Lie algebras graded by the weight system \((\Theta_n,sl_n)\) (Q2029189)

In this paper, the authors introduce and analyze the properties of a new class of Lie algebras graded by weight systems, extending the construction proposed in [\textit{Yu. Bahturin} and \textit{G. Benkart}, J. Lie Theory 14, No. 1, 243--270 (2004; Zbl 1138.17311)]. Specifically, a Lie algebra \(\mathfrak{g}\) is called \((\Theta_n,\mathfrak{sl}_n)\)-graded if it has a subalgebra isomorphic to \(\mathfrak{sl}_n\) such that, as a module, \(\mathfrak{g}\) decomposes into a sum of copies of the adjoint, the trivial module and the natural modules, as well as the symmetric and exterior squares of the latter. Explicit models for the \(\Theta_n\)-graded algebra \(\mathfrak{u}\) associated to the coordinate algebra \(\mathfrak{b}\) are constructed, and it is shown that for any \(\Theta_n\)-graded algebra \(\mathfrak{g}\) with coordinate algebra \(\mathfrak{b}\), the factor algebra \(\mathfrak{g}/Z(\mathfrak{g})\) is a cover of \(\mathfrak{u}/Z(\mathfrak{u})\), from which it follows that \(\mathfrak{g}\) is centrally isogenous to \(\mathfrak{u}\), i.e., the algebras are classified up to central extensions. To complete the study, the central universal extension of \(\mathfrak{g}/Z(\mathfrak{g})\) is studied using the skew-dihedral homology group of \(\mathfrak{b}\), a result that allows to classify \(\Theta_n\)-graded algebras up to isomorphism.