Transcribed Text
Problem 4. Let V denotes the vector space Rntm with basis e1
Consider the
following bilinear form on V
+   
Let O(m,n n) denote the set of all matrices A in GLm+n which preserve the form B, i.e.,
for
any vectors u. e E V and SO(m.n) be the set of matrices A in O(m,n which also
satisfy the condition det(A) = 1.
a) Show that O(m. n) and SO(m, n) are subgroups of Glatai
b) Find a condition which is satisfied by all matrices in O(m,n);
c) Explain why the Lie algebras of O(m. n) and SO(m,n) are the same;
d) Describe the Lie algebra so(m,n) of O(m,n);
Problem 2. Let G denotes the group GL,(R) for n > 2. Let H be a normal subgroup of
G.
a) Find all possibilities for H assuming it is discrete;
b) Find all possibilities for H assuming it is closed subgroup, i.e., if lim h, E H and h,

g
for some g E G, then g € H;
c) Give an example of non closed normal subgroup;
d) Describe all normal subgroups of GL"(R).
e) Find all values of m and n such that the Lie algebra so(m, n) is simple;
f) Give (infinitely many) examples of pairs (m,n) such that the group SO(m,1 n) is not
connected.
Hint: For parts b) and d) you need to find conditions like AAT = Id and A + AT = 0.
Part f) is harder than the rest.
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