We construct an abelian category and exact functors in which on the Grothendieck group descend to the action of a simply laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of . The category is the direct sum of a semisimple category and the category of modules over a certain algebra associated with a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root systems  2001 Elsevier Science

We categorify representations of quantum 𝔰𝔩n whose highest weight is twice a fundamental weight.

We show that the connection responsible for any abelian or non abelian Aharonov-Bohm effect with n parallel "magnetic" flux lines in R 3 , lies in a trivial G-principal bundle P → M , i.e. P is isomorphic to the product M × G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering spaceM → M , where path integrals are computed, and the associated bundle P × G C m → M , where the wave function and its covariant derivative are sections.Key words: Aharonov-Bohm effect; fibre bundle theory; gauge invariance.As is well known, the magnetic Aharonov-Bohm (A-B) effect 1,2 is a gauge invariant, non local quantum phenomenon, with gauge group U (1), which takes place in a non simply connected space. It involves a magnetic field in a region where an electrically charged particle obeying the Schroedinger equation cannot enter, i.e. the ordinary 3-dimensional space minus the space occupied by the solenoid producing the field; in the ideal mathematical limit, the solenoid is replaced by a flux line. Locally, the particle couples to the magnetic potential A but not to the magnetic field B; however, the effect is gauge invariant since it only depends on the flux of B inside the solenoid.