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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.
85
votes
Motivating the Casimir element
Maybe I should try to defend myself, or at least the self I was four decades ago when I improvised my graduate text. But first I should disclaim any originality in the proof of Weyl's theorem, which …
- 52.9k
50
votes
5
answers
9k
views
What role does the "dual Coxeter number" play in Lie theory (and should it be called the "Ka...
While trying to get some perspective on the extensive literature about highest weight modules for affine Lie algebras relative to "level" (work by Feigin, E. Frenkel, Gaitsgory, Kac, ....), I run into …
- 52.9k
47
votes
Accepted
What is significant about the half-sum of positive roots?
I don't think there is a one-line answer to this question, since it depends a lot on the direction from which you approach semi-simple Lie theory. For one thing, it's probably best at first to empha …
- 52.9k
41
votes
2
answers
2k
views
Implications of non-negativity of coefficients of arbitrary Kazhdan-Lusztig polynomials?
In their seminal 1979 paper Representations of Coxeter groups and Hecke algebras (Invent. Math. 53, doi:10.1007/BF01390031),
Kazhdan and Lusztig studied an arbitrary Coxeter group $(W,S)$ and the corr …
- 52.9k
31
votes
4
answers
3k
views
What was Casimir's precise role in describing the center of the universal enveloping algebra...
This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
37602. Like some others who started graduate study in …
- 52.9k
30
votes
0
answers
997
views
Follow-up to Steinberg's problem (12) in his 1966 ICM talk?
Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with posi …
- 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
22
votes
Accepted
On the full reducibility of representations of reductive Lie algebras
The statement is false. The standard definition of "reductive" for a finite dimensional Lie algebra $\mathfrak{g}$ over an arbitrary field of characteristic 0 is given in a number of equivalent ways …
- 52.9k
22
votes
Accepted
Lie algebras of algebraic groups
In the classical theory of Lie groups and Lie algebras, the exponential map defined in terms of the usual power series is a standard tool for passing from the Lie algebra to the group. This makes sen …
- 52.9k
21
votes
Accepted
Why the BGG category O?
I'd add to what Ben says the observation that people have found a number of different module categories valuable for different purposes within this same Lie algebra context. (And some Lie theory pe …
- 52.9k
19
votes
Accepted
Complex root systems
To supplement what Pete says, I'd emphasize that "root systems" have been defined in a variety of ways for a variety of purposes related to Lie theory or to some type of "reflection" group. (For in …
- 52.9k
19
votes
2
answers
2k
views
Dual versions of "folding" symmetric ADE Dynkin diagrams?
Start with the Dynkin diagram of an irreducible root system, typically associated with a simple
Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE
diagrams admi …
- 52.9k
18
votes
Accepted
Low dimensional nilpotent Lie algebras
Classification of nilpotent Lie algebras in characteristic 0 is an old problem,
with a lot of literature. For the dimensions up to 6 there is a finite list.
Among the many relevant papers on MathSci …
- 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
17
votes
Generators of invariant polynomials of semisimple Lie algebra
Especially for exceptional types it seems quite difficult to exhibit any explicit basic set of generators for the algebra of invariant polynomial functions on $\mathfrak{g}$. They are of course theo …
- 52.9k