Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type. II: Unipotent classes in symplectic groups (Q2809263)

This is the second paper in a series intended to determine the finite-dimensional complex pointed Hopf algebras \(H\) with group \(G(H)\) of group-like elements isomorphic to a finite simple group of Lie type.NEWLINENEWLINEA finite group \(G\) is said to collapse if every finite-dimensional pointed Hopf algebra \(H\), with \(G(H)\) isomorphic to \(G\), is isomorphic to \(\mathbb{C}G\), \(\mathbb{C}\) the complex numbers. The first author et al. showed that \(G\) collapses if and only if for every (irreducible) \(V\) in the category of Yetter-Drinfeld modules over \(\mathbb{C}G\), the Nichols algebra \(\mathfrak{B}(V)\) is infinite-dimensional [Ann. Mat. Pura Appl. (4) 190, No. 2, 225--245 (2011; Zbl 1234.16019)]. All irreducible \(V\) are of the form \(M(\mathcal{O},p)\), \(\mathcal{O}\) a conjugacy class of \(G\), \(p\) an irreducible character. \(\mathcal{O}\) is a rack. A rack \(X\) is said to collapse if \(\mathfrak{B}(X,s)\) is finite-dimensional for every faithful 2-cocycle \(s\). Thus the issue is to determine all pairs \((X,s)\), \(X\) a finite rack and \(s\) a non-principal 2-cocycle, such that \(\mathfrak{B}(X,s)\) is finite-dimensional.NEWLINENEWLINEIn Part I [J. Algebra 442, 36--65 (2015; Zbl 1338.16034)], the authors of this paper discussed racks of types D and F and unipotent classes in \(\mathrm{SL}_n(q)\) (\(q\) a power of a prime), and showed the unipotent classes in \(G=\mathrm{PSL}_n(q)\) which collapse (with a few exceptions when \(n=2\)). Now in part II, they give an analogous discussion for the projective symplectic linear group \(\mathrm{PSp}_{2n}(q)\). The result is that if \(\mathcal{O}\) is a unipotent conjugacy class in \(\mathrm{PSp}_{2n}(q)\), then \(\mathcal{O}\) collapses if it is not listed in a table of exceptions. To do this, the authors give criterion to deal with unipotent classes of general finite simple groups of Lie type and apply it to regular classes of Chevalley and Steinberg groups, using an isogeny argument and a reduction argument.NEWLINENEWLINEFor Part III, see [the authors, Rev. Mat. Iberoam. 33, No. 3, 995--1024 (2017; Zbl 1377.16024)].