The special linear group for nonassociative rings

Abstract

We generalise a definition of the special linear group due to Baez to arbitrary rings. At the infinitesimal level we get a Lie ring. We give a description of these special linear rings over some large classes of rings, including all associative rings and all algebras over a field.

Derivations of octonion matrix algebras

“…One barrier to generalisation is that the commutator of two left multiplications may fail to be a derivation, for while in the 3 × 3 case the derivation algebra has dimension dim(f 4 ) = 52, in the 4 × 4 case its dimension is only dim(g 2 ⊕ so 4 (F)) = 20. One remedy would be to include products of multiplication maps, and some work in this vein is done in [10].…”