K: Define an algebraic group. Give some examples. (GL(n), SL(n), O(n), Sp(n)). Why is one of these not a variety in the sense we discussed earlier? (O(n) is not connected, so I changed it to SO(n).) Do you know what a rational variety is? Are any of the examples you wrote down rational? (GL(n) is, because it is A_k^{n^2} minus a hypersurface. SL(n) is because it admits a map to A_k^{n^2-1} given by writing out all but one of the matrix entries.) Are the others rational? (“I don’t think so.” “Wrong.”) Do you know what the Cayley transform is? (I hadn’t heard the name, but realized he meant the mapping sending a skew-symmetric matrix S to (I S)(I-S)^{-1}.)
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