# 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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### representation theory of two-step nilpotent Lie algebras

Does anyone know of any good reference about the representation theory of two-step nilpotent Lie algebras, like whether their irreducible representations can be classified?
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### Weightspace decomposition of a semisimple Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a (finite dimensional) semisimple Lie algebra. Let $H$ be a maximal toral subalgebra of $L$. Consider a representation $\pi: L \to \mathfrak{gl}(V)$. It is ...
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### dimension of lie algebra

I am studying lie algebra myself and question is about finding dimension of lie algebra . While i read Wikipedia link about lie algebra and lie group i saw statement Lie algebra $\mathfrak{g}$ is ...
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### Fundamental group of a Root System and determinant of the Cartan matrix

This is the first time I am posting, so I hope I didn´t get the formatting wrong. I am currently reading J. E. Humphreys "Introduction to Lie Algebras and Representation Theory" and got stuck at ...
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### Solvable equivalent to nilpotency of first derived Lie algebra?

The Wikipedia "Solvable Lie Algebra" page lists the following property as a notion equivalent to solvability: $\mathfrak{g}$ is solvable iff the first derived algebra $[\mathfrak{g},\,\mathfrak{g}]$ ...
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### Etingof problem 2.16.2: Irreps of Two-dimensional Lie algebra over a field of positive characteristic

This is problem 2.16.2 in Etingof's introduction to representation theory. Note that problem 2.16.1 is a proof of Lie's theorem. I'm having trouble with the second case, where the base field has ...
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### Description of free Lie algebra in Weibel's book

In Exercise 7.3.2 in Weibel's book An Introduction to Homological algebra the following description of the free Lie algebra over some $k$-module $M$ is given (where $k$ is any commutative ring): ...
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### How to understand Weyl chambers?

Recall the definition of the Weyl Chambers: A Weyl Chamber is a region of $V \setminus \bigcup_{\alpha \in \Phi} H_{\alpha}$, where $V$ is underlying Euclidean space, and $H_\alpha$ the hyperplane ...
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### How can Clebsch-Gordan Decompositions be combined?

In section 4 of this paper the authors use a given list of Clebsch-Gordan coefficents for the $27 \otimes 27$ of $E_6$ from an old paper and combine it with their own list of Clebsch-Gordan ...
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### What is corresponding Lie group for Lie algebra of vector fields in dynamical systems?

According to Ado's theorem, for every finite dimensional abstract Lie algebra there is a Lie group. Finite dimensional analytic (or meromorphic) Vector fields (in dynamical systems) over the filed of ...
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### A Lie Algebra $L$ is reductive iff it is completely reducibile as an $\operatorname{ad}_L(L)$-module

Given a Lie Algebra $L$ we say it is reductive if $\operatorname{Rad}L=Z(L)$. How can we prove that $L$ is reductive iff it is an $\operatorname{ad}_L(L)$-module completely reducibile? Suppose $L$ ...
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### Question about the theorem of highest weights

I have some confusion from reading Theorem 7.3 in Sepanski's Compact Lie groups and would appreciate it if someone could clarify. In part (e) the book says "for $w\in W$, $wV_\lambda=V_{w\lambda}$, ...
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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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### Rootspace decomposition of a Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a non-zero Lie algebra which is semi simple. Then $L$ contains a toral element and hence a non-trivial toral subalgebra. Let $H$ denote a maximal toral ...
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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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### Preimage of singular points of smooth map between manifolds

Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what ...
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### Is there a name for this kind of space?

Assume a Riemannian symmetric space $G/H$ where the decomposition of the Lie algebra of $G$ is $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$. It is a known fact that if $\mathfrak{h}$ is the Lie ...
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### Matrix of Killing form a Lie algebra

Let $L$ be the Lie algebra with basis $B = \{u,v,w\}$, with $[u,v] = w, [v,w] = u, [w,u] = v$. Question : Find the matrix of the Killing form $\kappa$ of $L$ with respect to $B$. I have come across ...
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### Unknown proof Lie Algebra

I have a calculation where I do not know what it actually shows. I think it tells me that for right invariant vector fields, the commutator is again right invariant. Maybe somebody here could help me ...
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### High Dimensional Rotation Matrices As Product of In-Plane Rotations

Lately I've been thinking a lot about how to find high-dimensional rotation matrices. In particular, can any rotation in $n$-dimensional space be represented as the product of $2$D plane rotations? I'...
### Recipe to compute dimension and decompose product of $SO(N)$ group representations
As it is well known Young tableaux (YT) provide an efficient and very useful way to treat $SU(N)$ representation. This is principally based on these facts: There is a correspondence between irreps ...