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SAYD modules over Lie-Hopf algebras | B. Rangipour
; S. Sutlu
; | Date: |
31 Aug 2011 | Abstract: | In this paper a general van Est type isomorphism is established. The
isomorphism is between the Lie algebra cohomology of a bicrossed sum Lie
algebra and the Hopf cyclic cohomology of its Hopf algebra. We first prove a
one to one correspondence between stable-anti-Yetter-Drinfeld (SAYD) modules
over the total Lie algebra and SAYD modules over the associated Hopf algebra.
In contrast to the non-general case done in our previous work, here the van Est
isomorphism is found at the first level of a natural spectral sequence, rather
than at the level of complexes. It is proved that the Connes-Moscovici Hopf
algebras do not admit any finite dimensional SAYD modules except the unique
one-dimensional one found by Connes- Moscovici in 1998. This is done by
extending our techniques to work with the infinite dimensional Lie algebra of
formal vector fields. At the end, the one to one correspondence is applied to
construct a highly nontrivial four dimensional SAYD module over the Schwarzian
Hopf algebra. We then illustrate the whole theory on this example. Finally
explicit representative cocycles of the cohomology classes for this example are
calculated. | Source: | arXiv, 1108.6101 | Services: | Forum | Review | PDF | Favorites |
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