Minimal quasitriangular Hopf algebras
A quasitriangular Hopf algebra is minimal if it has no quasitriangular sub-Hopf algebra. The Drinfeld double \(D(H)\) of a finite-dimensional Hopf algebra \(H\) is minimal. The author shows that every minimal quasitriangular Hopf algebra is finite-dimensional, and is the quotient of a \(D(H)\). Every quasitriangular Hopf algebra \((A,R)\) has a unique minimal quasitriangular sub-Hopf algebra \((A_ R,R)\). Write \(R = \sum^ n_{i=1} h_ i \otimes b_ i\) with \(n\) minimal. Then \(H=\text{span}\{h_ i\}\) and \(B = \text{span}\{b_ i\}\) are sub-Hopf algebras, \(B\cong (H^*)^{\text{coop}}\) and \(A_ R = BH\). There is a morphism of quasitriangular Hopf algebras \(F:(D(H),{\mathfrak R})\to (A,R)\) with \(A_ R = \text{Image }F\). Thus \(F\) is surjective if \((A,R)\) is minimal. Solutions to the quantum Yang-Baxter equation arising from representations of \((A,R)\) arise from \((A_ R,R)\), and thus from \((D(H),{\mathfrak R})\). If \((A,R)\) is finite dimensional, the author shows that \((\text{rank }R)|(\dim A)\), and if \((A,R)\) is minimal, then \((\dim A)|(\text{rank }A)^ 2\). The equality \(\dim A = (\text{rank }R)^ 2\) characterizes Drinfeld doubles. For a finite-dimensional \(H\), \(D(H)\) has certain properties (e.g. unimodular, quasitriangular). The author shows that \(D(H)^*\) has such a property if and only if both \(H\) and \(H^*\) do. Finally he uses properties of \(D(H)\) to study analogous properties of arbitrary minimal quasitriangular Hopf algebras.
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