Friday, February 22, 2013

1302.5216 (Serkan Karaçuha et al.)

Integral Calculus on Quantum Exterior Algebras    [PDF]

Serkan Karaçuha, Christian Lomp
Hom-connections and associated integral forms have been introduced and studied by T.Brzezi\'nski as an adjoint version of the usual notion of a connection in non-commutative geometry. Given a flat hom-connection on an differential calculus $(\Omega, d)$ over an algebra $A$ yields the integral complex which for various algebras has been show to be isomorphic to the de Rham complex. The latter property had been coined the Poincar\'e duality for $A$ with respect to $\Omega$. In this paper we shed further light on the question when an algebra satisfies the Poincar\'e duality. We specialise our study to the case where an $n$-dimensional differential calculus can be constructed on a quantum exterior algebra over a bimodule of $A$. Criteria are given for free bimodules with diagonal or upper triangular bimodule structure. Our results are illustrated for a differential calculus on a multivariate quantum polynomial algebra and for a differential calculus on Manin's quantum $n$-space.
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