We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. As an application we obtain moduli spaces of flat connections on an oriented compact 2-manifold with boundary as quasi-Hamiltonian quotients of the space G 2 × · · · × G 2 .
Abstract. A quasi-Poisson manifold is a G-manifold equipped with an invariant bivector field whose Schouten bracket is the trivector field generated by the invariant element in ∧ 3 g associated to an invariant inner product. We introduce the concept of the fusion of such manifolds, and we relate the quasi-Poisson manifolds to the previously introduced quasi-Hamiltonian manifolds with groupvalued moment maps.
Abstract. For general compact Kähler manifolds it is shown that both Toeplitz quantization and geometric quantization lead to a well-defined (by operator norm estimates) classical limit. This generalizes earlier results of the authors and Klimek and Lesniewski obtained for the torus and higher genus Riemann surfaces, respectively. We thereby arrive at an approximation of the Poisson algebra by a sequence of finite-dimensional matrix algebras gl(N), N → ∞. IntroductionIn a couple of papers titled "Quantum Riemann Surfaces" [24] S. Klimek and A. Lesniewski have recently proved a classical limit theorem for the Poisson algebra of smooth functions on a compact Riemann surface Σ of genus g ≥ 2 (with Petersson Kähler structure) using the Toeplitz quantization procedure:{f,g} || = 0 .(1-2)Here, 1 = 1, 2, . . . are tensor powers of the quantizing Hermitian line bundle (L, h) over M , and the Toeplitz operators act on the Hilbert space of holomorphic sections of L 1/ as the holomorphic part of the operator that multiplies sections with f . As usual (1-2) gives the connection between the Poisson bracket of functions and the commutator of the associated operators and (1-1) prevents the theory from being empty. Compared to Berezin's covariant symbols [3] and to the concept of star products [2,6,9,11], where the basic idea is the deformation of the algebraic structure on C ∞ (M ) using as a formal deformation parameter, the emphasis lies here more on the approximation of C ∞ (M ) by operator algebras in norm sense.1991 Mathematics Subject Classification. 32J81, 58F06, 17B65, 47B35, 81S10.
Abstract. For any compact Lie group G, together with an invariant inner product on its Lie algebra g, we define the non-commutative Weil algebra W G as a tensor product of the universal enveloping algebra U (g) and the Clifford algebra Cl(g). Just as the usual Weil algebra W G = Sg * ⊗ ∧g * , W G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology H G (B) for any G-differential algebra B. The main result of this paper is the construction of an isomorphism Q : W G → W G of the two Weil algebras as G-differential spaces. Furthermore, we prove that the corresponding vector space isomorphism from the usual equivariant cohomology H G (B) to the equivariant cohomology H G (B) is in fact a ring isomorphism. This generalizes the Duflo isomorphism (Sg) G ∼ = U (g) G between the ring of invariant polynomials and the ring of Casimir elements. We extend our considerations to Weil algebras and equivariant cohomology with generalized coefficients, where the algebra U (g) is replaced by the convolution algebra E ′ (G) of distributions on G.
Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The "quantization commutes with reduction" theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M , provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.
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