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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.
3
votes
Matrices of Lie algebra of Dynkin diagram B2
While I'm not sure this question is appropriate for this site, here goes.
First, you need a maximal torus. Inside SO(5) we have SO(4) and then SO(2) x SO(2).
Write your antisymmetric matrices as
$$\be …
- 115
2
votes
Verma modules and Borel–Weil
I don't think the $\pm$ issue is too deep, and I'm punting on it in favor of answering the other question.
You can get a hold of dual Verma modules by considering distributions on $G/B$ supported on a …
- 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
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 …
- 4,713
2
votes
Real and quaternionic representations according to weights
$\mathrm{Hom}_G(V^*,V) \cong \mathrm{Hom}(V^*,V)^G \cong (V\otimes V)^G \cong (\mathrm{Sym}^2 V\oplus \mathrm{Alt}^2 V)^G$, so the first is nonzero ($V$ is self-dual) iff $V$ possesses a symmetric or …
- 63.9k
4
votes
Weyl's Branching Rule for $SU(N)$-Setting
Every irrep of $SU(n)$ extends to irreps of $U(n)$, and conversely, the restriction of any irrep of $U(n)$ to $SU(n)$ remains irreducible. If your dominant weight of $SU(n)$ is $(a_1,\ldots,a_{n-1})$ …
- 27.9k
81
votes
26
answers
7k
views
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, …
CommunityBot
- 1
1
vote
Question on irreducible representation of tensor products
Your question is about the vectors in $V_1\otimes V_2$ that provide $1$-dimensional $B_\Delta$-subrepresentations of weight $\mu$, where $B_\Delta$ is the diagonal in $B\times B \leq G\times G$. Let's …
- 27.9k
4
votes
The Analog of Borel Subgroup in a Compact Real Form
It sounds like you want a datum to associate to a compact Lie group and chosen torus, that's functorially the same as a choice of Borel containing that torus if one were to complexify, without using t …
- 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
5
votes
Dimension of the zero weight space in $V_{2\rho}$
In general, $V_{k\rho} \cong \bigotimes\limits_{\beta\in \Delta_+} (\mathbb C_{k\beta/2} \oplus \mathbb C_{(k-2)\beta/2} \oplus \ldots \oplus \mathbb C_{-k\beta/2})$ as $T$-representations, provable v …
- 27.9k
3
votes
How can one show $G/T$ is a coadjoint orbit for a compact Lie group $G$ and $T$ its maximal ...
Use the Haar measure on $G$ (compact!) to average a metric, obtaining a $G\times G$-invariant metric, and thus an identification $\mathfrak g \cong \mathfrak g^*$. Also, the geodesic spray $\mathfrak …
- 27.9k
6
votes
Accepted
Motivating the existence of Canonical Bases for Representations
I don't really see how to get there from just compact groups, so in that sense this is not an answer. My take on the question is something like: how might one have guessed the existence of canonical b …
- 27.9k
22
votes
1
answer
2k
views
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 …
- 52.9k
2
votes
Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?
This is not a complete answer, but grew large for a comment.
Duality acts on the simple weights $\{\omega_i\}$, by $-w_0$ (where $w_0$ is the long element of the Weyl group), so we should really grou …
- 43.2k