Questions tagged [lie-groups]
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
78 questions from the last 365 days
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Spherical representations of $(G,H)$ with $H$ not connected
Let $G$ be a compact connected semisimple Lie group.
Let $H$ be a closed subgroup of $G$ not necessarily connected.
Let us denote by $H^0$ the connected component of $H$ containing the trivial element ...
- 2,225
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Verification request: three geometric steps in a parameter-free derivation of α⁻¹ [closed]
I am seeking verification of three specific mathematical claims arising from a geometric framework. I am not asking for evaluation of the broader physical interpretation — only whether these three ...
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When Instant-on weights for A2 sub-algebra embedded in E8 via E8 ⊃ E6 × A2 — what selects the vacuum angle?
Consider the maximal subgroup decomposition of $E_8$:
$$\mathbf{248} = (\mathbf{78},\mathbf{1}) + (\mathbf{1},\mathbf{8}) + (\mathbf{27},\mathbf{3}) + (\overline{\mathbf{27}},\bar{\mathbf{3}})$$
under ...
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Definition of a semisimple p-adic analytic group
I would like to understand what it means for a $p$-adic analytic group $G$ to be semisimple.
I am especially interested in the case when $G$ is a subgroup of $\mathrm{GL}(n,\mathbb{Q}_p)$ that is ...
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Lie group one cocycle coming from a Manin triple induces a Poisson tensor
I am currently reading the paper on Poisson Lie groups, dressing actions, and Bruhat decompositions from Lu and Weinstein. Given a Manin triple $(\mathfrak g,\mathfrak g_+,\mathfrak g_-)$ by Theorem 3....
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Basic de Rham cohomology and integral classes
$\newcommand{\bR}{\mathbb{R}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\dR}{\text{dR}}\newcommand{\iu}{\mathrm{i}}\newcommand{\du}{\mathrm{d}}\newcommand{\fg}{\mathfrak{g}}\newcommand{\bas}{\text{bas}}$...
- 2,605
4
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Simplicity of $\mathfrak{g}$-sub-modules generated by a single vector
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $V$ a finite-dimensional $\mathfrak{g}$-module. Take a highest weight element $v$ in $V$ and consider the submodule generated by $v$ that is ...
3
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Is a group with a lattice of type I?
Let $G$ be a locally compact group. A lattice is a discrete subgroup $\Gamma$ of finite covolume. A type I group is a group whose $C^*$-algebra is type I, i.e., each factor representation is ...
- 873
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Representation category that contains both $(\mathfrak g,K)$-modules and unitary reps
Let $G$ be a connected, semi-simple real Lie group. Let $\mathfrak g$ be its Lie algebra. And let $K\subset G$ be a maximal compact. I am looking for a category $C$ with the following properties. I ...
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For which $n$ is the minimum max entry of an $n$ by $n$ orthogonal matrix known? Is $n=3$ already an open problem?
Crossposted on Mathematics SE, where the question Orthogonal matrices with small entries was brought to my attention, though it is about bounds rather than exact values.
Let $\| A \|_{\max} := \max\...
- 4,887
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The forward orbit equidistributes
I am working through some exercises in Homogeneous Dynamics and I am stuck on the following problem regarding the equidistribution of orbits in a simple Lie group.
The Setup
Let $G$ be a simple Lie ...
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1
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Reductive fixed points in simple algebraic groups
Let $G$ be a connected simple algebraic group over $\mathbb{C}$.
Then $\mathrm{Aut}(G)$ is a linear algebraic group, so every
automorphism $\sigma \in \mathrm{Aut}(G)$ admits a Jordan
decomposition
$$
...
- 3,035
12
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1
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$G$-equivariant coherent reflexive sheaves on $X$ and coherent sheaves on $X/G$
Let $G$ be a reductive algebraic group acting on an affine variety $X$ properly with finite stabilizers and free at general point. Assume that the quotient space $Y:=X/G$ is affine. As shown in Kollár'...
- 10.3k
4
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Vector stabilizers in the 273D irrep of $\mathrm{F}_4$
According to the LieART 2.0 manual, the restriction of the 273-dimensional irreducible representation of $\mathrm{F}_4$ to $\mathrm{Spin}(8)$ includes a single 1-dimensional summand (and the ...
- 3,461
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Lie subgroups of the group of volume-preserving diffeomorphisms
Which connected Lie groups are known to appear as subgroups of $\text{SDiff}(D)$, the group of volume-preserving diffeomorphisms of some relatively compact open subset $D$ of $\mathbb{R}^2$ or $\...
- 25.1k