Invariant subalgebras of involutorial quaternion division algebras.
The authors continue their study of involutions on algebras and their invariant subalgebras. For a quaternion division algebras \(Q\) over a field \(K\) of characteristic different from \(2\), it is easy to describe the \(K\)-subalgebras of \(Q\) which are invariant under an involution \(\tau\) of the first kind on \(Q\); they are \(Q\), \(K\), \(K(s)\) with \(s\not\in K\), \(s^\tau=\pm s\). When \(\tau\) is an involution of the second kind on \(Q\), with fixed field \(k\), they show the following result (although this may be known in a sense, it is useful to clearly spell it out as the authors have done in this paper): Let \(Q\) be an algebra with a \(K/k\)-involution with \(K\neq k\). For a fixed \(K/k\), the set of quaternion subalgebras of \(Q\) is in bijection with the \(K/k\)-involutions of \(Q\), where an involution \(\tau\) corresponds to a unique quaternion subalgebra \(Q_\tau\) over \(k\) such that \(\tau\) restricts to the canonical involution of \(Q_\tau\). There are also other interesting, useful results proved in the paper.
scientific article; zbMATH DE number 6110979