PlanetPhysics/Quantum Algebra

A quantum algebra over a [[../CosmologicalConstant2/|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 [[../IsomorphicObjectsUnderAnIsomorphism/|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 [[../Groupoid/|Hopf algebras]] are a basic source of quantum algebras.