\(R\)-matrices in rime (Q622294)

The main goal of this paper is to introduce and study a particular class of solutions for the classical Yang-Baxter equation (YBE for short). It is defined by a condition called rime which is a relaxation of the well-known ice condition on matrix solutions of YBE. It is shown that a strict rime non-unitary solution of YBE is parametrized by a projective vector \(\vec{\phi}\) and transforms to the Cremmer-Gervais \(R\)-matrix by a change of basis with a matrix depending on \(\vec{\phi}\), while a strict rime unitary solution is equivalent to a quantization of a classical boundary \(r\)-matrix of Gerstenhaber and Giaquinto. There are also studied cases when the Bézout operators satisfy the (non-)homogeneous associative YBE and are calculated the Rota-Baxter operators corresponding to them. The paper ends with the classification of the rime Poisson brackets and of the orderable quadratic rime associative algebras, respectively.