Highest weight irreducible representations of rank 2 quantum tori (Q1769976)

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.