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.
Abstract. The parastatistics algebra is a superalgebra with (even) parafermi and (odd) parabose creation and annihilation operators. The states in the parastatistics Fock-like space are shown to be in one-to-one correspondence with the Super Semistandard Young Tableaux (SSYT) subject to further constraints. The deformation of the parastatistics algebra gives rise to a monoidal structure on the SSYT which is a super-counterpart of the plactic monoid.a Michel découvreur de senties inconnus où la beauté mathématique rejoint la simplicité des lois de la physique.
Deformed parabose and parafermi algebras are revised and endowed with Hopf structure in a natural way. The noncocommutative coproduct allows for construction of parastatistics Fock-like representations, built out of the simplest deformed bose and fermi representations. The construction gives rise to quadratic algebras of deformed anomalous commutation relations which define the generalized Green ansatz.
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