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Automorphismes d’une algèbre de Bernstein. (Automorphisms of a Bernstein algebra). (French) Zbl 0753.17044
Algèbres génétiques, Cah. Math., Montpellier 38, 109-116 (1989).
The authors characterize the automorphisms of a Bernstein algebra $$A$$ over a field $$K$$ of characteristic $$\neq 2$$ and prove that the automorphism group is isomorphic to a certain group $$E_ k(A)$$, consisting of triplets $$(u,f,g)$$, which are elements of the product $$U\times GL_ K(U)\times GL_ K(V)$$, and satisfy a number of special conditions. $$U$$ and $$V$$ are the subspaces which appear in the Peirce decomposition $$A=Ke\oplus U\oplus V$$ with regard to an idempotent $$e$$. $$A$$ is said to be of type $$(r+1,s)$$, if $$r=\dim_ K(U)$$ and $$s=\dim_ K(V)$$. The principal theorem is: for an algebra $$A$$ of type $$(n+1,0)$$ there is an isomorphism $$\text{Aut}_ K(A)\to U\times GL_ K(U)$$ and for an algebra of type $$(1,n)$$ there is an isomorphism $$\text{Aut}_ K(A)\to GL_ K(V)$$.

##### MSC:
 17D92 Genetic algebras 17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
##### Keywords:
Bernstein algebra; automorphism group; Peirce decomposition