Symplectic reduction at zero angular momentum

Abstract

We study the symplectic reduction of the phase space describing $k$ particles in ${\mathbb{R}}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $O_n$ on $k$ copies of ${\mathbb{C}}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate ${\mathbb{Z}}^{+}$-graded regular symplectomorphisms among the $O_n$- and ${\mathrm{SO}}_n$-symplectic quotients and determine which of these quotients are graded regularly symplectomorphic to linear symplectic orbifolds. We demonstrate that when $n\le k$, the zero fibre of the moment map has rational singularities and hence is normal and Cohen-Macaulay. We also demonstrate that for small values of $k$, the ring of regular functions on the symplectic quotient is graded Gorenstein.

22 pages

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Cited by in corpus (6)

• Symplectic quotients have symplectic singularities
• Hilbert series associated to symplectic quotients by ${\mathrm{SU}}_2$
• Higher Koszul brackets on the cotangent complex
• When does the zero fiber of the moment map have rational singularities?
• Hilbert measures on orbit spaces of coregular $O_m$-modules
• Multigraded Hilbert series of invariants, covariants, and symplectic quotients for some rank $1$ Lie groups