Questions tagged [lie-groups]
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-theory) tag.
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Unitary representations of non-compact Lie Groups
I have often read the following statement:
Let $G$ be a connected, simple, non-compact Lie Group of dimension $n \geq 2$. Let $ρ: G \to U(H)$ be a unitary representation of $G$ on the Hilbert Space $H$...
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What could have been a good motivation / intuition behind the Dorfman Bracket condition?
I'm a graduate student in Mathematics, currently learning Dirac structures in Differential Geometry. However, I cannot make sense of how the Dorfman bracket condition comes all of a sudden. It ...
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How to prove that $\operatorname{GL}_n(\mathbb C)$ is Lie subgroup of $\operatorname{GL}_{2n}(\mathbb R)?$ [closed]
How to prove that $\operatorname{GL}_n(\mathbb C)$ is Lie subgroup of $\operatorname{GL}_{2n}(\mathbb R)?$
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What exactly is the adjoint representation - confused by varying definitions [closed]
In class, we first only defined the adjoint representation as a matrix of structure constants. We proved everything only using this. I tried to review the class material with other resources but I'm ...
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Why isn't $\mathbb{RP}^n$ always orientable?
If $G$ is a Lie group and $H$ is a closed subgroup, the homogeneous space $G/H$ admits a $G$-invariant volume form if and only if ${\Delta_G}_{|H} = \Delta_H$ (where $\Delta_G$ and $\Delta_H$ are the ...
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Understanding the application of FTC in the proof that $\dim \left( T_{p}M \right) = n$
Consider the following proof I am trying to break down
There are two things I don't understand in the proof.
I do not understand the way the author uses the Fundamental Theorem of Calculus. I know ...
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Confusion about differentiating with respect to local coordinates, and dimension of $T_{p}M$
Background and definitions:
At the moment I am taking a basic course discussing Lie groups and Lie algebras. In the last lecture we have defined the following;
Let $R$ be ring and let $A$ be an ...
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How much can the Haar measure on an $n$-dimensional Lie group be expanded by a Lipschitz map?
I'll start with the question itself:
Let $G$ be an $n$-dimensional Lie group, $\mu$ be the Haar measure on $G,ドル $d$ be a translation-invariant metric on $G$ generating the topology on $G$. Let $f: G \...