Parthasarathy, R. Quantum analogues of a coherent family of modules at roots of unity: \(A_2,B_2\). (English) Zbl 0901.17008 Adhikari, S. D. (ed.), Current trends in mathematics and physics. A tribute to Harish-Chandra. New Delhi: Narosa Publishing House. 87-115 (1995). Let \({\mathfrak g}\) be a semisimple complex Lie algebra and denote by \(U_q({\mathfrak g})\) the corresponding quantized enveloping algebra at a complex root of unity (obtained via Lusztig’s divided power construction). In a previous paper [R. Partharasathy, J. Math. Soc. Japan 48, No. 1, 187-194 (1996; Zbl 0853.17013)], the author has associated to a given coherent family of virtual representations of \({\mathfrak g}\) a corresponding family for \(U_q({\mathfrak g})\). He conjectures that in a certain positive cone the members of such families are all actual modules. In this paper he proves his conjecture for type \(A_2\) and \(B_2\).For the entire collection see [Zbl 0864.00048]. Reviewer: H.H.Andersen (Aarhus) Cited in 1 Document MSC: 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Keywords:quantum groups; virtual modules; quantized enveloping algebra; coherent family of virtual representations Citations:Zbl 0853.17013 PDFBibTeX XMLCite \textit{R. Parthasarathy}, in: Current trends in mathematics and physics. A tribute to Harish-Chandra. New Delhi: Narosa Publishing House. 87--115 (1995; Zbl 0901.17008)