Constructing symplectomorphisms between symplectic torus quotients
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Constructing symplectomorphisms between symplectic torus quotients Hans-Christian Herbig1 · Ethan Lawler2
· Christopher Seaton3
Received: 23 March 2019 / Accepted: 21 January 2020 © The Managing Editors 2020
Abstract We identify a family of torus representations such that the corresponding singular symplectic quotients at the 0-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may have graded isomorphic algebras of real regular functions and graded Poisson isomorphic complex symplectic quotients yet not be graded regularly diffeomorphic nor graded regularly symplectomorphic. Keywords Symplectic reduction · Singular symplectic quotient · Hamiltonian torus action · Graded regular symplectomorphism Mathematics Subject Classification Primary 53D20; Secondary 13A50 · 14L30
H.-C.H. was supported by CNPq through the Plataforma Integrada Carlos Chagas, E.L. was supported by a Rhodes College Research Fellowship, and C.S. was supported by the E.C. Ellett Professorship in Mathematics.
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Christopher Seaton [email protected] Hans-Christian Herbig [email protected] Ethan Lawler [email protected]
1
Departamento de Matemática Aplicada, Instituto de Matemática, UFRJ, Av. Athos da Silveira Ramos 149, Centro de Tecnologia-Bloco C, CEP: 21941-909 Rio de Janeiro, Brazil
2
Department of Mathematics and Statistics, Dalhousie University, 6316 Coburg Road, PO BOX 15000, Halifax, NS B3H 4R2, Canada
3
Department of Mathematics and Computer Science, Rhodes College, 2000 N. Parkway, Memphis, TN 38112, USA
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Beitr Algebra Geom
1 Introduction Let G be a compact Lie group and G → U(V ) a finite dimensional unitary representation of G. Here U(V ) stands for the unitary group of V , i.e., the group of automorphisms preserving the hermitian inner product ·, ·. To describe the orbit space V /G, i.e., the space of G-orbits in V , invariant theory is employed as follows. There exists a system of fundamental real homogeneous polynomial invariants ϕ1 , ϕ2 , . . . , ϕm ; we refer to the system ϕ1 , ϕ2 , . . . , ϕm as a Hilbert basis. This means that any real invariant polynomial f ∈ R[V ]G can be written as a polynomial in the ϕ’s, i.e., there exists a polynomial g ∈ R[x1 , x2 , . . . , xm ] such that f = g(ϕ1 , ϕ2 , . . . , ϕm ). More generally, by a theorem of Schwarz (1975), for any smooth function f ∈ C ∞ (V )G there exists g ∈ C ∞ (Rm ) such that f = g(ϕ1 , ϕ2 , . . . , ϕm ). The vector-valued map ϕ = (ϕ1 , ϕ2 , . . . , ϕm ) gives rise to an embedding ϕ of V /G into euclidean space Rm , which is called the Hilbert embedding. We denote its image by X := ϕ(V ). It turns out that ϕ is actually a diffeomorphism
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