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A categorification of quantum \(\text{sl}(n)\). (English) Zbl 1206.17015
Quantum Topol. 1, No. 1, 1-92 (2010); erratum ibid. 2, No. 1, 97-99 (2011).
The categorification program was initiated by I. Frenkel with the aim of extending 3-dimensional topological field theories to dimensions 4 and higher [L. Crane and I. Frenkel, J. Math. Phys. 35, No. 10, 5136–5154 (1994; Zbl 0892.57014)]. This program was extended by the first author in his work on categorified tangle invariants [M. Khovanov, Algebr. Geom. Topol. 2, 665–741 (2002; Zbl 1002.57006)].
The present paper is a part of ongoing research by the authors on categorification of quantum groups and their representations. A categorification of quantum \(sl(2)\) obtained previously by the second author is generalised to \(sl(n)\). More precisely the authors construct a linear 2-category whose Grothendieck category coincides with the idempotent form of quantum \(sl(n)\). Note that the interpretation of elements of Lustig’s canonical basis of the idempotent form as classes of indecomposible ob jects established for \(sl(2)\) is still an open problem for \(sl(n)\). This category has potential applications in representation theory. The authors expect this category to manifest itself as a symmetry of various categories of interest in representation theory, ranging from derived categories of coherent sheaves on quiver varieties to categories of modules over cyclotomic and degenerate affine Hecke algebras.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
14M15 Grassmannians, Schubert varieties, flag manifolds
16T20 Ring-theoretic aspects of quantum groups
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