Quantization of Hamiltonian-type Lie algebras
type: Item
instance of: scholarly article
title: Quantization of Hamiltonian-type Lie algebras
zbMATH Open document ID: 1239.17013
author: Guangai Song
author: Yucai Su
author: Bin Xin
published in: Pacific Journal of Mathematics
publication date: 2009-04-08T00:00:00Z
review text: Let \(\Gamma\) be a nondegenerate additive subgroup of \(\mathbb{F}^{2n}\) containing an \(\mathbb{F}\)-basis \(\{ \epsilon_p : 1\leq p \leq 2n\}\) of \(\mathbb{F}^{2n}\). Set \(\Sigma_p = \varepsilon_p +\varepsilon_{\bar{p}}\) for \(1\leq p \leq n\) and where \(\bar{p}=p+n\). The Hamiltonian Lie algebra \(\mathcal{H}\) is the quotient \(\mathcal{\overline{H}}/\mathbb{F}\cdot 1\), where \(\mathcal{\overline{H}}\) is the Lie algebra spanned by the set \(\{ x^\alpha : \alpha \in \Gamma \}\) with the product \[ [x^\alpha , x^\beta]=\sum_{i=1}^n (\alpha_i \beta_{\bar{i}} - \alpha_{\bar{i}} \beta_{i} )x^{\alpha + \beta - \Sigma_i}, \] for any \(\alpha , \beta \) in \( \Gamma\). Hamilton Lie algebras were introduced by \textit{X. Xu} [J. Algebra 224, No. 1, 23--58 (2000; Zbl 0955.17019)] as a generalization of the simple Lie algebras of Cartan type over a field with characteristic zero. The classification of the Lie bialgebras structures on such Lie algebras have been presented by the authors in a previous paper [Sci. China, Ser. A 50, No. 9, 1267--1279 (2007; Zbl 1136.17018)]. The quantization of Lie algebras is closed related to the construction of Lie bialgebras. The paper under review provides a nice quantization of Hamilton Lie algebras. It deals with a quantization which uses the method of exchanging the coproduct by a Drinfeld twist element and keep the product unchanged. The authors present the Drinfeld twist element in a explicit way, which is very important, it allows to obtain explicit expressions for the new coproduct and antipode. The proof of this result is performed in several lemmas.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17B62
zbMATH DE Number: 5542000
zbMATH Keywords: quantization
zbMATH Keywords: Lie bialgebras
zbMATH Keywords: Drinfel'd twist element
zbMATH Keywords: Hamiltonian-type Lie bialgebras
reviewed by: Maria Elisabete Félix Barreiro Carvalho
MaRDI profile type: Publication
full work available at URL: pjm.2009.240.371
OpenAlex ID: W2063323767
DOI: 10.2140/PJM.2009.240.371
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description: scientific article; zbMATH DE number 5542000