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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)