The quantum group of a preregular multilinear form. (Q2377386)

Let \(V\) be a finite-dimensional vector space of dimension \(n\), \(b\) a nondegenerate bilinear form on \(V\). A Hopf algebra \(H(b)\) was defined by the second author and \textit{G. Launer} [Phys. Lett., B 245, No. 2, 175-177 (1990; Zbl 1119.16307)], given by \(n^2\) generators and relations. \(H(b)\) satisfies a universal property involving coactions of Hopf algebras on \(V\). The notion of a preregular multilinear form \(w\) on \(V\) was given by the second author [C. R., Math., Acad. Sci. Paris 341, No. 12, 719-724 (2005; Zbl 1105.16020)]. It is a multilinear form \(w\) on \(V\), non-degenerate in each variable, such that there is an element \(Q\) in \(\text{GL}(V)\) such that \(w(x_1,\ldots,x_m)=w(Qx_m,x_1,\ldots,x_{m-1})\) for all \(x_1,\ldots,x_m\) in \(V\). The authors define a universal Hopf algebra \(H(w)\) given by \(2n^2\) generators and relations. The authors relate \(H(w)\) to the Hopf algebra \(H(w,w')\) constructed by the second author [J. Algebra 317, No. 1, 198-225 (2007; Zbl 1141.17010)]. \(H(w,w')\) is a homomorphic image of \(H(w)\). Several examples of \(H(w)\) are given, one of which is neither commutative nor cosemisimple. The quantum group of the title is the continuous dual of \(H(w)\).