Unital quadratic quasi-Jordan algebras (Q539016)

The notion of quasi-Jordan algebra was introduced in 2008 by \textit{R. Velásquez} and \textit{R. Felipe} in [Commun. Algebra 36, No. 4, 1580--1602 (2008; Zbl 1188.17021)]. The relationship between associative and Lie algebras translated to the world of di-structures is just the one which can be established between associative dialgebras and Leibniz algebras. Thus, to extend the relation between associative and Jordan algebras, one needs to define quasi-Jordan algebras. Since the theory of Jordan algebras is essentially quadratic it seems natural to provide a definition of quadratic quasi-Jordan algebra. In the paper under review the authors give the mentioned definition (with unit). They prove that any quadratic Jordan algebra can be seen in a canonical way as a quadratic quasi-Jordan algebra. Also any (unital) associative dialgebra can be endowed with a unital quadratic quasi-Jordan algebra structure. Recall that a quasi-Jordan algebra is a vector space \(\mathfrak{J}\) over \(K\) (of characteristic other that \(2\)) provided with a bilinear product \(\triangleleft:\mathfrak{J}\times\mathfrak{J}\to \mathfrak{J}\) such that \[ x\triangleleft(y\triangleleft z)=x\triangleleft(z\triangleleft y) \] and \[ (y\triangleleft x)\triangleleft x^2=(y\triangleleft x^2)\triangleleft x \] for all \(x,y,z\in\mathfrak{J}\). For a quasi-Jordan algebra we can consider \[ Z^r(\mathfrak{J}):=\{z\in\mathfrak{J}: x\triangleleft z=0 \text{ for all } x\in\mathfrak{J}\}. \] Also one denotes by \(\mathfrak{J}^{\text{ann}}\) the subspace of \(\mathfrak{J}\) spanned by the elements of the form \(x\triangleleft y-y\triangleleft x\) with \(x,y\in\mathfrak{J}\) (this is the so-called annihilator ideal of \(\mathfrak{J}\)). It is not difficult to realize that \(\mathfrak{J}^{\text{ann}}\subset Z^r(\mathfrak{J})\) (both of them being ideals in \(\mathfrak{J}\)). For an ideal \(I\) of \(\mathfrak{J}\) such that \(J^{\text{ann}}\subset I\subset Z^r(\mathfrak{J})\), we say that \(\mathfrak{J}\) is split over \(I\) if there is a subalgebra \(J\) of \(\mathfrak{J}\) such that \(\mathfrak{J}=I\oplus J\) (as subspaces). An interesting result states that \(\mathfrak{J}\) is split over \(I\) if and only if \(\mathfrak{J}\) is the demisemidirect product of \(I\) and a Jordan algebra \(J\). As a corollary any unital split quasi-Jordan algebra \(\mathfrak{J}\) can be written in the form \(\mathfrak{J}=\mathfrak{J}^{\text{ann}}\oplus J\) where \(J\) is a unital Jordan algebra. Then it can be endowed with a unital quadratic Jordan algebra structure (Theorem 11). In the final section of the paper, some of these results are extended to a quadratic setting.