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quantum algebra (Definition)

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)

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

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



"quantum algebra" is owned by bci1.

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See Also: quantum group, non-Abelian Quantum Algebraic Topology

Keywords:  quantum algebra

Cross-references: Hopf algebras, isomorphism, field
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This is version 1 of quantum algebra, born on 2008-12-17.
Object id is 338, canonical name is QuantumAlgebra.
Accessed 712 times total.

Classification:
Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
Pending Errata and Addenda
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