On spectral representation of coalgebras and Hopf algebroids
Abstract
In this paper we establish a duality between etale Lie groupoids and a class of non-necessarily commutative algebras with a Hopf algebroid structure. For any etale Lie groupoid G over a manifold M, the groupoid algebra C_c(G) of smooth functions with compact support on G has a natural coalgebra structure over C_c(M) which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over C_c(M) we construct the associated spectral etale Lie groupoid G_sp(A) over M such that G_sp(C_c(G)) is naturally isomorphic to G. Both these constructions are functorial, and C_c is fully faithful left adjoint to G_sp. We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid C_c(G) of an etale Lie groupoid G. We also demonstrate that an analogous duality exists between sheaves on M and a class of coalgebras over C_c(M).
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- August 2002
- DOI:
- 10.48550/arXiv.math/0208199
- arXiv:
- arXiv:math/0208199
- Bibcode:
- 2002math......8199M
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematics - Differential Geometry;
- Mathematics - Operator Algebras;
- 16W30;
- 22A22
- E-Print:
- 33 pages