Heisenberg Lie bialgebras as central extensions
A bialgebra structure for a Lie algebra \(\mathfrak g\) is given by a 1-cocycle map \(\varepsilon\colon{\mathfrak g}\to\Lambda^2{\mathfrak g}\) whose transpose \(\varepsilon^*\colon\Lambda^2{\mathfrak g}^*\to{\mathfrak g}^*\) determines a Lie algebra structure on \({\mathfrak g}^*\). Bialgebra structures on the Heisenberg Lie algebra \(H=\mathbb{R}^{2n}\oplus\langle Z\rangle\) were classified by \textit{I. Szymczak} and \textit{S. Zakrewski} [J. Geom. Phys. 7, No. 4, 553-569 (1990; Zbl 0735.58021)] and are of two types. The authors show that only the first type is the central extension of a bialgebra \(\mathbb{R}^{2n}\) by \(Z\) and that the Lie structure on \((\mathbb{R}^{2n})^*\) is bound to be trivial in order that a bialgebra central extension of \(\mathbb{R}^{2n}\) by \(\langle Z\rangle\) exists. These central extensions are defined by an endomorphism \(f\in\text{End }\mathbb{R}^{2n}\): \(\varepsilon(x)=Z\wedge f(x)\) for \(x\in\mathbb{R}^{2n}\) and \(\varepsilon(Z)=0\). The authors also determine when the central extensions are isomorphic bialgebras.
scientific article; zbMATH DE number 1215964