The unitary irreducible representations of the quantum Heisenberg algebra
The quantum Heisenberg algebra \(H_{q,h}(x,y)\) over a field \(k\) for \(q,h\) in \(k^*\) is the noncommutative polynomial algebra in \(x\) and \(y\) with relation \(xy-qyx=h\). The author describes the irreducible representations of \(H_{q,h}(x,y)\), and for \(k=\mathbb{C}\), the unitary irreducible representations of one of its real forms. His main tool is the left spectrum of \(H_{q,h}(x,y)\), using previous results of his about spectra of generic algebras of skew differential operators [Commun. Math. Phys. 142, 567-588 (1991; see the preceding review)]. The left spectrum of a ring \(R\) is defined as follows: Order the left ideals of \(R\) by \(I<J\) iff there is a finite subset \(S\) of \(R\) such that \((J:S)=\{r\text{ in } R\mid r S\subset J\}\) is contained in \(I\). Then the left spectrum \(\text{Spec}_ \ell R\) consists of those left ideals \(P\) of \(R\) such that \(P<(P:x)\) for every \(x\) in \(R-p\). \(\text{Spec}_ \ell R\) contains the maximal left ideals of \(R\) and is preserved by localizations.
scientific article; zbMATH DE number 54735
scientific article