Hello everyone,

What is the relation between almost simple Lie groups and semisimple Lie groups? (Especially in the case of subgroups of $SO(2,n)$.)

Recall that:

Def1: A Lie groups $G$ is said to be semisimple if it's Killing form is non-degenerate.
Def2: A Lie groups $G$ is said to be almost simple if every proper normal subgroup of $G$ either is finite or has finite index.

Thanks in advance.


A Lie group is semisimple if and only if its Lie algebra is semisimple which in turn is equivalent to saying that the Lie algebra is a direct sum of simple ideals. This means that a Lie group is semisimple if and only if it is locally isomorphic to a direct product of simple Lie groups. If the Lie group in question is connected and simply connected, it follows that it then is a direct product of simple groups. A connected Lie group is simple if and only if its Lie algebra is simple and this is equivalent to saying that all its normal subgroups are discrete and one shows that every normal subgroup must be central then. With your definition, this means that every almost simple group is indeed simple, but not the otehr way round, as the universal covering group of $SL_2({\mathbb R})$ shows. This is not the way a mathematician wants it.

The notion of almost simple groups is not used in Lie theory, but in the theory of algebraic groups, see for instance the book of Margulis. Indeed, if your Lie group is a linear algebraic group, then its connected component is simple as a Lie group if and only if it is almost simple.

  • $\begingroup$ Two remarks: first, as you may have observed, being simple in Lie group theory is not quite the same as in abstract group theory: $SL_2(\mathbb{R})$ is simple, in spite of the non-trivial centre; second, there is the issue of connectedness: $SO(2,n)$ (for $n>2$) is almost simple, as its connected component of identity $SO_0(2,n)$ is simple (and of index 2). Due to the exceptional isomorphism $S0_0(2,2)\simeq SO_0(2,1)\times SO_0(2,1)$, the group $SO(2,2)$ is semi-simple but not almost simple. $\endgroup$ – Alain Valette Nov 1 '12 at 19:15
  • $\begingroup$ @Xogn: The context and definitions in the question and in your answer need more precision. The question is apparently posed over the reals, while the Lie algebra theory is developed first over $\mathbb{C}$ and then adapted to the real case. Much of this is routine but needs to be specified. For instance, simple over $\mathbb{R}$ is not the same thing as simple over $\mathbb{C}$. And as Alain points out, the notion of simplicity for abstract groups is another matter. $\endgroup$ – Jim Humphreys Nov 1 '12 at 23:55

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