Solutions to the quantum Yang-Baxter equation having certain bialgebras as their reduced FRT construction (Q1280727)

Let \(M\) be a finite dimensional vector space over a field \(k\), \(R\colon M\otimes M\to M\otimes M\) a solution of the quantum Yang-Baxter equation (QYBE). The Faddeev-Reshetikhin-Takhtadzhan (FRT) construction produces a bialgebra \(A(R)\), which has a quotient \(B(R)\), called the reduced FRT-construction, which may be a Hopf algebra. The paper under review considers the question: Which bialgebras \(A\) are isomorphic to \(B(R)\) for some \(R\)? Answers are given for semigroup algebras \(k[S]\) with the elements of \(S\) group-like, \(k[x_1,\dots,x_n]\) for \(k\) of characteristic zero and the \(x_i\) primitive, and finite-dimensional Hopf algebras. For \(k[S]\), the answer depends on the representation theory of \(S\). For \(k[x_1,\dots,x_n]\), the answer depends on the existence of certain families of linear transformations. Finally, it turns out that every finite-dimensional Hopf algebra is isomorphic to \(B(R)\) for some solution \(R\) of the QYBE.