Talk:PlanetPhysics/Quantum Algebra

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: quantum algebra %%% Primary Category Code: 03. %%% Filename: QuantumAlgebra.tex %%% Version: 1 %%% Owner: bci1 %%% Author(s): bci1 %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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A \emph{quantum algebra} over a \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} $k$ 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 \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}, subject to the following two axioms:

\begin{enumerate} \item (QA.1) $$\rho^{-1} = (s \otimes 1_A)(\rho)$$ \item (QA.2) $$\rho = (s \otimes s)(\rho)$$ \end{enumerate}

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

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

\textbf{Remark} Quasitriangular \htmladdnormallink{Hopf algebras}{http://planetphysics.us/encyclopedia/Groupoid.html} are a basic source of quantum algebras.

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