The Brauer group of modified supergroup algebras

The computation of the Brauer group BM of modified supergroup algebras over a field is performed, yielding, in particular, the computation of the Brauer group of all finite-dimensional triangular Hopf algebras when the base field is algebraically closed and of characteristic zero. The used techniques are also applied for the computation of lazy cohomology of modified supergroup algebras. We relate our construction with Yinhuo Zhang's exact sequence when the base ring is a field. As an example, we compute explicitly the Brauer group and lazy cohomology for modified supergroup algebras with (extensions of) Weyl groups of irreducible root systems as a group datum and their standard representation as a representation datum.