$R$-matrices in Rime

Abstract

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 φ. We show that in the finite dimension this solution transforms to the Cremmer-Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of φ. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) in the finite dimension 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 Bézout operators. The Bézout operators satisfy the (non-)homogeneous associative classical Yang-Baxter equation which is related to the Rota-Baxter operators. We calculate the Rota-Baxter operators corresponding to the Bézout operators. We classify the rime Poisson brackets: they form a three-dimensional pencil.

Cremmer‐Gervais quantum Lie algebra

“…Since the coefficients β and C are arbitrary, the classical Yang-Baxter equation for r splits into three components. The component proportional to β 2 is the classical YangBaxter equation for ρ; it is satisfied: ρ is the classical Cremmer-Gervais r-matrix [9,10]. Next, a straightforward verification shows that …”

“…Now (9) splits into four components. The component proportional to β 3 vanishes (ρ satisfies a stronger equation, see [10]). Next, a direct verification shows that where p(x, y) is a polynomial of degree not higher than n in x and not higher than n in y.…”

Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles

“…In this section, we recall the concept of a weighted infinitesimal unitary bialgebra [19,37], which generalize simultaneously the one introduced by Joni and Rota [29] and the one initiated by Loday and Ronco [35]. Based on the mixture of Eqs.…”

Weighted infinitesimal unitary bialgebras, pre-Lie, matrix algebras, and polynomial algebras

“…In 2010, Ogievetsky and Popov [35] showed that weighted infinitesimal unitary bialgebras play an important role in mathematical physics. Given a solution r ∈ A ⊗ A of the non-homogenous associative classical Yang-Baxter equation, one can construct a weighted infinitesimal unitary bialgebra [35], involving a coproduct given by ∆ r (a) := a · r − r · a − λ(a ⊗ 1) for a ∈ A.…”