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

**Source**: https://www.maths.tcd.ie/report_series/abstracts/tcdm0419.html
**Parent**: https://www.maths.tcd.ie/research/papers/

**The appearance of the resolved singular hypersurface
{x\_0}{x\_1}-{{x\_2}^n} =0 in the classical phase space of the Lie group
SU(n)**

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.