Lie Algebra

Consider V = C[t, t-]] the Laurent polynomials in the variable t. (Eg. +3 - 6t + 3, t+ 2 and 4, 1 - * + 7 are Laurent polynomials.) (a) Show that L = Span {t2 d, _d at' at: 2t-} is a Lie subalgebra of Der(V). (b) Find a Lie algebra isomorphic to L. (c) Find the eigenspaces and eigenvalues of 2t d (d) Show that Vo = C (constant polynomials) form an L-subrepresentation of V. (e) Show that V/Vo ~ V_1 0 Vi as L-modules where V-1 = C[t-']/Vo and V1 = C[t] / Vo