2007/04/16 by O. Ogievetsky, T. Popov, Ogievetsky, O. +2 · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Digital Image Processing Techniques #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.0704.1947
50 pages, typos corrected
openalex publication_date 2007/04/16 · arxiv created 2009/05/06 · arxiv updated 2009/12/01 · openalex created_date 2023/02/25 · openalex updated_date 2026/07/28
We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by a weaker condition which we call "rime". Rime solutions include the standard Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime Ansatz which are maximally different from the standard one we call "strict rime". A strict rime non-unitary solution is parameterized by a projective vector. We show that this solution transforms to the Cremmer-Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of the parameterizing vector. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to a quantization of a classical "boundary" r-matrix of Gerstenhaber and Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form. We discuss then connections of the classical rime solutions with the Bezout operators. The Bezout operators satisfy the (non-)homogeneous associative classical Yang--Baxter equation which is related to the Rota-Baxter operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each individual member of the pencil depends on the discriminant of a certain quadratic polynomial. We also classify orderable quadratic rime associative algebras. For the standard Drinfeld-Jimbo solution, there is a choice of the multiparameters, for which it can be non-trivially rimed. However, not every Belavin-Drinfeld triple admits a choice of the multiparameters for which it can be rimed. We give a minimal example.