Canonical bases replace arbitrary coordinates in a quantum group with basis elements selected by geometry, bar invariance and integrality. Lusztig’s quiver construction explains these basis elements through simple perverse sheaves on representation varieties, turning positivity into the shadow of actual geometric multiplicities.
This chapter continues KLR Algebras, Hall Algebras, Crystal Bases and Nakajima Quiver Varieties.
The basis problem
The negative half U_q^−(g) of a quantum group is generated by f_i subject to quantum Serre relations. PBW bases can be constructed after choosing an ordering of positive roots, but the result depends on choices.
A canonical basis seeks elements intrinsic enough to survive changes of presentation while retaining integral and positivity properties useful in representation theory.
Bar involution and integral form
The quantum group has a bar involution sending q to q^{-1} while fixing the Chevalley generators in the standard normalization. Canonical-basis elements are bar-invariant and belong to an integral form generated by divided powers.
Triangularity relative to a PBW-type lattice then singles out distinguished elements. Kashiwara’s global crystal basis gives an equivalent construction under standard hypotheses.
Quiver representation varieties
For a quiver Q and dimension vector ν, let E_ν be the affine variety of representations with fixed vertex spaces. The base-change group G_ν acts by changing bases.
Flags compatible with a representation define proper maps from flag-enhanced varieties to E_ν. Pushing forward constant sheaves and decomposing them by the decomposition theorem produces semisimple complexes built from simple perverse sheaves.
Convolution
Geometric induction combines representations of dimension vectors ν and μ into one of dimension ν+μ. On sheaves, the corresponding pull-push construction defines convolution.
After passing to a Grothendieck group, convolution becomes multiplication in U_q^−(g). Cohomological shifts become powers of q.
Simple perverse sheaves as basis vectors
Distinguished simple perverse sheaves appearing in Lusztig’s category correspond to canonical-basis elements. Because direct-sum multiplicities and graded cohomology dimensions are nonnegative, structure constants acquire positivity explanations.
This is categorification before the term became standard: an algebra basis is lifted to actual geometric objects.
A2 model
For the quiver 1→2, representations of dimension (1,1) are scalars. There are two orbit types: zero and nonzero maps. Orbit closures and their intersection-cohomology complexes already illustrate how strata and sheaves can encode basis information.
As dimension vectors grow, compatible flag varieties and their pushforwards produce the higher root vectors and their canonical combinations.
Positivity
Multiplying canonical-basis elements produces Laurent-polynomial coefficients with nonnegative geometric interpretations in many standard settings. Positivity is therefore not merely an algebraic accident.
The same principle appears in Soergel bimodules, Hall algebras and KLR categories: once coefficients are dimensions or graded multiplicities, negative cancellation disappears.
Crystal limit
Kashiwara’s crystal basis is the q→0 combinatorial skeleton of the global basis. Crystal operators record how basis elements move under e_i and f_i after the q-adic structure is stripped away.
The global/canonical basis restores the q-dependent information around that crystal skeleton.
KLR categorification
KLR algebras provide an algebraic categorification parallel to Lusztig’s geometric one. Indecomposable graded projectives correspond to canonical-basis elements under suitable hypotheses.
The geometric and diagrammatic models therefore describe the same quantum-group basis through different categorical realizations.
Canonical versus semicanonical
Lusztig also constructed a semicanonical basis using irreducible components of varieties of representations of preprojective algebras. Canonical and semicanonical bases share important features but are not identical in general.
This distinction is essential when translating between perverse-sheaf, constructible-function and preprojective-geometric models.
Canonical bases in representations
Highest-weight modules inherit canonical/global bases from U_q^−(g) after quotienting by the highest-weight relations. Their crystal graphs become B(Λ), while basis vectors retain integral and positivity properties.
This is why canonical bases connect algebraic quantum groups to actual integrable representations rather than remaining internal bases of U_q^− alone.
Verification workflow
- Fix the Cartan datum and quantum-group normalization.
- Distinguish PBW, canonical/global and semicanonical bases.
- Track cohomological shifts as q-powers.
- Specify the quiver orientation used for geometric construction.
- Do not infer canonical equals semicanonical.
- State field/characteristic hypotheses for KLR identifications.
- Separate crystal-basis combinatorics from the global basis itself.
Practice
1. Why are PBW bases not canonical? 2. What geometric objects label Lusztig basis elements? 3. What does convolution become after decategorification? 4. Why does positivity appear? 5. How does the crystal basis relate? 6. What is the KLR counterpart?
Answers. They depend on root orderings; distinguished simple perverse sheaves; quantum-group multiplication; coefficients become graded dimensions; it is the q→0 skeleton; indecomposable graded projective KLR modules.
Representation Mathematics — Batch 16
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