HOMEWORK ASSIGNMENTS: LIE GROUPS, FALL 2009

(1) Describe the center of the Heisenberg group of strictly upper triangular three by three matrices,

( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} )

(2) Show that

T_{a}f(x) = f(x − a), M_{b}f(x) = e^{2πixb}f(x), U_{c}f(x) = e^{2πic}f(x)

define an action of the Heisenberg group on the vector space of C^{∞}-functions on the real line.

(3) Show that conjugation by the quaternion g = cos θ + u sin θ is rotation about the axis u by an angle of 2θ.

(4) Show that the center of GL_{n}R is given by scalar matrices.

(5) Check the Jacobi identity for the cross product in R^{3}.

(6) Check the Jacobi identity for commutators of matrices, and more generally for commutators in an associative C-algebra.