Classification of involutive automorphisms and anti-automorphisms of the Lie algebra of quaternions

Abstract

In this article, we treat the space H of real quaternions as a Lie algebra equipped with its commutator product. We show that all involutions of this Lie algebra that are automorphisms (respectively, anti-automorphisms) and restrict to the identity on the centre R . 1 (sometimes called automorphisms of the first kind) are actually algebra automorphisms (resp. anti-automorphisms) of the division algebra of quaternions, which we characterized in an earlier paper. If we compose with scalar multiplication by -1, we obtain all involutive automorphisms and anti-automorphisms of the second kind, i.e. those for which the centre is contained in the -1-eigenspace. Together, we have a complete determination of all involutive (anti-)automorphisms on the quaternionic Lie algebra L(H). From this determination of all the involutive (anti-)automorphisms of L(H), one can identify via a standard bijective correspondence all the involutive (anti-)automorphisms for the corresponding simply connected multiplicative quaternion Lie group H - {0}. Wecarry out this determination explicitly.