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As said, semisimple Lie algebras over a field of characteristic zero do not admit an LSA-structure. There are several proofs. Basically it relies on Whitehead's lemma for semisimple Lie algebras, sayi …

Kostant and Sullivan proved that the Euler characteristic of a compact complete affine manifolds must vanish, affirming the Chern conjecture in the complete case (Bull. AMS 81 (1975)). Benzecri proved …

A. L. Onishchik has classified decompositions $G=G_1G_2$ of redcutive Lie groups (e.g., see his article "Decompositions of reductive Lie groups" in Math. USSR-Sbornik, Vol. 9 (1969), No. 4). We have $ …

Yes, the orthogonal group makes sense over any field $k$. It is an linear algebraic group. In fact the theory of linear algebraic groups generalizes that of linear Lie groups over the real or complex …

It depends on the definition of a reductive space $G/H$. Some authors require that in addition to $\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$, $[\mathfrak{h},\mathfrak{m}]\subseteq \mathfrak{m}$, we hav …

I have only some simple remarks here, valid also for non-compact Lie groups. Let $G$ be a connected Lie group with Lie algebra $\mathfrak{g}$. A first remark is that for $X\in Z(\mathfrak{g})$ and $Y\ …

For an example, let $N$ be a compact complex hypersurface of degree $m+1$ of the complex projective space $\mathbb{CP}^m$ with complex dimension $m\ge 3$ (for $m=3$ this is a complex $K3$ surface). Th …

Question $1$: The theorem of Mostow says that every connected algebraic group $G$ over a field $K$ of characteristic zero has a Levi decomposition. This means, $G$ has a reductive algebraic subgroup $ …

In 1950s A. Borel, R. Bott, J. L. Koszul, F. Hirzebruch et al. investigated the coadjoint orbits as complex homogeneous manifolds. It was proven that each coadjoint orbit of a compact connected Lie gr …