Projective Schur Algebras of Nilpotent Type Are Brauer Equivalent to Radical Algebras

Abstract

Let k be a field and G a finite group . For any element α ∈ H(G, k∗) (k∗ denotes the units of k , here as a trivial G-module) we can form the twisted group algebra kG. It is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xuσyuτ = xyf(σ, τ)uστ where f : G × G → k∗ is a 2-cocycle representing α. It is easily checked that the 2-cocycle condition satisfied by f insures the associativity of the multiplication in kG. A projective Schur algebra over k is by definition a k-central simple algebra which is a homomorphic image of a twisted group algebra kG for some finite group G and some α ∈ H(G, k∗). Equivalently, a k− central simple algebra A is a projective Schur algebra (over k) if it contains a subgroup Γ of the units of A, which is finite modulo k∗ and spans A as k-vector space. We often use the notation k(Γ). Clearly a projective Schur algebra over k determines an element in the Brauer group Br(k) of the field k . The subgroup in Br(k) generated by (and in fact consisting of) such elements is called the projective Schur group of k and denoted by PS(k). The notions of projective Schur algebra and projective Schur group were introduced in [6] by Lorenz and Opolka. This construction can be viewed as the projective analogue of Schur algebras and the Schur group (denoted by S(k)) of the field k. (See [10]) It is natural to restrict attention to special types of groups G. Our interest here is in k-central simple algebras that are homomorphic images of twisted group algebras kG with G nilpotent (resp. abelian) (see [6]). We call these algebras “projective Schur algebras of nilpotent type (resp. abelian type)”. Clearly, as above, these algebras represent classes in Br(k) and we denote by PNil(k) ( resp. PAb(k)) the subgroup they generate. In [6] it is shown that if k is a number field and n is the number of roots of unity in k then PAb(k) = nBr(k), where nB denotes the n-torsion subgroup of B. More generally, since a symbol algebra of degree n over k is a homomorphic