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

1. (QA.1)
   
2. (QA.2)
   

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

(QA.3)

.

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