Centroids of finite dimensional associative dialgebras (Q2796322)

Loday introduced the notion of dialgebras, which are triples \((A,\vdash,\dashv)\), where \(A\) is a vector space and \(\vdash,\dashv\) are bilinear products on \(A\) such that for all \(a,b,c\in A\): NEWLINE\[NEWLINEa\dashv (b\dashv c)=a\dashv (b\vdash c),\:(a\vdash b)\dashv c=a\vdash(b\dashv c),\; (a\vdash b)\vdash c=(a\dashv b)\vdash c.NEWLINE\]NEWLINE Generalizing the notion of centroid of a Lie algebra, the centroid of a dialgebra \(A\) is the space of linear endomorphism \(\varphi\) of \(A\) such that for all \(a,b\in A\), NEWLINE\[NEWLINE\varphi(a\vdash b)=\varphi(a)\vdash b=a\vdash \varphi(b),\: \varphi(a\dashv b)=\varphi(a)\dashv b=a\dashv \varphi(b).NEWLINE\]NEWLINE If \(A\) is finite-dimensional, the centroid of \(A\) can be computed by solving a linear system involving the structure constants of \(A\). This is done for all complex dialgebras of dimension \(\leq 3\), using the classification of \textit{I. M. Rikhsiboev}, \textit{I. S. Rakhimov} and \textit{W. Basri} [Classification of 3-dimensional complex diassociative algebras, Malays. J. Math. Sci. 4, No. 2, 241--254 (2010)].