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Lauda, Aaron D.

A categorification of quantum \(\mathfrak{sl}(2)\). (English) Zbl 1219.17012

Adv. Math. 225, No. 6, 3327-3424 (2010).
The author categorifies Lusztig’s algebra \(\dot{\mathbf U}\) – a version of the quantized enveloping algebra \({\mathbf U}_q(\mathfrak{sl}_2)\). He constructs a 2-category \(\dot{\mathcal U}\) such that the split Grothendieck ring \(K_0(\dot{\mathcal U})\) is isomorphic to \(\dot{\mathbf U}\) and shows that the indecomposable 1-morphisms of \(\dot{\mathcal U}\) with no shift correspond to elements in Lusztig’s canonical basis. The Homs between 1-morphisms are graded lifts of a semilinear form defined on \(\dot{\mathbf U}\). For each positive integer \(N\) a representation of \(\dot{\mathcal U}\) is constructed using iterated flag varieties that categorifies the irreducible \((N+1)\)-dimensional representation of \(\dot{\mathbf U}\).
Reviewer: Florin Nicolae (Berlin)

Cited in 6 Reviews
Cited in 97 Documents

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

Lusztig canonical basis; categorification; partial flag varieties; 2-representation
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References:

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