Series of the solutions to Yang-Baxter equations: Hecke type matrices and descendant R-, L-operators

TL;DR

This paper constructs series of spectral parameter dependent solutions to the Yang-Baxter equations based on Hecke type matrices and explores their applications in quantum integrable models with complex spin interactions.

Contribution

It introduces new series of solutions to Yang-Baxter equations derived from Hecke type matrices and extends these to inhomogeneous operators and Lax operators for quantum models.

Findings

Constructed series of solutions from $sl_q(2)$-invariant Hecke matrices.

Derived analogues with quantum super-algebra $osp_q(1|2)$ symmetry.

Developed Hamiltonian operators for complex spin interactions.

Abstract

We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant solutions from the $sl_{q}(2)$-invariant Hecke type $R^{r\;r}(u)$-matrices. The analogues of the matrices of Hecke type with the symmetry of the quantum super-algebra $osp_q(1|2)$ are obtained precisely. For the homogeneous solutions $R^{r^2-1\;r^2-1}$ there are constructed Hamiltonian operators of the corresponding one-dimensional quantum integrable models, which describe rather intricate interactions between different kind of spin states. Centralizer operators defined on the products of the composite states are discussed. The inhomogeneous series of the operators $R^{r\mathcal{R}}(u)$, extended Lax operators of Hecke type, also are suggested.