Cyclic Cohomology of Lie Algebras

We define and completely determine the category of Yetter-Drinfeld modules over Lie algebras. We prove a one to one correspondence between Yetter-Drinfeld modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. We associate a mixed complex to a Lie algebra and a stable-Yetter-Drinfeld module over it. We show that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

2010 Mathematics Subject Classification: 17B56, 16T15, 16E45, 19D55

Keywords and Phrases: Yetter-Drinfeld modules, Lie algebras, mixed complex, Weil algebra, Koszul complex, Hopf cyclic cohomology

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