Canonical bases and higher representation theory

Abstract

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case.

55 pages; DVI may not compile correctly, PDF is preferred. v2: added section on dual canonical bases. v3: improved exposition in line with new version of 1309.3796. v4: final version, to appear in Compositio Mathematica. v5: corrected references for proof of Theorem 4.5

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• Knot invariants and higher representation theory
• Tensor product categorifications and the super Kazhdan-Lusztig conjecture
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• Semi-infinite highest weight categories
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• On the definition of quantum Heisenberg category
• Categorification of quantum symmetric pairs I
• Categorified skew Howe duality and comparison of knot homologies
• Current algebras and categorified quantum groups
• Koszul duality between Higgs and Coulomb categories $\mathcal{O}$
• Representations of the oriented skein category
• A categorical action on quantized quiver varieties
• Heisenberg and Kac-Moody categorification
• On generalized category $\mathcal{O}$ for a quiver variety
• Gelfand-Tsetlin modules in the Coulomb context
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Bar operators for quasiparabolic conjugacy classes in a Coxeter group
• Unfurling Khovanov-Lauda-Rouquier algebras
• Positivity vs negativity of canonical bases
• Folding KLR algebras
• Character formulae for queer Lie superalgebras and canonical bases of types $A/C$
• Gelfand-Tsetlin modules: canonicity and calculations
• On category $\mathcal{O}$ for affine Grassmannian slices and categorified tensor products
• Gelfand $W$-graphs for classical Weyl groups
• Representations of weakly triangular categories
• On a symplectic quantum Howe duality
• Comparing and characterizing some constructions of canonical bases from Coxeter systems
• A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$
• Representations of cyclotomic oriented Brauer categories
• Strands algebras and the affine highest weight property for equivariant hypertoric categories
• PBW bases for modified quantum groups
• Cellularity of KLR and weighted KLRW algebras via crystals