Quantization of non-unitary geometric classical $r$-matrices

Abstract

In this paper we explicitly attach to a geometric classical r-matrix $r$ (not necessarily unitary), a geometric (i.e., set-theoretical) quantum R-matrix $R$, which is a quantization of $r$. To accomplish this, we use the language of bijective cocycle 7-tuples, developed by A. Soloviev in the study of set-theoretical quantum R-matrices. Namely, we define a classical version of bijective cocycle 7-tuples, and show that there is a bijection between them and geometric classical r-matrices. Then we show how any classical bijective cocycle 7-tuple can be quantized, and finally use Soloviev's construction, which turns a (quantum) bijective cocycle 7-tuple into a geometric quantum R-matrix.