Genetic algebras and quantum mutation (Q1913643)

The authors investigate \((r,s)\)-mutations of certain genetic algebras. \(A= G(n+1,2)\) is a gametic \(K\)-algebra of a diploid population with \(n+1\) alleles, and weight-function \(\omega\). Characteristic \(K\neq 2\). Then \(A(r,s)\) is associative iff \(A(r,s)\) is commutative and either (i) characteristic \(K=3\) or 5, or (ii) \(r\) and \(s\in \text{Ker} \omega\). The following conditions are equivalent: (a) \(A(r,s)\) is power-associative and (b) \(A(r,s)\) is associative or \(r=s\). Moreover, if characteristic \(K=0\) or \(>5\), then the following conditions are equivalent: (i) \(A(r,s)^+\) is Jordan, (ii) \(A(r,s)^+\) is associative, (iii) \(\omega (r-s) = 0\). In the final section the derivations and automorphisms of \(A(r,s)\) are calculated.