#### Hiroyuki Yamane : Shapovalov determinants of Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

In the talk, the speaker will explain that the Matsumoto-type theorem of the Weyl groupoid and Heckenberger's Lusztig-type isomorphisms can be applied to give Lusztig-type PBW bases and factorization formula of Shapovalov determinants of Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type. For Lusztig's finite dimensional quantum groups at roots of unity, a generalization of Kumar's and Letzter's factorization formula is obtained for an arbitrary base field and arbitrary finite order of the deformation parameter. This is joint work with Istvan Heckenberger.