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- W2024691867 abstract "We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation bya weaker condition which we call rime. Rime solutions include the standardDrinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rimeAnsatz which are maximally different from the standard one we call strictrime. A strict rime non-unitary solution is parameterized by a projectivevector. We show that this solution transforms to the Cremmer-Gervais R-matrixby a change of basis with a matrix containing symmetric functions in thecomponents of the parameterizing vector. A strict unitary solution (the rimeAnsatz is well adapted for taking a unitary limit) is shown to be equivalent toa quantization of a classical boundary r-matrix of Gerstenhaber andGiaquinto. We analyze the structure of the elementary rime blocks and find, asa by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformlydescribed in a rime form. We discuss then connections of the classical rime solutions with the Bezoutoperators. The Bezout operators satisfy the (non-)homogeneous associativeclassical Yang--Baxter equation which is related to the Rota-Baxter operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. Anormal form of each individual member of the pencil depends on the discriminantof a certain quadratic polynomial. We also classify orderable quadratic rimeassociative algebras. For the standard Drinfeld-Jimbo solution, there is a choice of themultiparameters, for which it can be non-trivially rimed. However, not everyBelavin-Drinfeld triple admits a choice of the multiparameters for which it canbe rimed. We give a minimal example." @default.
- W2024691867 created "2016-06-24" @default.
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- W2024691867 date "2010-01-01" @default.
- W2024691867 modified "2023-09-27" @default.
- W2024691867 title "$R$-matrices in Rime" @default.
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- W2024691867 doi "https://doi.org/10.4310/atmp.2010.v14.n2.a3" @default.
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