Quantization of Hamiltonian-type Lie algebras
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.
scientific article; zbMATH DE number 5542000
scientific article