What are some examples of simple or semisimple non-split real Lie algebras? By non-split, I mean that no Cartan subalgebra $\mathfrak{h}$ is such that $\mathrm{ad}(X)$ is diagonalizable for each $X \in \mathfrak{h}$.
Is there a list somewhere? Or how can non-splitness be detected?
Let V_\lambda be a highest weight rep of a lie algebra g, and let W be the weyl group of g. let v_0 \in V_\lambda be a highest weight vector. let w \in W. why is it the case that w.v_0 has weight w.\lambda?
Let $G$ be a compact semisimple Lie group, and let $\mathfrak{g}$ be its Lie algebra. Let $X \in \mathfrak{g}$. Why is it the case that the centralizer $Z_\mathfrak{g}(X)$ contains a Cartan subalgebra?
I believe it's related to the fact that for any Cartan subalgebra $\mathfrak{h}$ of $\mathfra...
@TedShifrin My apologies, but I believe at least one of us misunderstood. We are working on the level of the Lie algebra. The centralizer contains the center, but the center need not contain a cartan subalgebra.
Let S = {(a^4, 4a^3 b, 6a^2 b^2, 4a^3 b, b^4) : a,b \in R}. How can I find the convex hull of this set, hopefully in terms of some inequalities or something?
I know no one cares .. but should I try to (somehow) compute the extreme points of the convex hull (by magic) and then use the krein milman theorem or something (along with some fancy computer algorithm)?
Anyway .. this is almost certainly a stupid question but I want some help. Let S = {(b^4, 4ab^3, 6a^2b^2, 4a^3b, a^4) : a,b \in R}. How can I compute the convex hull of this thing in R^5?
can someone help me with representation theory? i'm totally confused about those weight lattice diagrams. my question is here math.stackexchange.com/questions/1242472/…
I never learned Lie theory properly. I learned basic things about Lie algebras, and Fourier analysis more in depth, and now my MSc supervisor seems to expect me to be some type of expert
Yes, I am familiar with it. It's the representation on the space of degree $d$ homogeneous polynomials in $2$ variables, with highest weight $x^d$ (depending on exactly how one defines the action).
What are some examples of simple or semisimple non-split real Lie algebras? By non-split, I mean that no Cartan subalgebra $\mathfrak{h}$ is such that $\mathrm{ad}(X)$ is diagonalizable for each $X \in \mathfrak{h}$.
Is there a list somewhere? Or how can non-splitness be detected?
Discussion between nigelvr and Qiaoch…
Discussion between nigelvr and Qiaoch…