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Les algèbres quasi-constantes. (The quasi-constant algebras). (French) Zbl 0752.17039
Algèbres génétiques, Cah. Math., Montpellier 38, 47-64 (1989).
A quasi-constant (qc) algebra \(A\) is a baric algebra over a commutative ring \(K\) with unity and weight-function \(\omega: A\to K\) such that there exists an element \(e\) in \(A\) with \((x^ 2)^ 2=\omega(x)^ 4e\) for all \(x\in A\). In many cases this algebra coincides with the Bernstein algebra. There are 4 types of qc-algebras if the dimension is 3 and \(K\) is an infinite field. Moreover, if \(K\) is a field of characteristic \(\neq 2\), then \(\omega\circ d=0\) for any derivation \(d\) of \(A\). The automorphisms of \(A\) satisfy the property \(\text{Aut}_ KA\cong GL_ K(\text{ker }\omega)\), if the characteristic of \(K=2\). (\(L_ K\) is a subalgebra of a certain Lie algebra, connected with the vector spaces \(U\) and \(V\) in the decomposition \(A=Ke\oplus U\oplus V\)).
Finally: each qc-algebra over a ring without nilpotent elements has a unique weight-function and in a qc-algebra \(A\) over a ring without zero divisors, with characteristic 0 and with unique weight-function, \(\omega\circ d=0\) for every derivation \(d\) in \(A\).

MSC:
17D92 Genetic algebras
92D10 Genetics and epigenetics