Khovanov–Lauda–Rouquier algebras, also called quiver Hecke algebras or KLR algebras, categorify the negative half of a quantum group. Their graded module categories turn Chevalley generators into induction functors, quantum Serre relations into explicit complexes and decompositions, and canonical-basis coefficients into graded multiplicities.
This chapter continues BTT’s Quantum Groups, Crystal Bases, Categorification and Hall Algebras.
From a Cartan datum to diagrams
Start with a symmetrizable Cartan datum, equivalently a collection of simple roots α_i and integers a_{ij}. The KLR algebra R(ν) is attached to a positive root combination ν=Σ_i ν_i α_i of height n=Σ_iν_i.
Idempotents e(i_1,…,i_n) are indexed by sequences of vertices whose root sum is ν. Diagrammatically, they are n vertical strands coloured by the chosen vertices.
Generators
- e(i): orthogonal idempotents for colour sequences;
- y_r: dots on the r-th strand;
- ψ_r: crossings of adjacent strands r and r+1.
The relations depend on the colours and on chosen polynomials Q_{ij}(u,v). In simply-laced symmetric types, one may choose conventions in which crossings of distinct adjacent colours satisfy quadratic relations proportional to y_{r+1}−y_r when the corresponding vertices are joined.
Degrees are assigned so that dots have positive degree and crossings have degrees determined by the Cartan matrix. This grading is not decoration: it is the source of the quantum parameter q after decategorification.
A one-colour example
For sl_2 there is one colour. The algebra R(nα) becomes the nilHecke algebra. It is generated by commuting polynomial variables y_1,…,y_n and divided-difference operators ψ_r satisfying ψ_r²=0, braid relations and the nilHecke Leibniz-type relations.
The nilHecke algebra acts on polynomials by divided differences. This makes the sl_2 categorification completely explicit and gives the model for the general KLR construction.
Induction and restriction
If ν and μ are positive root combinations, there is an embedding R(ν)⊗R(μ) into a corner of R(ν+μ). This produces induction and restriction functors between graded module categories.
On Grothendieck groups, induction categorifies multiplication in U_q^−(g), while restriction reflects the coproduct-like structure.
Categorifying Chevalley generators
Adding one strand of colour i defines an induction functor F_i. Removing one i-coloured strand defines a restriction-type functor E_i.
After passing to Grothendieck groups, these functors become the quantum-group generators f_i and e_i in suitable highest-weight quotients. The categorical relations among E_i and F_i lift the quantum-group relations.
Quantum Serre relations
The quantum Serre relations are polynomial identities among the f_i. In KLR theory they are realized by graded decompositions among iterated induction functors.
Thus a cancellation identity in an algebra becomes a positive decomposition statement in a category. This is the recurring advantage of categorification: coefficients are explained by actual graded objects and maps.
Projectives and the canonical basis
The Grothendieck group of finitely generated graded projective R-modules identifies with the integral form of U_q^−(g). Distinguished indecomposable projective modules correspond, under suitable hypotheses and conventions, to canonical-basis elements.
Grading shifts become powers of q. Decomposition numbers become Laurent polynomials with nonnegative coefficients because they count graded multiplicities.
Simple modules and crystals
Simple graded KLR modules carry crystal operators induced by restriction and induction at the ends of colour sequences. Their isomorphism classes form the crystal B(∞) in the standard finite/symmetrizable setup.
This gives a categorical model for Kashiwara’s crystal graph: an edge is realized by an actual functor between module categories rather than an abstract combinatorial rule.
Cyclotomic quotients
For a dominant weight Λ, imposing a cyclotomic relation on the leftmost dot yields the cyclotomic KLR algebra R^Λ(ν).
The direct sum of module categories over all ν categorifies the integrable highest-weight representation V(Λ). The cyclotomic condition truncates the infinite negative-half category to the finite highest-weight region allowed by Λ.
Type A and symmetric groups
In type A, cyclotomic KLR algebras connect deeply with cyclotomic Hecke algebras. Brundan and Kleshchev established graded isomorphisms that place modular symmetric-group and Hecke representation theory inside the KLR framework.
This adds a grading invisible in the classical presentation and explains decomposition numbers through canonical-basis structures.
Geometry behind KLR algebras
KLR algebras admit geometric realizations through Ext algebras of sheaves on quiver-representation varieties and through convolution in suitable flag-type correspondences.
This places them beside Hall algebras and Nakajima quiver varieties: all three extract quantum-group structure from quiver geometry, but at different categorical levels.
2-representations
KLR algebras are the endomorphism algebras inside a 2-category categorifying U_q(g). Objects represent weights, 1-morphisms represent E_i and F_i, and 2-morphisms are generated by dots and crossings.
The familiar strand diagrams are therefore not merely a presentation of one algebra; they are local pieces of a higher representation theory.
Verification workflow
- Fix the Cartan datum and Q_{ij} normalization.
- State the root ν and sequence idempotents.
- Track grading conventions for dots and crossings.
- Separate ordinary KLR algebras from cyclotomic quotients.
- Distinguish projective Grothendieck groups from simple-module Grothendieck groups.
- Do not identify canonical-basis statements without the required field and characteristic hypotheses.
- Translate diagrammatic relations before comparing different papers.
Practice
1. What labels the idempotents? 2. What do dots and crossings represent? 3. What algebra appears for sl_2? 4. What does induction categorify? 5. What do cyclotomic quotients categorify? 6. Why does q appear?
Answers. Colour sequences; polynomial and braid-like generators; the nilHecke algebra; multiplication in U_q^−(g); highest-weight modules V(Λ); grading shifts become powers of q.
Representation Mathematics — Batch 15
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