  The Shapovalov determinant for a class of pointed Hopf algebras is
calculated, including quantized enveloping algebras, Lusztig's small quantum
groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl
groupoids, and Lusztig type isomorphisms. We elaborate powerful novel
techniques for the algebras at roots of unity, and pass to the general case
using a density argument.
  Key words: Hopf algebra, Nichols algebra, quantum group, representation
