Derivations in Bernstein algebras
The structure of Bernstein algebras has been frequently studied in the last twenty years. However, the automorphisms and derivations of such algebras have not been deeply investigated. In 1993, C. Martínez proposed a certain characterization of the group of the automorphisms. In this paper, Der(\(A\)) is studied following a similar approach to the one used by the last two authors in the description of Aut(\(A\)). Let \(A\) be a Bernstein algebra over a commutative field \(K\) of characteristic different two, and let \(A=Ke\oplus U\oplus V\) be the Peirce decomposition relative to an idempotent \(e\in A\). Every idempotent \(e_s\) of \(A\) can be obtained from an element \(s\in U\) making \(e_s=e+s+s^2\). The new proper Peirce subspaces relative to \(e_s\) are \(U_s=\{ u_s=u+su/u\in U\}\), and \(V_s=\{ v_s=v-2(s-s^2)v/v\in V\}\). The invertible linear map \(T_s:A\to A\) given by \(T_s(\lambda e+u+v)=\lambda e_s+u_s+v_s\) is called the Peirce transformation and this maps the original Peirce decomposition into the new one. Section two is devoted to the study of the structure of derivations of \(A\) in the case of any Peirce transformation is an automorphism. This section finishes relating the subgroup \(F\) of Aut(\(A\)) which fix the idempotent with the subalgebra \(\Delta\) of Der(\(A\)) satisfying \(\Delta (e)=0\). Explicitly, if \(A\) is finite-dimensional, then \(F\) is algebraic and its Lie algebra is \(\Delta\). In the last section, the above results are used to obtain the structure of Aut(\(A\)) and Der(\(A\)) in some interesting cases (gametic, of mutation, normal, exceptional Jordan, algebras). In these cases, Aut(\(A\)) is the semidirect product of \(F\) and \(U\), and Der\((A)=\Delta\times U\). Moreover, the computation of \(F\) and \(\Delta\) is made concrete in dimension \(4\).
scientific article; zbMATH DE number 923705