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Centroids of finite dimensional associative dialgebras. (English) Zbl 1346.17004

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 \(a,b,c\in A\): \[ a\dashv (b\dashv c)=a\dashv (b\vdash c),\:(a\vdash b)\dashv c=a\vdash(b\dashv c),\; (a\vdash b)\vdash c=(a\dashv b)\vdash c. \] 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 \(a,b\in A\), \[ \varphi(a\vdash b)=\varphi(a)\vdash b=a\vdash \varphi(b),\: \varphi(a\dashv b)=\varphi(a)\dashv b=a\dashv \varphi(b). \] 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 I. M. Rikhsiboev, I. S. Rakhimov and W. Basri [Classification of 3-dimensional complex diassociative algebras, Malays. J. Math. Sci. 4, No. 2, 241–254 (2010)].

MSC:

17A30 Nonassociative algebras satisfying other identities
16W25 Derivations, actions of Lie algebras

Citations:

Zbl 1394.16053
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