Invariance conditions on substructures of division rings. (Q2797014)

Let \(D\) be an associative division ring, \(F\) the centre of \(D\), and \(D^*\) the multiplicative group of \(D\). The Cartan-Brauer-Hua theorem states that if \(R\) is a division subring of \(D\), such that \(R^*\) is a normal subgroup of \(D^*\), then \(R=D\) or \(R\) is included in \(F\).NEWLINENEWLINE The Lie product in \(D\) is defined by the formula \([a,b]=ab-ba\), for any \(a,b\in D\). An additive subgroup \(I\) of \(D\) is said to be a Lie ideal, if \([d,i]\in I\) whenever \(i\in I\) and \(d\in D\). A subset \(A\subseteq D\) is called \(B\)-invariant, where \(B\) is a subset of \(D\), if \(bAb^{-1}\subseteq A\), for every \(b\in B\setminus\{0\}\); \(A\) is called self-invariant, if \(A\setminus\{0\}\) equals the normalizer \(N_D(A)=\{\delta\in D^*:\delta A\delta^{-1}\subseteq A\}\).NEWLINENEWLINE The paper under review shows that if \(D\neq F\) and \(T\) is a self-invariant subfield of \(D\) with a nontrivial \(F\)-automorphism, then \(T\) possesses a non-central proper subfield; in addition, if \(D\) is an algebraic division \(F\)-algebra, then finite-dimensional \(T\)-invariant subalgebras of \(D\) are contained in \(T\).NEWLINENEWLINE Henceforth, we assume that \(\mathrm{char}(D)\neq 2\) and \(D\) is an algebraic division \(F\)-algebra endowed with an involution \(*\), and we write \(S=S(D)=\{a\in D:a^*=a\}\) and \(K=K(D)=\{a\in D:a^*=-a\}\), for the set of symmetric elements and the set of skew-symmetric elements, respectively.NEWLINENEWLINE The reviewed paper shows that if \(M\) is an \(S\)-invariant \(F\)-subspace of \(D\), and \(F\) includes neither \(S\) nor \(M\), then \([a,b]\in M\), for all \(a,b\in D\); in particular, \(M\) is a Lie ideal of \(D\). Similarly, it proves that \([a,b]\in M\): \(a,b\in D\), provided that \(*\) is an involution of the second kind (i.e. \(F\) is not included in \(S\)), \(M\) is a \(K\)-invariant \(F\)-subspace of \(D\), and neither \(S\) nor \(M\) is a subset of \(F\).