Questions tagged [simple-lie-algebras]
9 questions
3
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What is meant by saying that $\theta$ is orthogonal?
I am reading the materials discussed in lecture $5$ from the lecture notes on quantum groups about Belavin-Drinfeld classification theorem written by Pavel Etingof and Oliver Schiffmann.
In the first half of this lecture the authors proved that any…
ACB
- 3,068
3
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1 answer
What is an orthonormal basis of a Lie algebra?
Let $\mathfrak g$ be a simple Lie algebra. Then what is an orthonormal basis of $\mathfrak g\ $? It has been written in one of the notes that if $(I_{\nu})$ is an orthonormal basis of $\mathfrak g$ then the Casimir tensor $\Omega$ corresponding to…
Anacardium
- 2,716
2
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1 answer
Are there any other examples of simple not absolutely simple lie algebra besides $so(3,1)$?
I just learned the distinction between simple and absolutely simple Lie algebras and was wondering if there are other well known examples other than the Lorentz algebra.
bonif
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Simple Lie algebras invariants
Do all simple Lie algebras have just one quadratic Casimir invariant as the Harish-Chandra isomorphism seems to imply or are there counter-examples?
bonif
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What are the $\mathfrak {g}$-invariant elements of $\mathfrak g \otimes \mathfrak g\ $?
Let $\mathfrak g$ be a simple Lie algebra. Is there any way to find the $\mathfrak g$-invariant subspace of $\mathfrak g \otimes \mathfrak g\ $? I am familiar with the result for $\mathfrak g = sl_2(\mathbb C)$ in which case the subspace is…
Anacardium
- 2,716
0
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1 answer
Lie algebra homomorphism of simple Lie algebra
Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, $\mathfrak{h}$ any Lie algebra over $\mathbb{C}$. Consider $\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ a Lie algebra homomorphism. My question; is $\phi$ an injective Lie algebra…
QuantizedObject
- 333
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1-dimensional representation of a Lie algebra
Let $\mathfrak{g}$ denote a Lie algebra. A proposition of a course of mine on Lie algebra claims that any one-dimensional representaiton of a simple Lie algebra is $0$. However, doesn't this extends to any any Lie algebra $\mathfrak{g}$?
Indeed, a…
QuantizedObject
- 333
0
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1 answer
Show that this irreducible representation is faithful
For a simple Lie algebra $\mathfrak{g}$ who contains a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$, I’m trying to show that a nontrivial irreducible representation $\pi:\mathfrak{g}\to\mathrm{End}(V)$ is faithful.
I tried to apply the…
itkyitfbku
- 614
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Understanding presentation of simple Lie algebras by means of Chevalley-Serre relations.
I came across the Chevalley-Serre Relations while studying semisimple Lie algebras. My question is how do the last two conditions regarding adjoint operator make sense? For $i = j$ we have $A_{ii} = 2,$ being the $i$-th diagonal entry of a Cartan…
Anacardium
- 2,716