Modules over \(\mathfrak U_ q(\mathfrak s\mathfrak l_ 2)\)
The author is interested in the decomposition of tensor products of simple modules of the restricted quantum universal enveloping algebra \({\mathfrak U}_q (\mathfrak {sl}_2)\); the aim is to understand the structure of the corresponding fusion rule algebra. As a first step in his new approach, the author describes a canonical decomposition of \({\mathfrak U}_q (\mathfrak {sl}_2)\) into PIMs. A decomposition of \({\mathfrak U}_q (\mathfrak {sl}_2)\) into indecomposable left ideals is in general not unique, but the author defines a ``nice'' gradation such that the corresponding decomposition is homogeneous. This condition makes it unique. The author uses these indecomposable left ideals to determine the direct summands which occur in the tensor product of two simple \({\mathfrak U}_q (\mathfrak {sl}_2)\)-modules. In addition, the author classifies the indecomposable finite-dimensional \({\mathfrak U}_q (\mathfrak {sl}_2)\)-modules.
restricted quantum universal enveloping algebra
scientific article; zbMATH DE number 617391
scientific article