Automorphisms and derivations in a class of Jordan and Malcev admissible algebras. (English) Zbl 0871.17030

If \(A\) is a nonassociative algebra over a field \(F\) of characteristic \(\neq 2\), then the \((p,q)\)-mutation \(A(p,q)\) is an algebra with multiplication \(x\circ y=(xp) y- (yq)x\) \((p\) and \(q\) are fixed elements of \(A)\). The primary concern is to investigate \(\operatorname{Aut} A(p,q)\) and \(\text{Der} A(p,q)\). For the associative case the isomorphisms of \(A(p,q)\) to \(B(a,b)\) are determined in terms of isomorphisms or anti-isomorphisms of \(A\) to \(B\): If \(A\) and \(B\) are prime with unity, \(p\neq q\), \(a\) and \(b\) a normal pair in \(B\) (i.e. \(B=aB+ bB=Ba +Bb)\) with \(a \neq b\), \(\varphi: A(p,q) \to B(a,b)\) is an isomorphism with \(\varphi (1)=1\), then \(\varphi\) is an isomorphism or an anti-isomorphism of \(A\) to \(B\).
\(\operatorname{Aut} A(p,q)\) is determined for any simple \(A(p,q)\) of a central simple artinian associative algebra over \(F\). And if \(g=\{(a,b) \in A \times A\); \(pa= bp,\;qa= bq\}\) and \(\Delta: g\to\text{Der} A(p,q)\) with \((a,b) \to\Delta (a,b)\) and \(\Delta (a,b)\): \(A(p,q) \to A(p,q)\) with \(x\to ax-xb\), then \(\text{Der} A(p,q) =\Delta (g) \cong g/Z(A)\), if \(A\) is a finite dimensional central simple associative algebra over \(F\), such that \(A(p,q)\) is simple \((Z(A)\) is the centre).
The quaternion and octonion cases are discussed in the closing sections.


17D25 Lie-admissible algebras
17D99 Other nonassociative rings and algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras