7
votes
Accepted
Does every Lie algebra appear as centralizer of an element in a semisimple Lie algebra?
Your title question and your body are different: your title asks what Lie algebras arise as fcss centralisers, and your body asks whether all Lie algebras arise this way. I answer the easier latter ...
5
votes
Accepted
Multiplication factors in folding root systems and Lie algebras by automorphisms
Suppose we fold a root system $(\Phi,\Delta)$ to a root system $(\Phi^\sigma,\Delta^\sigma)$. There exists two conventions:
long roots of $(\Phi^\sigma,\Delta^\sigma)$ correspond to multiple roots in ...
4
votes
Does the first fundamental representation of $\frak{sp}_n$ generates all the other fundamental representations
(Copied from my own comments.) Yes. In fact, if $V_k$ (for $1\leq k\leq n$) denotes the $k$-th fundamental representation in the order of the nodes of the Dynkin diagram, and $V_0$ the trivial ...
3
votes
Accepted
Which Lie groups admit finite generation by a set of Lie algebra elements? And what are some known choices of generators which realize this?
Let $G$ be a connected simple Lie group with finite center. Let $S$ be a generating subset of $G$ (symmetric with $1$) such that $S^n$ has nonempty interior for some $n$ (this is automatic if $S$ is $\...
1
vote
Accepted
$8 \times 31 = 8 \times 31$?
The decompositions are not conjugate. Pick (any) two Cartans in the direct sum decomposition. For the decomposition coming from $2^8 \subset E_8$, two Cartans together generate a subalgebra isomorphic ...
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