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Let $G$ be a compact Lie group of dimension $n$. Then we can embed $G$ (topologically) into a connected compact Lie group $H$. (One may choose $H=U(m)$, the unitary group, for example.)

The question is: Are there are any (non trivial) lower bounds on the dimensions of possible groups $H$? (In the case $H=U(m)$, how compares a minimal $m$ to $n$?)

If for arbitrary compact Lie groups $G$ such an answer seems not possible, are there subclasses of the class of compact Lie groups, where one can obtain lower bounds in the above sense? (The class of connected groups excluded...)

I imagine some bounds, which for example may utilize the number of components, the rank of $G$, or other Lie group data. But as I am not so experienced with Lie groups my imagination might prove unrealistic.

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Don't know the general answer, but might one be able to do this on an almost case-by-case analysis? e.g., one gets at once that $O(n)$ can be embedded in $SO(n+1)$. –  José Figueroa-O'Farrill Jun 7 '11 at 14:28
    
A simpler question is: is there a a function $f$ such that a compact Lie group of dimension n has a faithful representation of dimension bounded by $f(n)$? –  Mariano Suárez-Alvarez Jun 7 '11 at 17:40
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Once you have a faithful representation of the connected component of the identity, you can induce it up to a faithful representation of the whole group. –  André Henriques Jun 7 '11 at 20:24

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