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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.
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
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
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
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 …
- 27.9k
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 …
- 27.9k
8
votes
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
5
votes
Convention about "long" roots for simple Lie algebras of types ADE?
I like being able to say "the highest root is always long".
- 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 …
- 27.9k
4
votes
Highest weights of irreducible components of tensor product of irreducible sl(3)-module
I'm assuming that you meant "is it true... satisfies that". In which case the answer is yes, not just for $\mathfrak{sl}(3)$ but any semisimple Lie algebra. One of the many ways to see this uses the L …
- 27.9k