A new class of nonassociative algebras with involution
Given a unital nonassociative algebra with involution \(({\mathcal A}, -)\) and \(\delta=\pm 1\), the operator \(V_{x,y}^\delta\) is defined by \(V_{x,y}^\delta(z)=(x\bar y)z+\delta((z\bar y)x-(z\bar x)y)\) for any \(x,y,z\in{\mathcal A}\). Then \(({\mathcal A},-)\) is said to be a \(\delta\)-structurable algebra in case \[ [V_{u,v}^\delta, V_{x,y}^\delta]=V_{V_{u.v}^\delta(x),y}^\delta-V_{x,V_{v,u}^\delta(y)}^\delta \] for any \(u,v,x,y\in {\mathcal A}\). For \(\delta=1\) this is the definition of structurable algebras, while for \(\delta=-1\) a new class of algebras appear. The paper under review is devoted to study the first properties of these algebras and related anti-Lie triple systems and Lie superalgebras.
scientific article; zbMATH DE number 5362013
scientific article