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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