Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras
Given a field \(k\), a finite group \(G\) and \(\alpha\in H^2(G,k^\times)\), where \(G\) acts trivially on \(k^\times\), the authors form the twisted group algebra \(k^\alpha G\) and define a projective Schur algebra over \(k\) as a \(k\)-central simple algebra which is a homomorphic image of some \(k^\alpha G\). Such an algebra determines an element in the Brauer group \(\text{Br}(k)\) and the set of these elements forms a subgroup, called the projective Brauer group of \(k\) and denoted by \(\text{PS}(k)\) [see \textit{F. Lorenz} and \textit{H. Opolka}, Math. Z. 162, 175-182 (1978; Zbl 0373.16006)]. The authors study projective Schur algebras defined by a nilpotent group \(G\). The subgroup of \(\text{PS}(k)\) generated by the corresponding elements is written \(\text{PNil}(k)\), with the subgroup \(\text{PAb}(k)\) arising from Abelian groups, i.e. algebras of Abelian type. They show that every projective Schur algebra of Abelian type is isomorphic to a full matrix ring over a tensor product of symbol algebras. -- Next take a finite extension \(K/k\), let \(F\) be the set of elements of \(K^\times\) of finite order \(\text{mod }k^\times\), and form the subfield \(k(F)\) spanned by \(F\). If \(K=k(F)\), \(K/k\) is called a radical extension and a central simple \(k\)-algebra \(A\) is called a radical algebra if it is a crossed product with a radical extension as splitting field. The authors prove that every element of \(\text{PNil}(k)\) can be represented by a radical algebra and they conjecture that this result holds for every member of \(\text{PS}(k)\). The proofs are by a series of reductions, using results from their earlier papers [\textit{E. Aljadeff} and \textit{J. Sonn}, J. Algebra 175, No. 1, 179-187 (1995; Zbl 0828.16017), ibid. 178, No. 2, 530-540 (1995; Zbl 0852.16011)].
scientific article; zbMATH DE number 1384465