$q$-Deformed Chern Characters for Quantum Groups $SU_{q}(N)$

Details $q$-Deformed Chern Characters for Quantum Groups $SU_{q}(N)$ $q$-Deformed Chern Characters for Quantum Groups $SU_{q}(N)$

1994-07-11

arxiv.org/abs/hep-th/9407053v1

J.Math.Phys.36:5110-5138,1995

Physics

High Energy Physics

High Energy Physics - Theory

Scientific paper

10.1063/1.531217

In this paper, we introduce an $N\times N$ matrix $\epsilon^{a\bar{b}}$ in the quantum groups $SU_{q}(N)$ 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 $SU_{q}(N)$, such as the $q$-deformed structure constant ${\bf C}_{IJ}^{~K}$ 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 $SU_{q}(N)$. 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 $dQ_{2m-1}=P_{m}$, Furthermore, the $q$-deformed cocycle hierarchy, the $q$-deformed gauge covariant Lagrangian, and the $q$-deformed Yang-Mills equation are derived.

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