On the Deformation Parameter in SLq(2) Models of the Elementary Particles

When the fundamental invariant of $SLq(2)$ is expressed as $\epsilon_q = (\matrix{0 & \alpha_2 \cr -\alpha_1 & 0})$,thenthedeformationparameter,$q$,definingtheknotalgebrais$q = \frac{\alpha_1}{\alpha_2}$. We consider models in which the elementary particles carry more than one kind of charge with running coupling constants, ${\alpha }_1$ and ${\alpha }_2$, having different energy dependence and belonging to different gauge groups. Let these coupling constants be normalized to agree with experiment at hadronic energies and written as ${\alpha }_1=\frac{e}{\sqrt{\mathrm{\hslash }c}}$ and $\alpha_2 = \frac{g}{\sqrt{\hbar c}}$\mathrm{.}Then$q = \frac{e}{g}$\mathrm{.}If$e$ is an electroweak coupling and $g$ is a gluon coupling, $q$ will increase with energy. In previous discussions of $SLq(2)$ it has been assumed that $\epsilon_q^{2} = -1$\mathrm{.}Ifthisconditionismaintained,then$eg = \hbar c$. If the elementary particle is like a Schwinger dyon and therefore the source of magnetic as well as electric charge, $eg=\mathrm{\hslash }c$ is the Dirac condition for magnetic charge.