$q$-Deformed Chern Characters for Quantum Groups $S{U}_q(N)$

Abstract

In this paper, we introduce an $N\times N$ matrix ${\epsilon }^{a\bar{b}}$ in the quantum groups $S{U}_q(N)$ to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on $S{U}_q(N)$, such as the $q$-deformed structure constant ${\mathbf{C}}_{IJ}^K$ and the $q$-deformed transposition operator $\Lambda$. From the $q$-gauge covariant condition we define the generalized $q$-deformed Killing form and the $m$-th $q$-deformed Chern class $P_m$ for the quantum groups $S{U}_q(N)$. Some useful relations of the generalized $q$-deformed Killing form are presented. In terms of the $q$-deformed homotopy operator we are able to compute the $q$-deformed Chern-Simons $Q_{2m-1}$ by the condition $d{Q}_{2m-1}=P_m$, Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived.