# 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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### How does a Lie algebra act on a tensor product of L-modules?

What is the $L$-module structure on $V\otimes W$ where $L$ is a Lie-algebra and $V,W$ are $L$-modules? The following question is related but I can't find the definition in there: tensor product of ...
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### Local Lie derivative on $G$-space at the zero of a vector field

Let a Lie group $G$ act on a manifold $M$ and let $X\in Lie(G)$. For now suppose $G=T$ is a torus (but the answer to this question should hold for $G$ abelian). $L_X$ is a vector field on $M$, at a ...
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### Derivatives of solution to Schrödinger equation

Consider the differential equation (Schrödinger, but rewritten to be pleasing to Lie algebraic eyes): $\frac{d U(t)}{dt} = c(t)U(t)$ where $c(t)=a+w(t)b(t)$, $a,b \in \mathfrak{su}(n)$ and $w$ is a ...
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### Relationship between solutions of two matrix differential equations

Given a ($4\times4$ in the important case) matrix differential equation: $\frac{d U_t}{dt}= A_t U_t$ where $U_t \in SU(n)$ and $A_t \in \mathfrak{su}(n)$. What is the relationship between the ...
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### Why doesn't the “naive” scalar product for $SO(n)$ yield something invariant?

By definition, for $SO(n)$ we have $g^T g=1$ for $g \in SO(n)$. Given some vector $v \in V$ and some representation $R: SO(N) \rightarrow \mathrm{Lin}(V)$, the defining condition above tells us ...
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### Question related to the Campbell-Baker-Haussdorf formula

I have two operators $A$ and $B$ such that $[A,B] = C$ $[A,C] = -2A$ $[B,C] = +2B$ and I would like to obtain an expression for $\log(\exp(A+B)\exp(-B)\exp(-A))$. Is it a linear combination of ...
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### Tangent space of coadjoint orbit

Let $\xi \in T_xOx$ be a tangent vector at $x \in O_x :=\{\mathrm{ Ad} _{u}^*(x); u \in G\}$ for $x \in g^*.$ ($g$ is the Lie-Algebra) Then I read that this $\xi$ can be represented as the velocity ...
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### List of simple roots in the H-basis for various Lie algebras?

There are four usual bases one can use to express the roots and weights of a given algebra. The $\alpha$-basis, where we write the roots and weights in terms of the simple roots $\alpha_i$. The ...
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### Map for roots of a Lie group to roots of a special subalgebra?

For regular subalgebras $h$ of some group's Lie algebra $g$, $$h \subset h$$ the root system of the subalgebra is a subset of the root system of the original's group algebra. Subalgebras whose ...
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### signature function of Weyl group element in LieArt

I am currently using LieArt Mathematica package for some calculations in Lie algebra, I am wondering if there is a way to know what is the signature of a Weyl group element, it seems the package can ...
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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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### 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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### Standard cyclic module of sl2

Let $L=\mathfrak{sl}(2, \mathbb{F})$, $B$ a standard Borel subalgebra. I am trying to solve exercise 20.4 from J.E. Humphreys "Introduction to Lie Algebras and Representation Theory", but I am stuck. ...
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### Prove that actions commute

I am trying to understand a proof from Kobayashi and Nomizu (foundations of differential geometry, p. 280). Suppose that we have Lie subalgebras $a<b<g$, with $g$ the Lie subalgebra of $SO(n)$ ...
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### Lie groups and Lie algebras for matrices

Recently, I stumbled over a few things in very basic Lie group / Lie algebra theory concerning matrix groups. Basically, my question is: Is there a way to canonically understand all the Lie groups ...
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### Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
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### Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $[ \;. ]$ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
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### commutation relation of angular momentum operator in non cartesian coordinates

The angular momentum operator $J$ in quantum mechanics with the commutation relation \begin{equation*} [J_l,J_m]=i\hbar\epsilon_{lmn}J_n \end{equation*} has the structure of a Lie-algebra. It is ...
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### Dominant Weight

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra. Question 1: It says that ...
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### Exercise 11, chapter 2 in Lie Groups, Lie Algebras, and Representations: An Elementary Introduction

I am reading the book: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction by Brian C. Hall. I am stuck at the following exercise: exercise 11, chapter 2 . Can you help me? ...
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### cohomology of general linear group over finite fields

Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. Let $\mathrm{GL}_n(\mathbb{Z}_2)$ be the group consisting of all $n\times n$ matrices with entries in $\mathbb{Z}_2$ with non-zero determinant. What is the ...
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### Do I have the right idea for this isomorphism of Lie algebras of matrix groups?

I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying ...