Representations on train algebras of rank 3
Let \(F\) be an infinite field of characteristic not 2. A train algebra of rank 3 is a commutative not necessarily associative algebra over \(F\) together with a nonzero algebra homomorphism \(w:A\to F\) such that \(x^3-(1+\gamma)w(x)x^2+\gamma w(x)^2x\), with \(\gamma\in F\) and 3 is the minimal degree of such an identity. If \(A\) is a train algebra of rank 3 and if \(M\) is a vector space over \(F\), a linear map \(\mu:A\rightarrow\text{End}(M)\) is said to be a representation if the algebra \(\overline{A}=A\oplus M\) with multiplication given by \((a+m)(b+n)=ab+\mu_a(n)+\mu_b(m)\) is a train algebra of rank 3 with homomorphism given by \(\overline{w}(a+m)=w(a)\). In this setting the main result in the paper states that the subalgebra of \(\text{End}(M)\) generated by \(\mu(N)\) is nilpotent where \(N=\text{Ker}(w)\). As an easy consequence it is proved that every finite-dimensional irreducible \(A\)-module has dimension 1 over \(F\).
scientific article; zbMATH DE number 2173018