Non-Abelian phase spaces

Abstract

For a vector space V, its phase space T*V is the vector space V(+)V* together with the canonical symplectic form on it. Since the vector space is the same as an Abelian Lie algebra, the natural question is: given a Lie algebra G, does there exist a phase space T*G? In general, the answer is negative. Below, for a large class of Lie algebras G, the phase space T*G is constructed. Three examples are treated in detail: G=gl(V); G=D(Rn), the Lie algebra of vector fields on Rn; and current algebras.