Maximal graded subalgebras of loop toroidal Lie algebras
The author generalizes a previous result by Shalev, Zelmanov and himself in 1998 regarding the study of which he calls the \textit{loop toroidal Lie algebra} \(L(n, \mathfrak g)\) associated to a central simple Lie algebra \(\mathfrak g\) over a field \(\mathbb{F}\), that is, the maximal \(\mathbb{Z}^n\)-graded subalgebra of \(\mathfrak g\otimes_\mathbb{F} \mathbb{F}[x_1, x_1^{-1}, \ldots, x_n, x_n^{-1}]\) and obtains a stronger conclusion even in the case \(n = 1.\) The author takes advantage of the proof of his result to complete the proof of another previous result by do Rocio and San Martin which states that \textit{if \(S\) is a subsemigroup of \(\mathbb{Z}^n\) which in not contained in any half-space, then \(S\) is a subgroup}. Indeed, one of the statements in the proof of this last result appeared without a proof, when it does not seem to be obvious, and the author provides a short proof of it, for completeness.
scientific article; zbMATH DE number 5009034