Exercise 17, § 4

We reconsider the Lie algebra $g_3$ of Exercise 9 (a) with its basis $(x_1, x_2, x_3)$.

(a) Let V be the vector space K[X]. Let D be the differential operator with respect to X on V and M the operator of multiplication by X on V. Show that if K is of characteristic 0 the mapping

is an infinite-dimensional simple representation $\rho$ of g on V.

(b) If K is of characteristic $p > 0$, the ideal $(X^p)$ of K[X] is stable under $\rho(g)$. Under the quotient representation of g on $K[X]/(X^p)$, no line is stable.