On PBW bases for Yangians
type: Item
instance of: scholarly article
title: On PBW bases for Yangians
zbMATH Open document ID: 0781.17004
DOI: 10.1007/BF00739587
published in: Letters in Mathematical Physics
publication date: 1993-06-29T00:00:00Z
review text: 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 \(Y\).
reviewed by: Andrew N. Pressley
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 81R50
Mathematics Subject Classification ID: 82B23
zbMATH DE Number: 220465
zbMATH Keywords: Yangians
zbMATH Keywords: quantized enveloping algebras
zbMATH Keywords: Poincaré-Birkhoff-Witt bases
author: Sergei Levendorskiĭ
MaRDI profile type: Publication
cites work: Yangians and R-matrices
cites work: An analogue of P.B.W. theorem and the universal R-matrix for \(U_ h\mathfrak{sl}(N+1)\)
cites work: Canonical Bases Arising from Quantized Enveloping Algebras
cites work: Algebras of functions on compact quantum groups, Schubert cells and quantum tori
cites work: q-Weyl group and a multiplicative formula for universal R-matrices
full work available at URL: https://doi.org/10.1007/bf00739587
OpenAlex ID: W1967849834
description: scientific article; zbMATH DE number 220465