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)

    $\displaystyle \rho^{-1} = (s \otimes 1_A)(\rho)$

  2. (QA.2)

    $\displaystyle \rho = (s \otimes s)(\rho)$

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

(QA.3)

$\displaystyle \rho^{-1} = (1_A \otimes s^{-1})(\rho)$

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



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