PlanetPhysics/Quantum Algebra

A quantum algebra over a field $k$ is defined as a triple $(A,\rho ,s)$ where $(A,\rho )$ is a Yang-Baxter algebra over the field $k$ and $s:A\to A^{op}$ is an algebra isomorphism, subject to the following two axioms:

1. (QA.1) $\rho ^{-1}=(s\otimes 1_{A})(\rho )$
2. (QA.2) $\rho =(s\otimes s)(\rho )$

Note also that(QA.1) and(QA.2) imply(QA.3):

(QA.3) $\rho ^{-1}=(1_{A}\otimes s^{-1})(\rho )$.

Remark Quasitriangular Hopf algebras are a basic source of quantum algebras.