54 research outputs found
A pairing between super Lie-Rinehart and periodic cyclic homology
We consider a pairing producing various cyclic Hochschild cocycles, which led
Alain Connes to cyclic cohomology. We are interested in geometrical meaning and
homological properties of this pairing. We define a non-trivial pairing between
the homology of a Lie-Rinehart (super-)algebra with coefficients in some
partial traces and relative periodic cyclic homology. This pairing generalizes
the index formula for summable Fredholm modules, the Connes-Kubo formula for
the Hall conductivity and the formula computing the K0-group of a smooth
noncommutative torus. It also produces new homological invariants of proper
maps contracting each orbit contained in a closed invariant subset in a
manifold acted on smoothly by a connected Lie group. Finally we compare it with
the characteristic map for the Hopf-cyclic cohomology.Comment: 11 page
Cyclic Homology and Quantum Orbits
A natural isomorphism between the cyclic object computing the relative cyclic
homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic
object computing the cyclic homology of a Galois coalgebra with SAYD
coefficients is presented. The isomorphism can be viewed as the
cyclic-homological counterpart of the Takeuchi-Galois correspondence between
the left coideal subalgebras and the quotient right module coalgebras of a Hopf
algebra. A spectral sequence generalizing the classical computation of
Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous
quotient-coalgebra-Galois extensions is constructed. A Pontryagin type
self-duality of the Takeuchi-Galois correspondence is combined with the cyclic
duality of Connes in order to obtain dual results on the invariant cyclic
homology, with SAYD coefficients, of algebras of invariants in homogeneous
quotient-coalgebra-Galois extensions. The relation of this dual result with the
Chern character, Frobenius reciprocity, and inertia phenomena in the local
Langlands program, the Chen-Ruan-Brylinski-Nistor orbifold cohomology and the
Clifford theory is discussed
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