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The Shapovalov determinant for a class of pointed Hopf algebras iscalculated, including quantized enveloping algebras, Lusztig's small quantumgroups, and quantized Lie superalgebras. Our main tools are root systems, Weylgroupoids, and Lusztig type isomorphisms. We elaborate powerful noveltechniques for the algebras at roots of unity, and pass to the general caseusing a density argument.  Key words: Hopf algebra, Nichols algebra, quantum group, representation

Drinfel'd doubles and Shapovalov determinants