Highest weight irreducible representations of rank 2 quantum tori
type: Item
instance of: scholarly article
title: Highest weight irreducible representations of rank 2 quantum tori
zbMATH Open document ID: 1094.17011
author: Kaiming Zhao
author: S. Eswara Rao
published in: Mathematical Research Letters
publication date: 2005-04-05T00:00:00Z
review text: The rank 2 quantum torus \(C_q\) associated with non-zero complex \(q\) is the unital associative algebra with \(q\)-commuting Laurent generators \(t_1^{\pm}\) and \(t_2^{\pm}\) viewed as a Lie algebra. The algebra \(C_q\) can then be extended by the addition of a central element \(c\) and degree derivation \(d\) with relations depending on whether \(q\) is a primitive root of unity. The resulting algebras have a \(Z\)-grading with respect to the derivation. This grading gives a triangular decomposition of the algebra and so allows the construction of highest weight modules. Representation theory for these algebras has generally been restricted to integral level (or central charge). Here the authors give a construction allowing for arbitrary central charge. Let \(\phi : C[t_2^{\pm}] \oplus Cc \oplus Cd \to C\). The highest weight module with repect to \(\phi\) is constructed as follows. Take a vector \(v_0\). Let \(x\cdot v_0 = \phi(x)v_0\) for all degree-zero elements \(x\) and \(y\cdot v_0 = 0\) for all positive degree elements \(y\) in the \(Z\)-grading coming from \(d\). Next, form the induced module. These modules have unique maximal proper submodules and the quotient \(V(\phi)\) is then an irreducible highest weight module. The authors give a series of necessary and sufficient conditions for the modules \(V(\phi)\) to have finite-dimensional weight spaces (or homogeneous components) depending on whether \(q\) is generic or a root of unity.
reviewed by: Duncan J. Melville
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B68
zbMATH DE Number: 2152364
zbMATH Keywords: quantum torus
zbMATH Keywords: highest weight representations
MaRDI profile type: Publication
DOI: 10.4310/MRL.2004.V11.N5.A6
description: scientific article; zbMATH DE number 2152364