A periodisation of semisimple Lie algebras
A periodisation of a Lie algebra \(L\) over a field \(K\) is a Lie algebra \(L\otimes tK[t]\) with a Lie bracket defined naturally by tensor components. (So, roughly speaking, periodisation is a positively-graded Lie algebra with an underlying Lie algebra in each degree.) In this interesting paper the author proves that a periodisation of a semisimple Lie algebra over an algebraically closed field of characteristic \(0\) without direct summands isomorphic to \(sl(2)\) possesses a presentation with only quadratic relations. Such a presentation is explicitly constructed in terms of a Chevalley basis of an underlying Lie algebra. The author promises to handle the analogous problem for Lie superalgebras in a subsequent paper. The motivation of study of presentations of periodisations of Lie superalgebras stems from the work of \textit{C. Löfwall} and \textit{J.-E. Roos} about some amazing Hopf algebras [Adv. Math. 130, 161-200 (1997; Zbl 0903.17012)].
scientific article; zbMATH DE number 1770080