Derivations of octonion matrix algebras

Abstract

It is well-known that the exceptional Lie algebras f4 and g2 arise from the octonions as the derivation algebras of the 3 × 3 hermitian and 1 × 1 antihermitian matrices, respectively. Inspired by this, we compute the derivation algebras of the spaces of hermitian and antihermitian matrices over an octonion algebra in all dimensions.

The special linear group for nonassociative rings

“…In fact, the same argument works for O 0 , and we similarly obtain SL m .O 0 / Š SL 8m .R/ for m 3. ural to ask what happens to this construction when n is greater than 3. In this case, it is shown in [9,Theorem 3.3] that the derivation algebra of h m .O/ is g 2˚s o m , but it is not clear how this interacts with the multiplication operators. In particular, when m > 3, the commutator of two such operators may fail to be a derivation.…”

One underlying mechanism for two piezoelectric effects in the octonion spaces

“…C. Maxwell first utilized the algebra of quaternions to describe the physical properties of electromagnetic fields. This method encourages the subsequent scholars to apply the quaternions and octonions [15,16] to explore the electromagnetic fields [17], gravitational fields [18,19], wave functions [20], black holes, dark matter [21], strong nuclear fields [22], weak nuclear fields [23], Gauge field [26], Hall effect, dyons [24,25], hydromechanics [27], quantum computing [28], and plasmas [29] and others.…”