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The categorical ${s l}_{2}$ commutator in a small cyclotomic weight

Example

Take $I = {i}$ with $(α_{i} ∣ α_{i}) = 2$ , $d_{i} = 1$ , $q_{i} = q$ , and $Λ = 2 Λ_{1}$ , so that $⟨ h_{i}, Λ ⟩ = 2$ and the standard cyclotomic quotient has relations $x_{1}^{2} e (ν) = 0$ . Then:

(a) $R^{Λ} (0) = k$ , $R^{Λ} (α_{i}) = k [x_{1}] / (x_{1}^{2})$ with graded dimension $1 + q^{2}$ , and $R^{Λ} (n α_{i}) = 0$ for $n \geq 3$ , so the functors on the nonzero weights are $F_{i}^{Λ} k = R^{Λ} (α_{i})$ and $F_{i}^{Λ} E_{i}^{Λ} k = 0$ ;

(b) the categorical ${s l}_{2}$ relation of The categorical ${s l}_{2}$ commutator for cyclotomic KLR functors at $β = 0$ , $λ = Λ$ (case $⟨ h_{i}, λ ⟩ = 2 \geq 0$ ) reads $q^{- 2} F_{i}^{Λ} E_{i}^{Λ} \oplus I d \oplus q^{2} I d ≅ E_{i}^{Λ} F_{i}^{Λ}$ ; on the vacuum it says $E_{i}^{Λ} F_{i}^{Λ} k ≅ k \oplus k {2}$ , which is exactly $R^{Λ} (α_{i}) ≅ k \oplus k {2}$ ;

(c) consequently, with the normalized operators of Cyclotomic KLR functors define an integrable quantum action on $K_{0}$ and $G_{0}$ , on $[P r o j (R^{Λ} (0))]$ : $E_{i} F_{i} [1_{Λ}] = (q + q^{- 1}) [1_{Λ}]$ and $F_{i} E_{i} [1_{Λ}] = 0$ , so $[E_{i}, F_{i}] [1_{Λ}] = (q + q^{- 1}) [1_{Λ}] = (K_{i} - K_{i}^{- 1}) / (q_{i} - q_{i}^{- 1}) [1_{Λ}]$ with $K_{i} [1_{Λ}] = q^{2} [1_{Λ}]$ ; on $[R e p (R^{Λ} (0))]$ the same commutator identity holds: $F_{i} [k] = q^{- 1} [k \oplus k {2}]$ , $E_{i}$ lowers back, $F_{i} E_{i} [k] = 0$ , and $E_{i} F_{i} [k] = (q + q^{- 1}) [k]$ . This is the mixed relation of Cyclotomic KLR functors define an integrable quantum action on $K_{0}$ and $G_{0}$ verified at the first nontrivial weight.

Verification

Given: The rank-one datum $I = {i}$ , $(α_{i} ∣ α_{i}) = 2$ , $d_{i} = 1$ , $q_{i} = q$ , $Λ = 2 Λ_{1}$ , with the standard quotient $x_{1}^{2} e (ν) = 0$ and the functors of Cyclotomic KLR induction and restriction functors.

[F1] The rank-one quotients: $R^{Λ} (0) = k$ , $R^{Λ} (α_{i}) = k [x] / (x^{2})$ of graded dimension $1 + q^{2}$ , $R^{Λ} (n α_{i}) = 0$ for $n \geq 3$ , and basic restriction of $F_{i} k$ is the vector space $k \oplus k {2}$ (Cyclotomic KLR quotients in ranks zero, one and two).

[F2] The categorical ${s l}_{2}$ relation at target weight $h = 2$ has identity shifts $0, 2$ , so its vacuum evaluation is exactly $k \oplus k {2}$ (The categorical ${s l}_{2}$ commutator for cyclotomic KLR functors).

[F3] The normalized operators on $K_{0}$ carry $q^{- 1}$ on projective $E$ at input root $α_{i}$ , while KK module $F$ carries $q^{- 1}$ and KK $E$ is raw, and the canonical module action shifts $E$ and leaves $F$ raw (Cyclotomic KLR functors define an integrable quantum action on $K_{0}$ and $G_{0}$ ).

Verification.

1.1F1

The blocks. The rank-zero block is $k$ and the rank-one quotient is $k [x] / (x^{2})$ with basis $1, x$ and degrees $0, 2$ . Restriction of the vacuum has zero target. Thus the relevant functor values on the vacuum are $F_{i}^{Λ} k = R^{Λ} (α_{i})$ and $F_{i}^{Λ} E_{i}^{Λ} k = 0$ .

2.1step 1.1F2

The commutator. The relation of [F2] evaluated on the vacuum gives $E_{i}^{Λ} F_{i}^{Λ} k ≅ k \oplus k {2}$ , which is $R^{Λ} (α_{i})$ by [F1], so $q^{- 2} F_{i}^{Λ} E_{i}^{Λ} \oplus I d \oplus q^{2} I d ≅ E_{i}^{Λ} F_{i}^{Λ}$ holds at the first nontrivial weight.

3.1step 2.1F3∎

The Grothendieck commutator. With the normalizations of [F3]: projective $E_{i} F_{i} [1_{Λ}] = q^{- 1} (1 + q^{2}) [1_{Λ}] = (q + q^{- 1}) [1_{Λ}]$ and $F_{i} E_{i} [1_{Λ}] = 0$ ; KK module $F$ from the vacuum also carries $q^{- 1}$ while KK $E$ is raw, and the canonical module action shifts $E$ and leaves $F$ raw, giving the same result in both module conventions. All agree with $(q^{2} - q^{- 2}) / (q - q^{- 1})$ , the mixed relation evaluated at $K_{i} = q^{2}$ . This verifies the commutator identity at the first nontrivial weight and is a finite rank-one computation using no choice.

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Sources

Seok-Jin Kang and Masaki Kashiwara, Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras, arXiv:1102.4677v4 (standard reference, not scraped)
Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Advances in Math. 228 (2011) 803-861, arXiv:0909.1810v2 (standard reference, not scraped)

The categorical sl2sl2​ commutator in a small cyclotomic weight