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The quasiprimitive Serre element in type $A_{2}$

Statement

In the Drinfeld–Jimbo algebra of type $A_{2}$ (the Cartan datum $I = {1, 2}$ , $A = (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ , $d_{1} = d_{2} = 1$ , so $q_{1} = q_{2} = q$ ) the Serre elements

${S e r r e}_{12}^{+} = E_{1}^{2} E_{2} - [2]_{q} E_{1} E_{2} E_{1} + E_{2} E_{1}^{2}, {S e r r e}_{12}^{-} = F_{1}^{2} F_{2} - [2]_{q} F_{1} F_{2} F_{1} + F_{2} F_{1}^{2}, [2]_{q} = q + q^{- 1},$

are quasiprimitive with explicit grouplike factors:

$Δ ({S e r r e}_{12}^{+}) = {S e r r e}_{12}^{+} \otimes K_{1}^{- 2} K_{2}^{- 1} + 1 \otimes {S e r r e}_{12}^{+}, Δ ({S e r r e}_{12}^{-}) = {S e r r e}_{12}^{-} \otimes 1 + K_{1}^{2} K_{2} \otimes {S e r r e}_{12}^{-} .$

In particular every mixed bidegree term cancels. The polynomial positive Serre expression at $q = 1$ is the classical Serre bracket $[e_{1}, [e_{1}, e_{2}]] = 0$ of type $A_{2}$ .

Facts & Assumptions

Given: The Cartan matrix has $a_{12} = a_{21} = - 1$ and symmetrizer $d_{1} = d_{2} = 1$ , so $q_{1} = q_{2} = q$ and the toral actions are those in Symmetrizable Cartan data for quantum groups and The Drinfeld-Jimbo quantized enveloping algebra by generators and relations. The Drinfeld–Jimbo definition supplies the two Serre words; the coproduct/Serre lemma supplies the positive and negative toral-action Borel maps, and their images give the formulas in the Drinfeld–Jimbo quotient. Tensor-product multiplication is as stated in The coproduct preserves the positive and negative quantum Serre ideals and The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ .

[F1]

$[2]_{q} = q + q^{- 1}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F2]

If $y x = q_{i}^{2} x y$ , then $(x + y)^{N} = \sum_{r = 0}^{N} q_{i}^{r (N - r)} {(\binom{N}{r})}_{i} x^{r} y^{N - r}$ (The quantum binomial expansion for $q$ -commuting elements).

[F3]

The normalized positive coproduct assignment is an algebra map on the toral-action Borel (The coproduct preserves the positive and negative quantum Serre ideals).

[F4]

For every $i \neq j$ , the normalized negative coproduct formula is $Δ_{-} ({S e r r e}_{i j}^{-}) = {S e r r e}_{i j}^{-} \otimes 1 + K_{i}^{m_{i j}} K_{j} \otimes {S e r r e}_{i j}^{-}$ (The coproduct preserves the positive and negative quantum Serre ideals).

[F5]

In the classical Kac–Moody algebra, the Serre presentation imposes $(ad e_{i})^{1 - a_{i j}} e_{j} = 0$ (Serre presentation of a kac moody algebra).

[F6]

Tensor-product multiplication is $(a \otimes b) (c \otimes d) = a c \otimes b d$ (The tensor product of $R$ -algebras has multiplication $(a \otimes b) (a^{'} \otimes b^{'}) = a a^{'} \otimes b b^{'}$ ).

[F7]

The toral action is $K_{h} E_{i} K_{h}^{- 1} = q^{⟨ α_{i}, h ⟩} E_{i}$ (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F8]

The quantum Cartan datum fixes $q_{i} = q^{d_{i}}$ , so $d_{1} = d_{2} = 1$ gives $q_{1} = q_{2} = q$ (Symmetrizable Cartan data for quantum groups).

[F9]

The positive and negative Serre words are the sums in the Drinfeld–Jimbo presentation (The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

Proof

technique · Expand the positive three-letter words by choosing, at each position, a left coproduct term or a right coproduct term, and collect the six mixed tensor words

1.1F3F6F7F8algebra

The toral rules give $K_{1} E_{1} K_{1}^{- 1} = q^{2} E_{1}$ , $K_{1} E_{2} K_{1}^{- 1} = q^{- 1} E_{2}$ , $K_{2} E_{1} K_{2}^{- 1} = q^{- 1} E_{1}$ , and $K_{2} E_{2} K_{2}^{- 1} = q^{2} E_{2}$ . For $x_{i} = E_{i} \otimes K_{i}^{- 1}$ and $y_{i} = 1 \otimes E_{i}$ , these imply $y_{i} x_{i} = q^{2} x_{i} y_{i}$ .

1.2F4given

Here $m_{12} = 2$ , so [F4] gives $Δ ({S e r r e}_{12}^{-}) = {S e r r e}_{12}^{-} \otimes 1 + K_{1}^{2} K_{2} \otimes {S e r r e}_{12}^{-}$ ; thus the negative expansion has no mixed bidegree terms either.

2.1step 1.1F2F3F6algebra

Applying [F2] at $N = 2$ gives $Δ (E_{i})^{2} = E_{i}^{2} \otimes K_{i}^{- 2} + (1 + q^{2}) E_{i} \otimes K_{i}^{- 1} E_{i} + 1 \otimes E_{i}^{2}$ .

3.1F1F3F6F7F9step 2.1algebra

To collect the positive expansion, for each original word choose $L$ for a position sent to $E_{i} \otimes K_{i}^{- 1}$ and $R$ for one sent to $1 \otimes E_{i}$ . The left word preserves the $L$ letters; the right word preserves the $R$ letters, followed by the $K^{- 1}$ factors from $L$ . Moving a $K_{i}^{- 1}$ across a later $E_{j}$ contributes $q^{- a_{i j}}$ . In bidegree $(2, 1)$ the coefficients of $E_{1} E_{2} \otimes E_{1} K_{1}^{- 1} K_{2}^{- 1}$ , $E_{1}^{2} \otimes E_{2} K_{1}^{- 2}$ , and $E_{2} E_{1} \otimes E_{1} K_{1}^{- 1} K_{2}^{- 1}$ are respectively $1 + q^{- 2} - [2]_{q} q^{- 1}$ , $q^{2} - [2]_{q} q + 1$ , and $- [2]_{q} + q + q^{- 1}$ .

3.2F1F3F6F7F9step 1.1step 2.1algebra

In bidegree $(1, 2)$ the coefficients of $E_{1} \otimes E_{1} E_{2} K_{1}^{- 1}$ , $E_{2} \otimes E_{1}^{2} K_{2}^{- 1}$ , and $E_{1} \otimes E_{2} E_{1} K_{1}^{- 1}$ are respectively $q^{- 1} + q - [2]_{q}$ , $1 - [2]_{q} q + q^{2}$ , and $- [2]_{q} q^{- 1} + q^{- 2} + 1$ . These six coefficient groups exhaust the non-extreme splits of the three-letter Serre words.

4.1F1step 3.1step 3.2F6algebra

Substituting $[2]_{q} = q + q^{- 1}$ makes all six coefficients in steps 3.1–3.2 zero. The all-left and all-right choices contribute exactly ${S e r r e}_{12}^{+} \otimes K_{1}^{- 2} K_{2}^{- 1}$ and $1 \otimes {S e r r e}_{12}^{+}$ , proving the positive formula.

5.1F5algebra∎

At $q = 1$ , the positive Serre polynomial becomes $e_{1}^{2} e_{2} - 2 e_{1} e_{2} e_{1} + e_{2} e_{1}^{2} = [e_{1}, [e_{1}, e_{2}]]$ , which vanishes by the classical type- $A_{2}$ Serre relation [F5]. This is the specialization of the polynomial Serre expression only; it does not assert specialization of the whole algebra over $Q (q)$ .

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Sources

Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (standard reference, not scraped)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Benjamin Enriquez, PBW and Duality Theorems for Quantum Groups and Quantum Current Algebras, Journal of Lie Theory 13 (2003), 21–64 (standard reference, not scraped)
Alexander Kleshchev, Lectures on Infinite Dimensional Lie Algebras (standard reference, not scraped)

The quasiprimitive Serre element in type A2A2​