2024/09/09 by Beggs, Edwin, Majid, Shahn
#FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2409.05253
We construct a quantum Dolbeault double complex ⊕p,qΩp,q on the quantum plane \Bbb Cq2. 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 \Bbb C-linear braided bar category of real type is a quantum complex space in this sense with a factorisable Dolbeault double complex. We combine the Chern construction on Ω1,0 in such a Dolbeault complex for an algebra A with its conjugate to construct a canonical metric compatible connection on Ω1 associated to 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 Ω(G) in this case, recovering the quantum Levi-Civita connection for any edge-symmetric metric on the integer lattice with Ω(\Bbb Z) now viewed as a quantum complex structure. We also show how to build natural quantum metrics on Ω1,0 and Ω0,1 separately where the inner product in the case of the quantum plane, in order to descend to ⊗A, is taken with values in an A-bimodule.