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Results tagged with lie-algebras
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user 4231
Lie algebras are algebraic structures which were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used. Related mathematical concepts include Lie groups and differentiable manifolds.
4
votes
On radicals of a lie algebra
An extended comment might be useful, though Ben and Dietrich have addressed the mathematical issues concisely. The question is implicitly about the history of Lie theory, which was still somewhat un …
3
votes
Accepted
length function of a coxeter group with respect to two different simple systems are equal or...
To supplement what David Speyer and Paul Garrett have written, I'd emphasize that the question itself is faulty: the terminology involved in "two simple systems are weyl conjugates of one another" doe …
- 52.9k
7
votes
Sign conventions for a Chevalley basis of a simple complex Lie algebra
As Florian suggests, there is no canonical choice of structure constants in Chevalley's approach (or any other I'm aware of). But for the irreducible root systems, especially those of exceptional ty …
- 52.9k
6
votes
Twisted affine Lie algebras
It's an unfortunate tendency in textbooks on Lie algebras to assume that the base field is $\mathbb{C}$ even though virtually everything done in the classical structure theory and finite dimensional r …
- 52.9k
0
votes
Recovering a Lie algebra from its affine Lie algebra
It's not clear exactly what "given" means here, but I assume you know the affine Dynkin diagram belonging to the given affine Lie algebra. In that case the ordinary Dynkin diagram of the original si …
- 52.9k
5
votes
One more question about PBW
While I can't immediately give a precise example, it may help to focus the question a little more in order to emphasize what is actually involved.
1) In most literature on Lie algebras (related t …
- 52.9k
8
votes
Exceptional Lie algebras
Like most other mathematicians, I am not an expert on the mathematical physics literature related to Lie algebras. But the E series has led further into Kac-Moody algebras: affine, hyperbolic, ... …
- 52.9k
5
votes
Lie algebras with abelian Cartan subalgebras
Maybe it should be emphasized that these questions deal with finite dimensional Lie algebras over a field of characteristic 0; in prime characteristic there are further complications. BR has addres …
- 52.9k
4
votes
Minimal dimension of maximal abelian subalgebras
Some of my previous comments were unfortunately too casual and unfocused, but I still suspect that the answer to the original question is yes. At the same time, I can't document precisely enough wha …
- 52.9k
25
votes
Accepted
Why are affine Lie algebras called affine?
It's not easy to separate out the purely mathematical from the historical question here: What is the mathematical justification for use of the label "affine" and how did this label get attached to cer …
- 52.9k
18
votes
Accepted
finding highest weight of dual of a representation of a semisimple lie algebra
To expand my short comment, the result itself (formulated by Sasha) has been around a long time and depends only on the definitions involved. Textbooks dealing with the highest weight classification …
- 52.9k
3
votes
What's the most simple proof of Kostant's version of Borel-Weil-Bott for Lie Algebra cohomol...
I'm not sure exactly what your header means, but maybe I can suggest partial answers to your questions. First of all, there are by now many ways to approach the original Borel-Weil theorem, dependin …
- 52.9k
3
votes
The annihilator of a Borel subalgebra being its nilpotent radical
Say $\mathfrak{g}$ is a semisimple Lie algebra over an algebraically closed field of characteristic 0 such as $\mathbb{C}$. The nondegeneracy of the Killing form $\kappa$ of $\mathfrak{g}$ is one wa …
- 52.9k
1
vote
Accepted
tensor product of two irreducibles having same maximal weight
The two modules in the question are actually isomorphic, since highest weights determine irreducibles. Anyway, the answer to your question is that there is usually no explicit decomposition known. …
- 52.9k
6
votes
Quadratic Casimir of fundamental irreps of simply-laced Lie algebras
Probably the best explanation will come from Freudenthal's method for recursive computation of weight multiplicities in any irreducible representation of a finite dimensional simple Lie algebra $\math …
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