Leibniz algebras graded by finite root systems of type \(C_l\)
Let \(k\) be a field of characteristic zero and let \(L\) be a Lie algebra over \(k\) graded by the root system \(\Delta\). Recall \textit{G. Benkart} and \textit{E. Zelmanov} [Invent. Math. 126, No. 1, 1--45 (1996; Zbl 0871.17024)] that \(\Delta\)-graded Lie algebras of type \(C_l\) (symplectic type) have been classified. The goal of the paper under review is to extend this classification to \(\Delta\)-graded Leibniz algebras of type \(C_l\). The classification in the Leibniz case (\(l \geq 4\)) is in terms of symplectic matrices over a dialgebra. From [\textit{J.-L. Loday} (ed.) et al., Dialgebras and related operads. Lecture Notes in Mathematics. 1763. Berlin: Springer. (2001; Zbl 0970.00010)], a dialgebra \(\mathcal{D}\) over \(k\) is a vector space supporting two associative binary operations \(\dashv\), \(\vdash\) so that additionally: \[ \begin{aligned} & a \dashv ( b \dashv c) = a \dashv ( b \vdash c) \\ & (a \vdash b) \dashv c = a \vdash ( b \dashv c) \\ & (a \vdash b) \vdash c = (a \dashv b) \vdash c \end{aligned} \] Proven is that for \(L\) a \(\Delta\)-graded Leibniz algebra of type \(C_l\), \(l \geq 4\), then there is a unital associative dialgebra \(\mathfrak{a}\) with involution \(* : \mathfrak{a} \to \mathfrak{a}\) such that \(L\) is centrally isogenous with the algebra \(\mathfrak{sp}_{2l}( \mathfrak{a}, \, - )\) of symplectic \((2l) \times (2l)\)-matrices over \(\mathfrak{a}\). Two (perfect) Leibniz algebras are centrally isogenous if they have isomorphic universal central extensions. For \(l = 3\), the authors prove that \(L\) is centrally isogenous with the symplectic Steinberg algebra \(\mathfrak{stl}(\mathfrak{sp}_6 (\mathfrak{a}, - ))\), where \(\mathfrak{a}\) is an alternative involutive dialgebra whose symmetric elements lie in the nucleus of \(\mathfrak{a}\).
scientific article; zbMATH DE number 6105698
scientific article