A categorification of quantum sl(2)

Abstract

We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for each positive integer N. This representation categorifies the irreducible (N+1)-dimensional representation of quantum sl(2).

89 pages, LaTeX2e with xypic and pstricks macros v3. updates notation to match arXiv:0807.3250 and corrects misprints

References in corpus (6)

• Derived equivalences for symmetric groups and sl\_2-categorification
• The Karoubi envelope and Lee's degeneration of Khovanov homology
• Frobenius algebras and planar open string topological field theories
• A geometric categorification of tensor products of $U_q(s{l}_2)$-modules
• A brief review of abelian categorifications
• The geometry of unitary 2-representations of finite groups and their 2-characters

Cited by in corpus (81)

• 2-Kac-Moody algebras
• A diagrammatic approach to categorification of quantum groups III
• Knot invariants and higher representation theory
• Crystals from categorified quantum groups
• Diagrammatics for Soergel categories
• Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products
• The two-color Soergel calculus
• Canonical bases and higher representation theory
• The odd nilHecke algebra and its diagrammatics
• Knot invariants and higher representation theory II: the categorification of quantum knot invariants
• On uniqueness of tensor products of irreducible categorifications
• Deformations of colored sl(N) link homologies via foams
• A Prehistory of n-Categorical Physics
• Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)
• Hecke algebras, finite general linear groups, and Heisenberg categorification
• An approach to categorification of some small quantum groups II
• The ${\mathfrak{sl}}_n$ foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality
• Derived equivalences for cotangent bundles of Grassmannians via categorical sl(2) actions
• Trace decategorification of the Hecke category
• Trace decategorification of categorified quantum sl(2)
• Annular Khovanov homology and knotted Schur-Weyl representations
• Current algebras and categorified quantum groups
• Vertex operators and 2-representations of quantum affine algebras
• Lecture notes on 2-dimensional defect TQFT
• Trace as an alternative decategorification functor
• Canonical basis for quantum osp(1|2)
• A categorification of quantum sl(2) at prime roots of unity
• sl(3)-Foams and the Khovanov-Lauda categorification of quantum sl(k)
• Clasp technology to knot homology via the affine Grassmannian
• A diagrammatic categorification of the q-Schur algebra
• Cyclicity for categorified quantum groups
• An approach to categorification of Verma modules
• Braid group actions via categorified Heisenberg complexes
• The differential graded odd nilHecke algebra
• Super Kac-Moody 2-categories
• Oddification of the cohomology of type A Springer varieties
• Thick Soergel calculus in type A
• On cubes of Frobenius extensions
• Categorified sl(N) invariants of colored rational tangles
• Categorification at prime roots of unity and hopfological finiteness
• ${\mathfrak{gl}}_n$-webs, categorification and Khovanov-Rozansky homologies
• Coxeter categories and quantum groups
• Skein Modules from Skew Howe Duality and Affine Extensions
• The Categorified Heisenberg Algebra I: A Combinatorial Representation
• Khovanov spectra for tangles
• On the structure of cyclotomic nilHecke algebras
• Cellularity and subdivision of KLR and weighted KLRW algebras
• p-DG cyclotomic nilHecke algebras
• A categorification of the Casimir of quantum sl(2)
• DG structures on odd categorified quantum sl(2)
• Monoidal categories, representation gap and cryptography
• Biadjointness in cyclic Khovanov-Lauda-Rouquier Algebras
• Odd Grassmannian bimodules and derived equivalences for spin symmetric groups
• Generalized trace and modified dimension functions on ribbon categories
• A categorification of a quantum Frobenius map
• A diagrammatic categorification of the fermion algebra
• Braid group actions from categorical symmetric Howe duality on deformed Webster algebras
• Associated graded of Hodge modules and categorical sl_2 actions
• A Categorification of Hall Algebras
• p-DG cyclotomic nilHecke algebras II
• Parameters in categorified quantum groups
• Cluster algebra structure on the finite dimensional representations of $U_q(\widehat{A_3})$ for $l$=2
• Representations of general linear groups and categorical actions of Kac-Moody algebras
• Categorified quantum sl(2) is an inverse limit of flag 2-categories
• Clifford-symmetric polynomials
• Categorifying Hecke algebras at prime roots of unity, part I
• DG-Enhanced Hecke and KLR Algebras
• Categorification of the internal braid group action for quantum groups I: 2-functoriality
• A groupoidification of the fermion algebra
• Tensor product categorifications, Verma modules and the blob 2-category
• Towards an ${\mathfrak{sl}}_2$ action on the annular Khovanov spectrum
• Towards functor exponentiation
• Enhanced nilHecke algebras and baby Verma modules
• Faithfulness of simple 2-representations of ${\mathfrak{sl}}_2$
• Cell 2-Representations and Categorification at Prime Roots of Unity
• Categorification of infinite-dimensional ${\mathfrak{sl}}_2$-modules and braid group 2-actions I: tensor products
• PBW bases for modified quantum groups
• Spin nilHecke algebras of classical type
• Trace and categorical sl(n) representations
• Asymptotic Grothendieck groups and c.b.l.f. positive dg-algebras
• Schubert Class and cyclotomic nilHecke algebras