Bimodule structure of central simple algebras (Q342855)

Let \(F\) be a field and \(A\) be a central simple algebra over \(F\). Let \(K\) a maximal separable field extension of \(F\) inside \(A\). It is a classical fact that \(A=K a K\) for some choice of \(a\) in \(A \setminus K\). However, this is not true for an arbitrary choice of \(a\). For example, if \(K\) is Galois over \(F\), and conjugation of elements in \(K\) by \(a\) induces an automorphism of \(K\), then \(K a K\) is equal to \(K a\).NEWLINENEWLINEThe goal of this paper is to study the subspaces \(K a K\) of \(A\) and their powers \((K a K)^m\), for different choices of \(a\). The main result (Theorem 15) is a semi-ring isomorphism between \(K\)-\(K\)-sub-bimodules of \(A\) and \(H\)-\(H\)-sub-bisets of \(\text{Gal}(L/F)\) where \(L\) is the Galois closure of the separable field extension \(K/F\), and \(H=\text{Gal}(L/K)\). One of the applications (Proposition 30) is that if \(K\) is cyclic, which implies that \(A\) is generated by \(K\) and an element \(y\) with \(y^n=\beta\) and \(y \ell y^{-1}=\ell^{\sigma}\), where \(\sigma\) generates \(\text{Gal}(K/F)\) and \([K:F]=n\), then every subalgebra of \(A\) of the form \(K a K\) is in fact \(K[y^d]\) for some \(d|n\).NEWLINENEWLINEThe appendix contains an alternative proof for the classical result mentioned above, that \(A=K a K\) for some choice of \(a \in A \setminus K\).