THE QUANTUM sl(n, ℂ) REPRESENTATION THEORY AND ITS APPLICATIONS

Abstract

In this paper, we study the quantum sl(<TEX>$n$</TEX>) representation category using the web space. Specially, we extend sl(<TEX>$n$</TEX>) web space for <TEX>$n{\geq}4$</TEX> as generalized Temperley-Lieb algebras. As an application of our study, we find that the HOMFLY polynomial <TEX>$P_n(q)$</TEX> specialized to a one variable polynomial can be computed by a linear expansion with respect to a presentation of the quantum representation category of sl(<TEX>$n$</TEX>). Moreover, we correct the false conjecture [30] given by Chbili, which addresses the relation between some link polynomials of a periodic link and its factor link such as Alexander polynomial (<TEX>$n=0$</TEX>) and Jones polynomial (<TEX>$n=2$</TEX>) and prove the corrected conjecture not only for HOMFLY polynomial but also for the colored HOMFLY polynomial specialized to a one variable polynomial.