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The correspondence between Lie algebras, Lie groups, and algebraic groups, onone side and commutative Hopf algebras on the other side are known for a longtime by works of Hochschild-Mostow and others. We extend this correspondence byassociating a noncommutative noncocommutative Hopf algebra to any matched pairof Lie algebras, Lie groups, and affine algebraic groups. We canonicallyassociate a modular pair in involution to any of these Hopf algebras. Moreprecisely, to any locally finite representation of a matched pair object asabove we associate a SAYD module to the corresponding Hopf algebra. At the end,we compute the Hopf cyclic cohomology of the associated Hopf algebra withcoefficients in the aforementioned SAYD module in terms of Lie algebracohomology of the Lie algebra associated to the matched pair object relative toan appropriate Levi subalgebra with coefficients induced by the originalrepresentation.

Lie-Hopf algebras and their Hopf cyclic cohomology