(\mathbb R)$, we have
$$\pi_{\lambda} (z,t)\varphi(\xi)=e^{i\...
- 650
0
votes
0
answers
87
views
Some details about Kirillov-Kostant Poisson bracket
Let $G$ be a finite dimensional Lie group with Lie algebra $\mathfrak{g}$. The Kirillov-Kostant Poisson bracket on $\mathfrak{g}^*$ is defined as $$\{\cdot ,\cdot \} :C^{\infty}(\mathfrak{g}^*)\times ...
- 287