The unitary irreducible representations of the quantum Heisenberg algebra
type: Item
instance of: scholarly article
title: The unitary irreducible representations of the quantum Heisenberg algebra
zbMATH Open document ID: 0753.17026
DOI: 10.1007/BF02099190
author: Alexander L. Rosenberg
published in: Communications in Mathematical Physics
publication date: 1992-09-26T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16S32
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 54735
zbMATH Keywords: quantum Heisenberg algebra
zbMATH Keywords: irreducible representations
zbMATH Keywords: unitary irreducible representations
zbMATH Keywords: left spectrum
zbMATH Keywords: algebras of skew differential operators
reviewed by: Earl J. Taft
MaRDI profile type: Publication
cites work: Q4065692
cites work: A q-difference analogue of \(U({\mathfrak g})\) and the Yang-Baxter equation
cites work: UNITARY REPRESENTATIONS OF NILPOTENT LIE GROUPS
cites work: Q3998175
cites work: The spectrum of the algebra of skew differential operators and the irreducible representations of the quantum Heisenberg algebra
cites work: The left spectrum, the Levitzki radical, and noncommutative schemes.
cites work: Q4010761
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description: scientific article; zbMATH DE number 54735