On PBW bases for Yangians
A Yangian \(Y\) is a deformation, in the category of Hopf algebras, of the universal enveloping algebra \(U\) of the Lie algebra of polynomial maps \(\mathbb{C}\to {\mathfrak g}\), where \({\mathfrak g}\) is a finite-dimensional complex simple Lie algebra. The aim of this paper is to define a vector space basis of \(Y\) analogous to the classical Poincaré-Birkhoff-Witt basis of \(U\). As an algebra, \(Y\) is generated by elements \(n^ \pm_{\alpha k}\), \(h_{\alpha k}\), where \(k\in\mathbb{N}\) and \(\alpha\in\Pi\), the set of simple roots of \({\mathfrak g}\). The main point is to define elements \(n^ \pm_{\alpha k}\), where \(k\in\mathbb{N}\) and \(\alpha\in\Delta^ +\), the set of all positive roots of \({\mathfrak g}\). This is done in an ad hoc fashion, and the resulting elements \(n^ \pm_{\alpha k}\) are by no means canonical. However, if one says that \(n^ \pm_{\alpha k}\), \(h_{\alpha k}\), for \(k\in\mathbb{N}\), \(\alpha\in\Pi\), have degree \(k\), one obtains a filtration \(Y_ 0\subset Y_ 1 \subset\dots\) of \(Y\) such that, if \(\widetilde{n}^ \pm_{\alpha k}\) is another choice, then \(\widetilde{n}^ \pm_{\alpha k}-n^ \pm_{\alpha k}\in Y_{k-1}\) for all \(k\in\mathbb{N}\), \(\alpha\in\Delta^ +\). The author proves that the set of ordered monomials in the set \(\{n^ \pm_{\alpha k}\}_{\alpha\in\Delta^ +,k\in\mathbb{N}} \cup \{h_{\alpha k}\}_{\alpha\in\Pi,k\in\mathbb{N}}\), with respect to any total order on this set, is a basis of \(Y\).
scientific article; zbMATH DE number 220465