Commutator identities of the homotopes of \((-1, 1)\)-algebras (Q350857)

Let \(F\) be a field of characteristic \(\neq 2,3\). Given a nonassociative (i.e.; not necessarily associative) algebra \(A\) over \(F\), consider the commutator \([x,y]=xy-yx\), associator \((x,y,z)=(xy)z-x(yz)\) and symmetrized product \(x\circ y=\frac{1}{2}(xy+yx)\), of elements in \(A\). Denote, as usual, by \(A^-\) (respectively \(A^+\)) the algebra defined on the vector space \(A\) but with multiplication \([x,y]\) (respectively \(x\circ y\)). Finally, given an element \(c\in A\), \(A^{(c)}\) will denote the algebra defined on \(A\) with multiplication \(x\cdot y=(xc)y\) (the \(c\)-\textit{homotope} of \(A\)). The algebra \(A\) is said to be \textit{right alternative} if it satisfies \((x,y,y)=0\) for any \(x,y\). Right alternative and Lie-admissible (i.e., \(A^-\) is a Lie algebra) algebras are called \((-1,1)\)-algebras. A \((-1,1)\)-algebra is said to be strict in case \([[A,A],A]=0\). Prime not associative \((-1,1)\)-algebras are always strict. The main results of the paper under review are the following: 1. If \(A\) is a \((-1,1)\)-algebra and \(c\in A\), then \((A^{(c)})^-\) is a Malcev algebra. 2. Moreover, if \(A\) is strict, then \((A^{(c)})^-\) satisfies Filippov's identity. (This is a homogeneous identity of degree \(5\) introduced by \textit{V. T.~Filippov} [Math. Notes 31, 341--345 (1982); translation from Mat. Zametki 31, 669--678 (1982; Zbl 0494.17012)] that plays a key role in the theory of Malcev algebras.) 3. Any third Engel Malcev algebra \(M\) satisfying Filippov's identity and \(xy^2z^2=0\) (right-normalized parentheses) for any \(x,y,z\), is nilpotent with \(M^6=0\). 4. If \(A\) is a prime not associative \((-1,1)\)-algebra and \(c\in A\) is invertible, then \(A^{(c)}\) is a prime alternative algebra, \((A^{(c)})^+\) is a prime Jordan algebra, and \((A^{(c)})^-\) is a Malcev algebra satisfying the hypotheses of the previous result. All these results are obtained through very detailed and technical computations with identities.