Complex structure on quantum-braided planes

Abstract

We construct a quantum Dolbeault double complex \(\oplus _{p,q}\Omega ^{p,q}\) on the quantum plane \({\mathbb {C}}_q^2\). This solves the long-standing problem that the standard differential calculus on the quantum plane is not a \(*\)-calculus, by embedding it as the holomorphic part of a \(*\)-calculus. We show in general that any Nichols–Woronowicz algebra or braided plane \(B_+(V)\), where V is an object in an Abelian \({\mathbb {C}}\)-linear braided bar category of real type, is a quantum complex space in this sense of a factorisable Dolbeault double complex. We combine the Chern construction on \(\Omega ^{1,0}\) in such a Dolbeault complex for an algebra A with its conjugate to construct a canonical metric-compatible connection on \(\Omega ^1\) associated with a class of quantum metrics, and apply this to the quantum plane. We also apply this to finite groups G with Cayley graph generators split into two halves related by inversion, constructing such a Dolbeault complex \(\Omega (G)\) in this case. This construction recovers the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with \(\Omega ({\mathbb {Z}})\), now viewed as a quantum complex structure on \({\mathbb {Z}}\). We also show how to build natural quantum metrics on \(\Omega ^{1,0}\) and \(\Omega ^{0,1}\) separately, where the inner product in the case of the quantum plane, in order to descend to \(\otimes _A\), is taken with values in an A-bimodule.