Grothendieck rings of universal quantum groups.
Let \(k\) be an algebraically closed field. The author considers Hopf algebras of the following types: 1) Images of matrix coalgebras \(M_n(k)^*\) through the left adjoint of the forgetful functor from Hopf algebras to coalgebras; these were introduced by \textit{M. Takeuchi} [J. Math. Soc. Japan 23, 561-582 (1971; Zbl 0217.05902)]; 2) Images of matrix coalgebras \(M_n(k)^*\) through the left adjoint of the forgetful functor from Hopf algebras with bijective antipode to coalgebras; these were introduced by \textit{P. Schauenburg} [Lect. Notes Pure Appl. Math. 210, 331-344 (2000; Zbl 0977.16019)]; 3) If \(d\) is a positive integer and \(F\) is an invertible \(n\times n\) matrix over \(k\), a new Hopf algebra \(H_d(F)\) is constructed as a free Hopf algebra generated by a matrix coalgebra \(u=(u_{ij})\) such that the \(2d\)-th power of the antipode acts on \(u\) as conjugation by \(F\) (this generalizes the universal cosovereign Hopf algebras defined by \textit{J. Bichon} [J. Pure Appl. Algebra 157, No. 2-3, 121-133 (2001; Zbl 0976.16027)]).NEWLINENEWLINE The Grothendieck ring of finite dimensional comodules over a Hopf algebra of one of these 3 types is determined, and it is shown to be isomorphic to the non-commutative polynomials over a certain set. An explicit multiplication table is given for these Grothendieck rings.
scientific article; zbMATH DE number 6037567
scientific article