Hecke symmetries
Let \(K\) be an algebraically closed field of characteristic zero and let \(V\) be a finite-dimensional vector space over \(K\). A Hecke symmetry is an invertible closed operator \(R\colon V\otimes V\to V\otimes V\), satisfying the braid equation and the Hecke equation: \((R+1)(R-q)=0\), for some \(q\in K\) (\(q\) is assumed not to be a root of \(1\), unless \(q=1\)). In a first part, this paper contains a very nice review of the main results in the study of Hecke symmetries and the associated Hopf (bi-) algebras, with emphasis on their representation theory. These results concern mainly the class of even (and odd) Hecke symmetries. In a second part, non-even Hecke symmetries are studied. New results are given; in particular, it is shown that the quantum rank of a Hecke symmetry is a \(q\)-integer. \textit{M. Takeuchi, D. Tambara}'s example [Hokkaido Math. J. 21, No. 3, 409-419 (1992; Zbl 0763.17017), Proc. Japan Acad., Ser. A 67, No. 8, 267-269 (1991; Zbl 0753.17032)] is discussed, serving as an illustration to open questions and conjectures.
scientific article; zbMATH DE number 1517296
scientific article