Geometric construction of crystal bases for quantum generalized Kac-Moody algebras (Q843233)

The classical geometric realization of the crystal graph for a quantized Kac-Moody algebra uses the cotangent geometry of the moduli space \(\mathcal{M}_{\alpha}\) of representations of the corresponding Dynkin quiver. The paper under review extends this approach to the case of quantum generalized Kac-Moody algebras. Similarly to the classical approach, the authors consider certain Lagrangian subvariety in \(T^* \mathcal{M}_{\alpha}\) and build a crystal graph out of its irreducible components and generic fibrations among them. However, in the case of generalized Kac-Moody algebras quivers might have loops and hence the fibrations in questions correspond to extensions with simple objects possibly having non-vanishing first self-extensions. To get around this the authors restrict the Lagrangian subvariety to the cotangent bundle of a certain open subset of \(\mathcal{M}_{\alpha}\).