# Questions tagged [lie-algebras]

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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### Definition of $Ad(r)$ the automorphism of a Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$.

Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$. Consider the Lie-algebra $\mathfrak{g}$ of $G$, which is semisimple and acts irreducible on $V$. Then i want to do the ...
• 249
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### Showing that the Lie-algebra $\mathfrak{g}$ of $G \subset SL(V)$ is semisimple and irreducible on $V$. [solved]

Let $G \subset SL(V)$ be a connected algebraic group, acting irreducible on $V$, where $V$ is a complex vectorspace of dimension $n$ I want to show that the Lie algebra $\mathfrak{g}$ of $G$ is ...
• 249
1 vote
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### Zero element in Lie algebra of a Lie group

On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth ...
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### Recovering the definition of exponential matrix from the abstract definition of Lie groups.

I am studying the exponential function of the book introduction to the smooth manifold by John Lee and the following question has arisen. Let $\exp:\mathcal{G}\to G$ exponential map, with $G$ a Lie ...
• 2,580
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### Time derivative of the blend of a pair of quaternion curves

I have two curves ${\bf q}_0(t), {\bf q}_1(t)$. Each curve maps time $t$ to a unit quaternion. Construction of these curves is not important here, although we do have the respective time derivatives ...
• 51
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### Proof of Lie's theorem [duplicate]

I am reading Humphrey's Introduction to Lie algebras and representation theory. I am stucked on the proof of theorem in Chapter 4.1 for this small point. Assume $L$ is a solvable subalgebra of $gl(V)$ ...
• 135
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### What is the KAK (Cartan) decomposition in $\text{SL}(d, \mathbb R)$ in terms of linear algebra language (and its relation with SVD)?

Let $G= \text{SL}(d,\mathbb R)$ and consider its Cartan decomposition in the Lie group level $G=KAK$. Here $K$ should be the compact group $\text{SO}(d,\mathbb R)$ and $A$ is a diagonal matrix (please ...
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• 197
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### Commuting generators of different Lie Algebras [closed]

Suppose that I have two different Lie algebras $\mathfrak{a}$ and $\mathfrak{b}$ with generators $M_{i}$ and $N_{j}$. Is it always the case that, the commutator (which is assumed to exist throughout ...
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### Tits' construction gives semisimple Lie algebras

Back to 1966, Jacques Tits gave a unified construction of the $5$ exceptional semisimple Lie algebras, and his work leads to the famous Freudenthal-Tits magic square. See, for example, https://arxiv....
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### Is the Maurer-form a 1-form?

I am studying Lie groups and lie algebras following Nakahara book. In 5.6.4 he introduces the concept of Maurer-Cartan one-form in this way: \theta: X \rightarrow {(L_{g})^{-1}} _{*} ...
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### Semisimple Lie algebra coincides with its commutant [duplicate]

I am trying to understand why for a semisimple Lie algebra $L$, we have $L = [L,L]$. here are my thoughts. Let $I:=[L,L]$ be a proper ideal in $L$, i.e., $I \ne L$, we then consider its orthogonal ...
1 vote
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• 179
1 vote
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### What is a simple component of a root system?

The above is from Knapp's Lie Groups; Beyond an introduction', 2ed, page 397. Question 1: What is a simple component of a reduced root system? Or, more specifically, given a root system $\Delta$ and ...
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### The eigenvalues of the generators of $SO(N)$
What are the eigenvalues of generators in the irreducible representations of the Lie algebra of $SO(N)$ (and its doublecover)? For me, the naive generators of $SO(N)$'s fundamental representation are ...