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Do free abelian groups embed into connected semisimple complex lie groups?
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mark zorro
Mar 3, 2010, 6:19:01 AM3/3/10
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Question:
Let A be a connected semisimiple complex lie group. Is it true that
any free abelian group G of finite rank can be embedded in A?
I know that it is the case that any free groups of finite rank can be
embedded into any connected complex semisimple lie group.
Do we have to make any assumptions about the dimension of A for the
question to have a positive answer?
Corbennick
Mar 9, 2010, 1:12:36 PM3/9/10
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> Let A be a connected semisimiple complex lie group. ? Is it true that
> any free abelian group G of finite rank can be embedded in ?A?
>
> I know that it is the case that any free groups of finite rank can be
> embedded into any connected complex semisimple lie group.
>
> Do we have to make any assumptions about the dimension of A for the
> question to have a positive answer?
> any free abelian group G of finite rank can be embedded in ?A?
>
> I know that it is the case that any free groups of finite rank can be
> embedded into any connected complex semisimple lie group.
>
> Do we have to make any assumptions about the dimension of A for the
> question to have a positive answer?
Any such Lie group contains a complex torus isomorphic to C^*.
This group contains a subgroup isomorphic to the additive group R
of reals. Now let n be a natural number and choose v_1,...,v_n in R
which are linearly independent over Q.
Then the group Z v_1 + ... + Z v_n is free abelian of rank n and
embeds into A.
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