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)

Details 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) 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)

2002-11-18

arxiv.org/abs/hep-th/0211159v1

J.Math.Phys. 45 (2004) 2124-2133

Physics

High Energy Physics

High Energy Physics - Theory

11 pages, 2 figures, LaTex

Scientific paper

10.1063/1.1723700

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.

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