Quasi-triangular structures on Hopf algebras with positive bases
This paper is a continuation of work by the authors [J. Algebra 237, No. 2, 421-445 (2001; Zbl 0991.16032)] concerning finite \(\mathbb{C}\)-Hopf algebras with positive bases. A basis \(B\) is positive if all of the structure constants of \(H\) relative to \(B\) are positive. In the original work, the authors proved that any such Hopf algebra is isomorphic to the bicrossproduct Hopf algebra \(H(G;G_+,G_-)\) arising from a factorization \(G=G_+G_-\). (The Hopf algebra structure on \(H(G;G_+,G_-)\) is recalled in the second section.) The positive basis in this situation is precisely \(G\).NEWLINENEWLINENEWLINEThe authors study quasi-triangular structures on such Hopf algebras. The quasi-triangular structures studied are also positive in that the coefficients of \(R\in H\otimes H\) when written with respect to the basis \(G\otimes G\) are all nonnegative. The main theorem is that positive quasi-triangular structures on \(H(G;G_+,G_-)\) correspond to pairs of homomorphisms \(\xi,\eta\colon G_+\to G_-\) satisfying certain conditions. The very long proof of this theorem is provided at the end of the paper.NEWLINENEWLINENEWLINEGiven two factorizations of \(G\), it is shown that their corresponding Hopf algebra structures are quasi-isomorphic. From this it follows that every positive quasi-triangular structure \(R\) on a Hopf algebra \(H\) is quasi-isomorphic to one where \(R\) is normal, i.e. where \(\xi(u)=e\) for all \(u\in G_+.\)NEWLINENEWLINENEWLINEThe conditions on the data \((G;G_+,G_-,\xi,\eta)\) can be augmented to classifying triangular structures (rather than quasi-triangular). Theorem 5.1 cites four equivalent conditions, for example \(uv=({^\xi(u)}v)(v^{\xi(u)})\) and \(\xi({^xu})x^u=x\xi(u)\). It is shown that, in this case, \(H\) is cocommutative. As a result, any finite dimensional positive triangular Hopf algebra is a twist of a group algebra.NEWLINENEWLINENEWLINEFinally, it is shown that a positive quasi-triangular structure is also ``set-theoretical''. It provides conditions for which \({\mathcal R}_{G_+}(u,v)=(u^{\eta(v)},{^\xi(u)}v) \) satisfy the Yang-Baxter equation. The conclusion is a brief paragraph which discusses Hopf groupoids that arise from a group factorization, and the notion of quasi-isomorphisms of such structures.NEWLINENEWLINEFor the entire collection see [Zbl 0955.00038].
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