Shape identities in train algebras of rank 3 (Q1290389)

The authors consider the possible shape identities satisfied by a commutative baric algebra \((A,\omega)\) over the field \(F\) \((\text{char } F\neq 2)\) that satisfies the identity \(x^3- (1+\gamma) \omega(x)x^2+ \gamma\omega(x)^2 x=0\), where \(\gamma\in F\). These algebras are called train algebras of rank 3. The authors classify train algebras of rank 3 according to their levels, up to level 5. The studying of train algebras of level \(6,7,\dots\) requires more sophisticated computational techniques. Therefore, they do not consider these algebras in this paper.