The appearence of the resolved singular hypersurface {x_0}{x_1}-{{x_2}^n} =0 in the classical phase space of the Lie group SU(n)

TL;DR

This paper constructs a classical phase space for SU(n) with a symplectic structure, revealing a geometric link between the resolved singular hypersurface {x_0}{x_1}-{x_2}^n=0 and the Lie group's algebraic structure.

Contribution

It establishes a geometric realization of the Lie algebra of SU(n) in a classical phase space, connecting singularity resolution with the group's algebraic and geometric properties.

Findings

The phase space contains fibers with spheres intersecting according to the Cartan matrix.

The variety of intersecting spheres is the resolution of the singular hypersurface {x_0}{x_1}-{x_2}^n=0.

A direct link between the singularity resolution and the classical phase space of SU(n) is demonstrated.

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 fibres that are complex lines containing spheres. There are n-1 spheres on a fibre 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 {x_0}{x_1}-{{x_2}^n}=0 in {C^3}. A direct connection between this singular variety and the classical phase space corresponding to the Lie group SU(n) is thus established.