3 fixed the end of the argument

This question is a follow-up on the answer given here http://mathoverflow.net/questions/62385/can-a-lie-group-as-an-abstract-group-be-given-more-than-one-topology-making-it-a

It is motivated by the following observations:

1. If $m,n$ are positive integers, then $\mathbb{R}^m$ is isomorphic to $\mathbb{R}^n$ as abstract groups since they are both $\mathbb{Q}$-vector spaces of the same dimension. So a Lie group structure on $\mathbb{R}^n$ is not unique.

2. On the other hand, a Lie group structure on a compact $n$-torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ is unique: if $m\neq n$, then $T^n$ can not be isomorphic to $T^m$ as abstract groups, e.g. since $T^n$ has $2^n-1$ elements of order 2, and $T^m$ has $2^m-1$; nor can $T^n$ be isomorphic to $\mathbb{R}^m$ since $\mathbb{R}^m$ has no elements of finite order at all (apart from 0).

3. Here is an ad hoc proof that a Lie group structure on $SU(2)$ is unique. Let $G$ be a Lie group isomorphic to $SU(2)$ as an abstract group. Then $G$ semi-simple since semi-simplicity can be described in group theoretic terms (there are no nontrivial solvable normal subgroups). Moreover, all maximal abelian subgroups of $G$ are $T^1$'s by 2. So the complexification of $G$ has Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. There are two Lie semi-simple Lie groups that fit all of the above: $SU(2)$ and $SO(3)$; the first has a center and the second does not, so $G$ must be $SU(2)$. [upd: as pointed out by Claudio, this only shows that $G_e$, the connected component of the unit $G$, is $SU(2)$ or $SO(3)$; but if $G$ has more than one connected component, then it has a normal subgroup, $G_e$, different from $\mathbb{Z}/2$, which $SU(2)$ can't have.] (I believe something similar should work for any semi-simple compact group.)

So I would like to ask: is there a reasonable way to characterize Lie groups that admit a unique Lie group structure? If not, then what happens if we restrict the attention to (real or complex) semi-simple Lie groups? (I would be particularly interested in a proof that did not rely to much on the classification.)

2 corrected a mistake

This question is a follow-up on the answer given here http://mathoverflow.net/questions/62385/can-a-lie-group-as-an-abstract-group-be-given-more-than-one-topology-making-it-a

It is motivated by the following observations:

1. If $m,n$ are positive integers, then $\mathbb{R}^m$ is isomorphic to $\mathbb{R}^n$ as abstract groups since they are both $\mathbb{Q}$-vector spaces of the same dimension. So a Lie group structure on $\mathbb{R}^n$ is not unique.

2. On the other hand, a Lie group structure on a compact $n$-torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ is unique: if $m\neq n$, then $T^n$ can not be isomorphic to $T^m$ as abstract groups, e.g. since $T^n$ has $n$ 2^n-1$elements of order 2, and$T^m$has$m$; 2^m-1$; nor can $T^n$ be isomorphic to $\mathbb{R}^m$ since $\mathbb{R}^m$ has no elements of finite order at all (apart from 0).

3. Here is an ad hoc proof that a Lie group structure on $SU(2)$ is unique. Let $G$ be a Lie group isomorphic to $SU(2)$ as an abstract group. Then $G$ semi-simple since semi-simplicity can be described in group theoretic terms (there are no nontrivial solvable normal subgroups). Moreover, all maximal abelian subgroups of $G$ are $T^1$'s by 2. So the complexification of $G$ has Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. There are two Lie semi-simple Lie groups that fit all of the above: $SU(2)$ and $SO(3)$; the first has a center and the second does not, so $G$ must be $SU(2)$. (I believe something similar should work for any semi-simple compact group.)

So I would like to ask: is there a reasonable way to characterize Lie groups that admit a unique Lie group structure? If not, then what happens if we restrict the attention to (real or complex) semi-simple Lie groups? (I would be particularly interested in a proof that did not rely to much on the classification.)

1

# Which groups admit a unique Lie group structure?

This question is a follow-up on the answer given here http://mathoverflow.net/questions/62385/can-a-lie-group-as-an-abstract-group-be-given-more-than-one-topology-making-it-a

It is motivated by the following observations:

1. If $m,n$ are positive integers, then $\mathbb{R}^m$ is isomorphic to $\mathbb{R}^n$ as abstract groups since they are both $\mathbb{Q}$-vector spaces of the same dimension. So a Lie group structure on $\mathbb{R}^n$ is not unique.

2. On the other hand, a Lie group structure on a compact $n$-torus $T^n=\mathbb{R}^n/\mathbb{Z}^n$ is unique: if $m\neq n$, then $T^n$ can not be isomorphic to $T^m$ as abstract groups, e.g. since $T^n$ has $n$ elements of order 2, and $T^m$ has $m$; nor can $T^n$ be isomorphic to $\mathbb{R}^m$ since $\mathbb{R}^m$ has no elements of finite order at all (apart from 0).

3. Here is an ad hoc proof that a Lie group structure on $SU(2)$ is unique. Let $G$ be a Lie group isomorphic to $SU(2)$ as an abstract group. Then $G$ semi-simple since semi-simplicity can be described in group theoretic terms (there are no nontrivial solvable normal subgroups). Moreover, all maximal abelian subgroups of $G$ are $T^1$'s by 2. So the complexification of $G$ has Lie algebra $\mathfrak{sl}_2(\mathbb{C})$. There are two Lie semi-simple Lie groups that fit all of the above: $SU(2)$ and $SO(3)$; the first has a center and the second does not, so $G$ must be $SU(2)$. (I believe something similar should work for any semi-simple compact group.)

So I would like to ask: is there a reasonable way to characterize Lie groups admit a unique Lie group structure? If not, then what happens if we restrict the attention to (real or complex) semi-simple Lie groups? (I would be particularly interested in a proof that did not rely to much on the classification.)