Yang–Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang–Baxter equation with the trigonometric $R$-matrix. This novel Hopf algebra together with its $q\to 1$ limit seems the most general Yang–Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter–radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect.