Mathematics – Quantum Algebra
Scientific paper
2011-08-11
Mathematics
Quantum Algebra
Complete rewrite of previous version
Scientific paper
We introduce a novel first-order differential calculus for the quantum unitary group, and a generalisation of the Woronowicz construction of higher forms which is applicable to it. The exterior algebra produced is shown to have classical dimension. Building on this we describe the quantum projective spaces (endowed with the Heckenberger--Kolb calculus) as the base space of a quantum homogeneous bundle with the quantum unitary group as a total space. The induced higher forms for each quantum projective space are then framed as associated bundles using a new general result. Next a notion of noncommutative complex structure is given, along with a general method for constructing such structures from quantum homogeneous bundles. The method is applied to the quantum projective spaces yielding a direct deformation of their classical complex structure.
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