The Covariant Picard Groupoid in Differential Geometry

Waldmann, Stefan

doi:10.1142/s0219887806001314

Cited by 2 publications

(3 citation statements)

References 28 publications

(60 reference statements)

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“…Remark 5.8. We note that the above result generalizes to other algebras A which have a sort of "exponential function", see [26] for details. We also note that the above statement gives a classification of how many inequivalent lifts of the Lie algebra action on M one has for a complex line bundle.…”

Section: Proposition 54supporting

confidence: 55%

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Covariant Strong Morita Theory of Star Product Algebras

Waldmann

2009

Preprint

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In this note we recall some recent progress in understanding the representation theory of * -algebras over rings C = R(i) where R is ordered and i 2 = −1. The representation spaces are modules over auxiliary * -algebras with inner products taking values in this auxiliary * -algebra. The ring ordering allows to implement positivity requirements for the inner products. Then the representations are required to be compatible with the inner product. Moreover, one can add the notion of symmetry in form of Hopf algebra actions. For all these notions of representations there is a well-established Morita theory which we review. The core of each version of Morita theory is the corresponding Picard groupoid for which we give tools to compute and determine both the orbits and the isotropy groups. * Contributions to the proceedings of the NoMap conference, 2008.

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Section: Proposition 54supporting

confidence: 55%

“…Let A = C ∞ (M) and let H = U (g) be the complexified universal enveloping algebra of a real Lie algebra g acting on M by vector fields. Then one can show [26] U 0 (H, A) = H 1 CE (g,C ∞ (M, iÊ)) H 1 dR (M, 2πi ),…”

Section: Proposition 54mentioning

confidence: 99%

Covariant Strong Morita Theory of Star Product Algebras

Waldmann

2009

Preprint

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“…The ring-theoretic situation is handled analogously replacing U(H, Z(A)) by Gl(H, Z(A)): we do not need to formulate this in detail. In the case of (C ∞ (M )[[λ]], ⋆) and U λ (g), the ambiguity in the existence of quantum momentum maps reduces to the one in [20], see also the discussion in [22].…”

Section: Momentum Mapsmentioning

confidence: 99%

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

et al. 2011

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In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, * -equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita

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The Covariant Picard Groupoid in Differential Geometry

Abstract: In this paper we discuss some general results on the covariant Picard groupoid in the context of differential geometry and interpret the problem of lifting Lie algebra actions to line bundles in the Picard groupoid approach.

Cited by 2 publications

References 28 publications

Covariant Strong Morita Theory of Star Product Algebras

Covariant Strong Morita Theory of Star Product Algebras

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

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