The appearance of the resolved singular hypersurface in the classical phase space of the Lie group SU(n)

Abstract

A classical phase space with a suitable symplectic structure is constructed together with functions which have Poisson brackets algebraically identical to the Lie algebra structure of the Lie group SU(n). In this phase space we show that the orbit of the generators corresponding to the simple roots of the Lie algebra give rise to fibers that are complex lines containing spheres. There are n−1 spheres on a fiber and they intersect in exactly the same way as the Cartan matrix of the Lie algebra. This classical phase space bundle, being compact, has a description as a variety. Our construction shows that the variety containing the intersecting spheres is exactly the one obtained by resolving the singularities of the variety x0x1−x2n=0 in C3. A direct connection between this singular variety and the classical phase space corresponding to the Lie group SU(n) is thus established.