Problem set 2. Due Monday, September 28.
Mathematics 534, Term 1, 2015. Instructor: Reichstein.
Assume that the base field is the field of complex numbers C throughout.
Problems 1, 2, 7, 8, 9, 10 on page 14 in the book.
Problem A: Show that part (b) of the Proposition on p. 11 fails if
“solvable” is replaced by “nilpotent”.
Problem B: Let L ⊂ gl3n be the vector subspace whose elements
are 3n × 3n-matrices of the form


0n
A
0n
λIn
0n
A ,
0n −λIn 0n
where A ranges over gln and λ ranges over C. Here In and 0n denote
the n × n identity matrix and and the n × n zero matrix, respectively.
(a) Check that B 3 = 03n for every B ∈ L.
(b) Check that no proper subspace W ⊂ C3n is preserved by every
B ∈ L.
(c) Show, without appealing to Engel’s theorem, that L is not closed
under the Lie bracket in gln .