Search Results

Another reference which was not yet mentioned, I think, is the article of M. Moskowitz, "Faithful Representations and a local property of Lie groups", Math. Z. $143$, 1975. There the question is disc …

Every matrix Lie group is a smooth manifold, hence it is path-connected if and only if it is connected. And $U(n)$ is compact and connected as a topological space (any unitary matrix can be diagonali …

Since a Lie group is semisimple if and only if its Lie algebra is semisimple, we may show that the semidirect sum of semisimple Lie algebras is semisimple. Indeed, if $L$ is a semidirect sum of two se …

Every discrete group of Euclidean isometries acts properly discontinuously and cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the …

A reference for this is also "Lectures on Real Semisimple Lie Algebras and Their Representations" by A. L. Onishchik. A first result here is: any irreducible real representation $\rho\colon \mathbb{g …

There are already uncountably many isomorphism classes of $3$-dimensional real Lie algebras. In fact, there are $1$-parameter families of $3$-dimensional solvable Lie algebras. The classification has …

They are the compact Lie algebras. Note that are two different definitions in the literature. One is that a compact Lie algebra is the Lie algebra of a compact Lie group. This includes tori, and the K …

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 $ …

Maximal (closed) subgroups of semisimple Lie groups have been first classified by E.B. Dynkin (see the reference above). There is indeed a way to "reduce" the problem to the case of a simple Lie group …

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\ …

Here are some more purely group theoretical conditions. This is also not a complete answer, since it gives just some necessary conditions for certain groups to be linear. Schur: Suppose that $G$ is …

There are exactly four non-solvable complex Lie algebras of dimension $6$ with nontrivial Levi decomposition, namely \begin{align*} \mathfrak{sl}_2(\Bbb C) & \ltimes_{\phi} \Bbb C^3\\ \mathfrak{sl}_2( …

As the comment shows the answer is negative in general. Perhaps it is worth to mention that for connected compact Lie groups the answer is yes, because its exponential map is surjective. In general, i …

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 $ …