All bicovariant differential calculi on Glq(3,C) and SLq(3,C)

TL;DR

This paper classifies all bicovariant first order differential calculi on quantum groups GLq(3,C) and SLq(3,C), revealing their structure, relations, and classical limits, with implications for quantum geometry.

Contribution

It provides a complete classification of bicovariant differential calculi on GLq(3,C) and SLq(3,C), including their relations and classical limits.

Findings

Two one-parameter families of calculi on GLq(3,C)

Six calculi on SLq(3,C) for generic q

Classical limit involves a deformation with the Cartan-Killing metric

Abstract

All bicovariant first order differential calculi on the quantum group GLq(3,C) are determined. There are two distinct one-parameter families of calculi. In terms of a suitable basis of 1-forms the commutation relations can be expressed with the help of the R-matrix of GLq(3,C). Some calculi induce bicovariant differential calculi on SLq(3,C) and on real forms of GLq(3,C). For generic deformation parameter q there are six calculi on SLq(3,C), on SUq(3) there are only two. The classical limit q-->1 of bicovariant calculi on SLq(3,C) is not the ordinary calculus on SL(3,C). One obtains a deformation of it which involves the Cartan-Killing metric.