# Tagged Questions

For questions about Lie algebras, an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds.

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### Every base of a root system arises as indecomposable positive roots of a regular element?

I'm confused about a line in the Theorem p48 in Humphrey's's book on Lie Algebras. He's proving that every base $\Delta$ of a root system $\Phi$ arises as the set of $\Delta(\gamma)$ of ...
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### Finding the tangent space of $Z(U(n))$

Recall that the center of the matrix unitary group for a given $n\in \mathbb N$ is\begin{equation*}Z(U(n))=\{\omega I:|\omega|=1\}\end{equation*} I'm trying to find the tangent space at the identity ...
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### What are the generators of the coset of SU(2)/U(1)?

I need to understand the coset, $SU(2)/U(1)$ for the fundamental representation. How would I go about doing so? From my understanding of coset it means that any transformation of $SU(2)$ mod a $U(1)$...
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### Coefficients of positive roots in term of simple roots

Let $\Phi$ be an irreducible root system and $\Phi^+$ be positive root system and $\Delta$ be base. For every positive root $\beta=\sum_{\alpha \in \Delta}m_\alpha\alpha$, the numbers $m_\alpha$ are ...
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### Symplectic group action

Let $(M,\omega)$ be a symplectic manifold. We say that a group action $\phi: G \times M \rightarrow M$ is symplectic if each $\phi(g,.)$ is a symplectomorphism. Now, I am going through some lecture ...
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### Lie algebra associated to Lie group of algebra automorphisms

I'm working through Fulton and Harris's Representation Theory, and I'm stuck on Exercise 8.27. I'm trying to show that if $A$ is an algebra and $G$ is the Lie group of algebra automorphisms (...
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### Equality on pg. 40 of Humphreys's Lie Algebras, $\kappa(t_\lambda,t_\mu)=\sum_{\alpha\in\Phi}\alpha(t_\lambda)\alpha(t_\mu)$?

I don't understand part of an equality on page 40 of Humphreys's book on Lie Algebras. Suppose $L$ is a semi-simple Lie algebra over an algebraically closed field of characteristic $0$, and $H$ a ...
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### Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
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### What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
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### Branching $U(2)$ with respect to $SU(2)$

By construction $SU(2)$ is contained in $U(2)$, the special unitary and unitary groups respectively. Thus, any representation of $U(2)$ will induce a representation of $SU(2)$. The irreducible irreps ...
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### Real version of Harish-Chandra-Itzykson-Zuber integral

I'm interested in an integral of the form $$\int_{O(d)} \exp\left(-\frac{1}{2}\mathrm{trace}(CUAU^T)\right)dU$$ where the integration is with respect to the Haar measure on the orthogonal group, i.e....
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### Why do roots span dual space of maximal toral subalgebra?

Suppose $\Phi$ is the root system of a semi simple Lie algebra with maximal toral subalgebra $H$. I read that $\Phi$ spans $H^\ast$. The Killing form on $H$ is nondegenerate, so $H\cong H^\ast$ by ...
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### Computing the Cohomology of Lie groups

In Bredons "Topology and Geometry" [Chapter V, section 12] the following theorem is proven: If $G$ is a compact connected Lie group its $k$-th cohomology $H^k(G,\mathbb{R})$ is isomorphic to the ...
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### Adding tori to semi-simple groups

Let $G$ be a complex, connected, semi-simple Lie group (throw in simply connected if you like) with Lie algebra $\mathfrak g$. Let $T \subseteq B$ be a maximal torus and choice of Borel, respectively. ...
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### How to embed $U(1)$ (or other groups) into a bigger group, using Dynkin diagrams

I am trying to find the embedding and the branching rules for some group decompositions. For example, I consider $E_7$ and its maximally compact subgroup $SU(8)$ and I want to "see" how the Dynkin ...
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### Compute variation left action subgroup

I consider a Lie group $G$, with a group element $g$ parametrised in some manner with parameter $\theta_i$, $i=1,\cdots, \dim G$. Suppose that $K\subset G$. I want to compute the variation of an group ...
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