Quivers with loops and generalized crystals

Abstract

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

44 pages

References in corpus (2)

• Quiver Varieties and Branching
• Quivers with loops and perverse sheaves

Cited by in corpus (17)

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• Counting absolutely cuspidals for quivers
• Introduction to quiver varieties -- for ring and representation theoriests
• Cohomological Hall algebra of Higgs sheaves on a curve
• Varieties of modules over the quantum plane
• Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers
• Ringel-Hall algebra construction of quantum Borcherds-Bozec algebras
• Continuum Kac-Moody algebras
• Anti-commuting varieties
• Addendum to "Quivers with loops and perverse sheaves"
• Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras
• Borcherds-Bozec algebras, root multiplicities and the Schofield construction
• Abstract crystals for quantum Borcherds-Bozec algebras
• Quantum Borcherds-Bozec algebras and their integrable representations