On weak crossed products, Frobenius algebras, and the weak Bruhat ordering.
Let \(K/F\) be a Galois field extension with Galois group \(G\). A map \(f\colon G\times G\to K\) is called a cosikel or weak cocycle if the \(K\)-algebra \(A_f=\bigoplus_{\sigma\in G}Ku_\sigma\), with multiplication rules \[ u_\sigma u_\tau=f(\sigma,\tau)u_{\sigma\tau};\qquad u_\sigma k=\sigma(k)u_\sigma \] is associative. Note that we do not require that \(f(\sigma,\tau)\neq 0\), as is the case with usual cocycles. In this note, it is studied when \(A_f\) is Frobenius or symmetric. A graph is associated to \(f\), and \(A_f\) is Frobenius if and only if this graph has a unique maximal element. In the case where \(f\) takes only values in \(\{0,1\}\) (that is, \(f\) is idempotent), we can consider the restricted subalgebra \(\overline A_f=\bigoplus_{\sigma\in G}Fu_\sigma\). The graph of \(f\) also allows to determine whether \(\overline A_f\) is Frobenius or symmetric.
scientific article; zbMATH DE number 2173130
scientific article