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 ...
LSpice's user avatar
  • 11.3k
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 ...
Kenta Suzuki's user avatar
  • 1,612
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 ...
Gro-Tsen's user avatar
  • 29.9k
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 $\...
YCor's user avatar
  • 60.1k
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 ...
Theo Johnson-Freyd's user avatar

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