Structure of root graded Lie algebras (Q2892860)

From the introduction: We give a complete description of the structure of root graded Lie algebras. We fix an infinite irreducible locally finite root system \(R\) and show that a Lie algebra \(\mathcal L\) graded by \(R\) can be described in terms of a locally finite split simple Lie subalgebra \(G\), some natural representations of \(\mathcal G\) and a certain algebra called the coordinate algebra. We also give the Lie bracket on \(\mathcal L\) in terms of the Lie bracket on \(\mathcal G\), the action of the representations and the product on \(\mathfrak b\). NEWLINENEWLINEMore precisely, depending on type of \(R\), we consider a quadruple \(\mathfrak c\) so called coordinate quadruple. We next correspond to \(\mathfrak c\), a specific algebra \(\mathfrak b_{\mathfrak c}\) and a specific Lie algebra \(\{\mathfrak b_{\mathfrak c}, \mathfrak b_{\mathfrak c}\}\). Then for each subspace \(K\) of the center of \(\{\mathfrak b_{\mathfrak c}, \mathfrak b_{\mathfrak c}\}\) satisfying a certain property called the uniform property, we define a Lie algebra \(\mathcal L(\mathfrak b_{\mathfrak c},K)\) and show that it is a Lie algebra graded by \(R\). Conversely, given a Lie algebra \(\mathcal L\) graded by \(R\), we prove that \(\mathcal L\) can be decomposed as \(\mathcal M_1\oplus \mathcal M_2\) where \(\mathcal M_1\) is a direct sum of certain irreducible nontrivial \(\mathfrak g\)-submodules for a locally finite split simple Lie subalgebra \(\mathfrak g\) of \(\mathcal L\) and \(\mathcal M_2\) is a specific subalgebra of \(L\). We derive a quadruple \(\mathfrak c\) from the \(\mathfrak g\)-module structure of \(\mathcal M_1\) and show that it is a coordinate quadruple. We also prove that there is a subspace \(K\) of \(\{\mathfrak b_{\mathfrak c}, \mathfrak b_{\mathfrak c}\}\) satisfying the uniform property such that \(\mathcal M_2\) is isomorphic to the quotient algebra \(\{\mathfrak b_{\mathfrak c}, \mathfrak b_{\mathfrak c}\}/K\) and moreover \(\mathcal L\) is isomorphic to \(\mathcal L(\mathfrak b_{\mathfrak c},K)\). If the root system \(R\) is reduced, our method suggests another approach to characterize Lie algebras graded by \(R\) compared with what is offered by \textit{E. Neher} [Am. J. Math. 118, 439--491 (1996; Zbl 0857.17019)].