Drinfeld coproduct, quantum fusion tensor category and applications.

*(English)*Zbl 1133.17010Given any symmetrizable Kac-Moody algebra \(\mathfrak{g}\), one has the associated quantum algebra \(U_q(\mathfrak{g})\). The Drinfeld affinization process can then be used to construct the affinization, \(U_q(\widehat{\mathfrak{g}})\), which contains \(U_q(\mathfrak{g})\) as a subalgebra. If \(\mathfrak{g}\) is a finite-dimensional simple Lie algebra, \(U_q(\widehat{\mathfrak{g}})\) is the quantum affine algebra. However, in general, \(U_q(\widehat{\mathfrak{g}})\) is not a quantum Kac-Moody algebra and does not have a Hopf algebra structure.

In an earlier paper [Transform. Groups 10, No. 2, 163–200 (2005; Zbl 1102.17009)], the author gave a triangular decomposition of \(U_q(\widehat{\mathfrak{g}})\) and was able to initiate a highest weight representation theory and to define a deformation of the Drinfeld coproduct. Here, the author uses the deformed coproduct to construct quantum fusion modules and togive a tensor product structure on an appropriate category of integrable modules, \(\text{Mod} (U_q(\widehat{\mathfrak{g}}))\). Modules in \(\text{Mod} (U_q(\widehat{\mathfrak{g}}))\) have composition series, but not necessarily of finite length. However, the author shows that the subcategory of modules with finite length composition series is stable under the tensor structure.

In an earlier paper [Transform. Groups 10, No. 2, 163–200 (2005; Zbl 1102.17009)], the author gave a triangular decomposition of \(U_q(\widehat{\mathfrak{g}})\) and was able to initiate a highest weight representation theory and to define a deformation of the Drinfeld coproduct. Here, the author uses the deformed coproduct to construct quantum fusion modules and togive a tensor product structure on an appropriate category of integrable modules, \(\text{Mod} (U_q(\widehat{\mathfrak{g}}))\). Modules in \(\text{Mod} (U_q(\widehat{\mathfrak{g}}))\) have composition series, but not necessarily of finite length. However, the author shows that the subcategory of modules with finite length composition series is stable under the tensor structure.

Reviewer: Duncan J. Melville (Canton)

##### MSC:

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

20G42 | Quantum groups (quantized function algebras) and their representations |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |