The algebraic and geometric classification of nilpotent assosymmetric algebras (Q2048313)

A (nonassociative) algebra is called \textit{assosymmetric} if for any \(3\) elements their associator \((x,y,z) = (xy)z - x(yz)\) is invariant with respect to any permutation of \(x,y,z\). The approach going back to Skjelbred and Sund, consisting of successive characterization of nilpotent algebras of higher dimension as central extensions of algebras of lower dimension, used previously to classify low-dimensional algebras in many varieties of nonassociative algebras, is utilized here to classify \(4\)-dimensional nilpotent assosymmetric algebras. Also, irreducible components of the corresponding algebraic varieties are described.