MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.