18.755 fourth problems, due October 5, 2015
I intend to do a special case of Problems 1 and 2 in class Friday, involving cosh
and sinh. Feel free to remind me if I forget.
1. By definition SL(n, R) consists of n × n real matrices of determinant 1.
You may assume that its Lie algebra sl(n, R) consists of all n × n real matrices
of trace 0; it therefore has dimension n2 − 1. The conjugation action of SL(n.R)
on Lie subgroups corresponds (by taking Lie algebras) to the conjugation action of
SL(n, R) on n × n matrices.
Find as many different (up to SL(2, R)-conjugation) Lie subalgebras of sl(2, R)
as you can. (Hint: there are six.) You should explain why your subalgebras are not
conjugate, and (if you can) why you have found all possibilities.
2. For each of your Lie subalgebras h in Problem 1, find as many different
CLOSED Lie subgroups H1 , H2 , . . . of SL(2, R) as you can, each having Lie algebra
h. (Hint: the numbers of possibilities are (I think) 1, 1, 2, 3, and the continuum
(twice). In the last cases you’re not required to index the possibilities by natural
numbers).
3. In class I defined GL(n, F) to be the group of invertible n × n matrices with
entries in a (possibly noncommutative) field F. If F is commutative, I defined a
subgroup
SL(n, F) = {g ∈ GL(n, F) | det(g) = 1}.
I discussed the fact that this definition doesn’t make sense for noncommutative F.
Find a reasonable definition of something like “SL(n, H)” inside GL(n, H). Explain what makes your definition reasonable. Try to compute the Lie algebra and
the dimension of your subgroup.
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