Canonical bases and higher representation theory

By Ben Webster. Compositio Mathematica, 2012

71 citations recorded by the source corpus.

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 of 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/5e168ee9bce53a96d023128355f5bbb298c3dca5

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Citations in the corpus

A basis theorem for the affine oriented Brauer category and its cyclotomic quotients (2014, background): “f level ‘for the Lie algebra sl 1. We expect that these categories are closely related to the categorications of tensor products of lowest and highest weight representations introduced by Webster in [W]. In the remainder of the introduction, we are going to explain in detail the de- nitions of all o”
Semi-Infinite Highest Weight Categories (2018, methodology): “nsor product categoriﬁcations. Until quite recently, most of the naturallyoccurringexampleswerehighest weightcategories(like the onesdescribedin the previous two subsections). But the work of Webster [Web1], [Web2] and Losev and Webster [LW] has brought to prominence a very general source of example”
Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m) (2012, background): “Furthermore, the images of the indecomposable 1-morphisms in U̇Q(slm) in K0(UQ(slm)) agree with the Lusztig canonical basis in AU̇(slm) [74].”
The $\mathfrak{sl}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality (2014, background): “Furthermore, Webster [53] has shown that the images of the indecomposable 1-morphisms in U̇Q(slm) in K0(UQ(slm)) agree with the Lusztig canonical basis in Z[q±]U̇(slm).”