## 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.

### MSC:

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