A categorification of finite-dimensional irreducible representations of quantum \({\mathfrak{sl}_2}\) and their tensor products. (English) Zbl 1188.17011
Summary: The purpose of this paper is to study categorifications of tensor products of finite-dimensional modules for the quantum group for \({\mathfrak{sl}_2}\). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra \({\mathfrak{gl}_n}\). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed.We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical bases and the (dual) standard bases in terms of projective, tilting, standard and simple Harish-Chandra bimodules.
MSC:
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 20G42 | Quantum groups (quantized function algebras) and their representations |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 16G10 | Representations of associative Artinian rings |
