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Results tagged with lie-algebras
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user 391
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.
81
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26
answers
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What would you want on a Lie theory cheat poster?
For some long time now I've thought about making a poster-sized "cheat sheet" with all the data about Lie groups and their representations that I occasionally need to reference. It's a moving target, …
35
votes
Accepted
Figure out the roots from the Dynkin diagram
Here's an answer in the simply-laced case. Its proof, and generalization to non-simply-laced, are left to the reader.
1) Start with a simple root, and think of it as a labeling of the Dynkin diagram …
- 27.9k
22
votes
1
answer
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Modern reference for maximal connected subgroups of compact Lie groups
What's the nicest place to see a list of the maximal connected subgroups of compact Lie groups? Is there anything on-line?
I looked at Tits' Bourbaki talk on Dynkin's and others' work, but he admits …
- 27.9k
17
votes
What is significant about the half-sum of positive roots?
While I appreciate Dave Ben-Zvi's half-densities answer, I'm going to
put forth the contrary opinion that it's largely a bookkeeping artifact.
The most familiar place that $\rho$ shows up is in the W …
- 27.9k
17
votes
Understanding moment maps and Lie brackets
I believe the following way (Kostant's, 1970) to be the best way to think about the Hamiltonian condition.
First, "why" is there a central extension $H^0(M; {\mathbb R}) \to C^\infty (M) \to symp(M)$ …
- 27.9k
14
votes
Necessary and sufficient conditions for Littlewood Richardson coefficients to be non zero
If $\lambda,\mu,\nu$ are considered as vectors in $\mathbb Z^n$, then the set of triples forms a convex cone, given by a finite (for each $n$) list of inequalities. (The corresponding statement does n …
- 27.9k
10
votes
1
answer
409
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Is there much theory of superalgebras acting on manifolds by alternating polyvector fields?
Usual story: vector fields on $M$, with their Lie bracket, form a Lie algebra. We can consider "actions" of some other Lie algebra ${\mathfrak g}$ on $M$ by looking at Lie homomorphisms ${\mathfrak g} …
- 27.9k
9
votes
Accepted
Which linear combinations of simple roots are roots
My favorite answer to #2 and #3 is Kostant's "Find the highest root game", which is written up in detail in section 5.4 of Balázs Elek's notes on reflection groups. It is not hard to show that all pla …
- 27.9k
8
votes
Accepted
How to find faces of polytope defined by a Weyl orbit
The faces are all of the following form: $w W_P / Stab_W(\xi)$, where $W_P$ varies over the subgroups generated by subsets of the simple reflections. In particular, for $\xi$ regular, the number of th …
- 27.9k
8
votes
HIgher Homotopy Groups and Representation Theory
If $G$ is compact, then we can think about its cohomology with Lie algebra cohomology, so be doing some kind of representation theory. And of course the first nontrivial homology is determined by the …
- 27.9k
8
votes
Is every finite-dimensional Lie algebra the Lie algebra of an algebraic group?
Let ${\mathbb R}$ act on ${\mathbb R}^2$ by rotation at unit speed, and on a second ${\mathbb R}^2$ by rotation at some irrational speed. Let $G$ be the (solvable) semidirect product ${\mathbb R} \lti …
- 27.9k
8
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Root in positive Weyl chamber
This is only a slight modification of the argument already given, but
I liked it enough to type it in.
Since $W$ acts with no stabilizer on the open Weyl chamber $K$,
for $K$ to contain a root $\beta …
- 27.9k
7
votes
Matrix representation for $F_4$
Answerlet: it's either in ${\mathfrak so}(26)$ or ${\mathfrak sp}(13)$, that is to say, it's either a complexification of a real representation or forgetful from a quaternionic representation. Indeed, …
- 27.9k
7
votes
Outer automorphisms of simple Lie Algebras
In characteristic 2, $B_2$ and $F_4$ each have an outer automorphism, and in characteristic 3, so does $G_2$. This is relevant when you want to construct the twisted Chevalley groups, which use a Chev …
- 27.9k
7
votes
Accepted
How to write down the map $T(V)_n \to S(Lie(V))_n$ explicitly?
The natural map is rather from $TV \to U(FreeLie(V))$: consider the forgetful functors $Assoc \to Lie \to Vec$ and compose their left adjoints to get the left adjoint $T$ of the composite. Then, as Al …
- 27.9k