LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-08

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The coproduct preserves the positive and negative quantum Serre ideals

Statement

Let $(I, A, D, P, P^{\lor}, q)$ be a symmetrizable Cartan datum, use the notation $q_{i}$ , $K_{h}$ and $K_{i} = K_{d_{i} h_{i}}$ of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations, and put $m_{i j} = 1 - a_{i j}$ for $i \neq j$ . Let $T^{+}$ and $T^{-}$ be the free $Q (q)$ -algebras on the symbols $E_{i}, K_{h}$ and $F_{i}, K_{h}$ , respectively. Let $R^{+}$ and $R^{-}$ be the two-sided ideals generated by $K_{0} - 1$ , $K_{h} K_{h^{'}} - K_{h + h^{'}}$ , and respectively $K_{h} E_{i} - q^{⟨ α_{i}, h ⟩} E_{i} K_{h}$ or $K_{h} F_{i} - q^{- ⟨ α_{i}, h ⟩} F_{i} K_{h}$ . Define the toral-action algebras $B^{+} = T^{+} / R^{+}$ and $B^{-} = T^{-} / R^{-}$ , and let $J^{+}$ and $J^{-}$ be their two-sided ideals generated by the images of the corresponding Drinfeld–Jimbo Serre elements.

The generator assignments

$Δ_{+} (K_{h}) = K_{h} \otimes K_{h}, Δ_{+} (E_{i}) = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}, Δ_{-} (K_{h}) = K_{h} \otimes K_{h}, Δ_{-} (F_{i}) = F_{i} \otimes 1 + K_{i} \otimes F_{i}$

define algebra homomorphisms $B^{\pm} \to B^{\pm} \otimes_{Q (q)} B^{\pm}$ . For every $i \neq j$ , the Serre elements are quasiprimitive:

$Δ_{+} ({S e r r e}_{i j}^{+}) = {S e r r e}_{i j}^{+} \otimes K_{i}^{- m_{i j}} K_{j}^{- 1} + 1 \otimes {S e r r e}_{i j}^{+}, Δ_{-} ({S e r r e}_{i j}^{-}) = {S e r r e}_{i j}^{-} \otimes 1 + K_{i}^{m_{i j}} K_{j} \otimes {S e r r e}_{i j}^{-} .$

Consequently $Δ_{\pm} (J^{\pm}) \subseteq J^{\pm} \otimes B^{\pm} + B^{\pm} \otimes J^{\pm}$ . The canonical maps $B^{\pm} \to U_{q} (g)$ send the displayed coproducts of the Serre elements to zero in $U_{q} (g) \otimes U_{q} (g)$ ; thus these formulas provide the Serre-family part of the check that the Drinfeld–Jimbo coproduct descends.

Facts & Assumptions

Given: A symmetrizable Cartan datum over $Q (q)$ and the defining relations and Serre sums in The Drinfeld-Jimbo quantized enveloping algebra by generators and relations.

[F1]

For $a = a_{i j}$ and $m = 1 - a$ , symmetrizability gives $d_{i} a_{i j} = d_{j} a_{j i}$ , so $K_{j} E_{i} K_{j}^{- 1} = q_{i}^{a} E_{i}$ ; also $K_{i} E_{j} K_{i}^{- 1} = q_{i}^{a} E_{j}$ (Symmetrizable Cartan data for quantum groups).

[F2]

The Drinfeld–Jimbo toral-action relations are those displayed in The Drinfeld-Jimbo quantized enveloping algebra by generators and relations.

[F3]

The positive and negative Serre sums are those displayed in The Drinfeld-Jimbo quantized enveloping algebra by generators and relations.

[F4]

The symmetric Gaussian coefficients are factorial quotients $C_{N, r} = {(\binom{N}{r})}_{i} = [N]_{i}! / ([r]_{i}! [N - r]_{i}!)$ and satisfy $C_{N, r} = C_{N, N - r}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F5]

If $x, y$ lie in an algebra and $y x = q_{i}^{2} x y$ , 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).

[F6]

For every $N \geq 1$ , $\sum_{r = 0}^{N} (- 1)^{r} q_{i}^{r (N - 1)} C_{N, r} = 0$ (The quantum Pascal recurrences, the Gauss product formula and Gaussian integrality).

[F7]

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

[F8]

The tensor algebra on the free generators admits the unique algebra extension of every generator assignment (Tensor algebra of a vector space, Universal property of the tensor algebra).

[F9]

Quotient algebras and factor maps are as in The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ and A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring.

[F10]

A two-sided ideal generated by a subset is as in The ideal generated by a subset and principal ideals.

Proof

technique · Work first in the toral-action quotients, expand each coproduct by the q-binomial theorem, and cancel every mixed bidegree by the alternating Gaussian identity [F6]

1.1givenF7F8construct

Write $a = a_{i j}$ , $m = 1 - a$ , $C_{n, r} = {(\binom{n}{r})}_{i}$ , and let $T^{\pm}, R^{\pm}, B^{\pm}, J^{\pm}$ be as in the statement. The algebra maps on $T^{\pm}$ prescribed by the displayed coproduct assignments exist by the tensor-algebra universal property in [F8], with the target tensor products made into algebras by [F7].

1.2F6algebra

For every $N \geq 1$ , the coefficient identity [F6] cancels the alternating sum with exponent $(N - 1) r$ . For $N = 0$ the corresponding one-term sum equals $1$ . These two cases will distinguish the mixed terms from the two extreme bidegrees below.

2.1step 1.1F2F7F9algebra

The images of $K_{0} - 1$ and $K_{h} K_{h^{'}} - K_{h + h^{'}}$ vanish because $Δ (K_{0}) = 1 \otimes 1$ and $Δ (K_{h} K_{h^{'}}) = K_{h} K_{h^{'}} \otimes K_{h} K_{h^{'}} = Δ (K_{h + h^{'}})$ . For $b = ⟨ α_{i}, h ⟩$ , in the positive algebra $Δ (K_{h}) Δ (E_{i}) - q^{b} Δ (E_{i}) Δ (K_{h}) = 0$ : its two summands cancel by $K_{h} E_{i} = q^{b} E_{i} K_{h}$ in the first tensor factor and in the second tensor factor, while the toral elements commute. In the negative algebra, $Δ (K_{h}) Δ (F_{i}) - q^{- b} Δ (F_{i}) Δ (K_{h}) = 0$ by $K_{h} F_{i} = q^{- b} F_{i} K_{h}$ in the first and second tensor factors and commutation of toral elements. Thus both maps kill $R^{\pm}$ and descend to $B^{\pm}$ by [F9].

3.1step 2.1F1F2F5algebra

In $B^{+}$ , put $x_{i} = E_{i} \otimes K_{i}^{- 1}$ and $y_{i} = 1 \otimes E_{i}$ . The toral-action relation gives $y_{i} x_{i} = q_{i}^{2} x_{i} y_{i}$ , so [F5] yields $Δ_{+} (E_{i})^{N} = \sum_{r = 0}^{N} q_{i}^{r (N - r)} C_{N, r} E_{i}^{r} \otimes K_{i}^{- r} E_{i}^{N - r}$ . Also $K_{i}$ is grouplike and invertible since $K_{h} K_{- h} = 1$ in $B^{\pm}$ .

4.1step 3.1F1F2F3F4F5algebra

Expand $Δ_{+} ({S e r r e}_{i j}^{+}) = \sum_{s = 0}^{m} (- 1)^{s} C_{m, s} Δ_{+} (E_{i})^{m - s} Δ_{+} (E_{j}) Δ_{+} (E_{i})^{s}$ and first take the $E_{j} \otimes K_{j}^{- 1}$ term from the middle factor. Fix $r, t \geq 0$ with $M = m - r - t \geq 0$ , and put $k = s - t$ ; the allowed indices are $0 \leq k \leq M$ . Using [F5] on the two powers of $Δ (E_{i})$ , the corresponding tensor word is $E_{i}^{r} E_{j} E_{i}^{t} \otimes K_{i}^{- (r + t)} K_{j}^{- 1} E_{i}^{M}$ , and its coefficient is $(- 1)^{t + k} C_{m, r} C_{m - r, t} C_{M, k} q_{i}^{(r + a + 2 t) M + (M - 1) k}$ . Here $C_{m, s} C_{m - s, r} C_{s, t} = C_{m, r} C_{m - r, t} C_{M, k}$ follows by cancelling the factorial quotients, and the exponent comes from moving $E_{i}^{M - k}$ past $K_{j}^{- 1}$ and $K_{i}^{- t}$ .

4.2step 3.1F1F2F3F4F5algebra

Now take the $1 \otimes E_{j}$ term from the middle factor. Fix $u, v \geq 0$ with $k = m - u - v \geq 0$ ; after moving toral factors to the left, the tensor word is $E_{i}^{k} \otimes K_{i}^{- k} E_{i}^{u} E_{j} E_{i}^{v}$ . Its coefficient, summed over $r + t = k$ with $t = k - r$ , is $(- 1)^{v + k} {(\binom{m}{u, v, k})}_{i} q_{i}^{k (u + 1 - k)} \sum_{r = 0}^{k} (- 1)^{r} C_{k, r} q_{i}^{(k - 1) r}$ , where ${(\binom{m}{u, v, k})}_{i} = [m]_{i}! / ([u]_{i}! [v]_{i}! [k]_{i}!)$ . Indeed the unsummed Gaussian factor is $C_{m, v + t} C_{u + r, r} C_{v + t, t} = {(\binom{m}{u, v, k})}_{i} C_{k, r}$ , and the two q-binomial expansions and toral crossings give exponent $r u + t v + t (a + 2 u) = k (u + 1 - k) + (k - 1) r$ , using $a + u + v = 1 - k$ . The prefactor is independent of $r$ and is $1$ when $k = 0$ .

5.1step 1.2step 4.1F4F6algebra

If $M > 0$ , summing the coefficients in step 4.1 over $k$ gives a scalar multiple of $\sum_{k = 0}^{M} (- 1)^{k} C_{M, k} q_{i}^{(M - 1) k} = 0$ by step 1.2. If $M = 0$ , only $k = 0$ remains; then $r + t = m$ , and summing over these pairs gives ${S e r r e}_{i j}^{+} \otimes K_{i}^{- m} K_{j}^{- 1}$ , using $C_{m, r} = C_{m, m - r}$ . Thus the $E_{j} \otimes K_{j}^{- 1}$ part contributes exactly the first term in the claimed formula.

6.1step 1.2step 4.2step 5.1F4F6algebra

For $k > 0$ , the sum in step 4.2 is zero by step 1.2. For $k = 0$ , the only term has $u + v = m$ , and its coefficient is $(- 1)^{v} C_{m, v}$ ; these terms sum to $1 \otimes {S e r r e}_{i j}^{+}$ . Combining this with step 5.1 proves the positive quasiprimitive identity.

7.1step 2.1step 6.1F2F7algebra

Define $ω : B^{+} \to B^{-}$ by $K_{h} \mapsto K_{- h}$ and $E_{i} \mapsto - F_{i}$ . The toral and positive action relations map to the toral and negative action relations (the action relation at $- h$ ), so this is a well-defined algebra map. Directly on generators, $Δ_{-} ω = f l i p \circ (ω \otimes ω) \circ Δ_{+}$ : on $K_{h}$ both sides equal $K_{- h} \otimes K_{- h}$ , and on $E_{i}$ both sides equal $- F_{i} \otimes 1 - K_{i} \otimes F_{i}$ . Moreover $ω ({S e r r e}_{i j}^{+}) = (- 1)^{m + 1} {S e r r e}_{i j}^{-}$ by applying the substitution to its $m + 1$ factors. Applying this identity to the positive quasiprimitive formula gives $Δ_{-} ({S e r r e}_{i j}^{-}) = {S e r r e}_{i j}^{-} \otimes 1 + K_{i}^{m} K_{j} \otimes {S e r r e}_{i j}^{-}$ .

8.1step 2.1step 6.1step 7.1F2F3F7F9F10algebra∎

The subspace $J^{+} \otimes B^{+} + B^{+} \otimes J^{+}$ is a two-sided ideal of $B^{+} \otimes B^{+}$ by the tensor multiplication in [F7], and likewise for the negative sign. The two quasiprimitive identities therefore imply $Δ_{\pm} (J^{\pm}) \subseteq J^{\pm} \otimes B^{\pm} + B^{\pm} \otimes J^{\pm}$ , since each $J^{\pm}$ is generated as a two-sided ideal by the Serre elements and $Δ_{\pm}$ is an algebra map. The canonical maps $B^{\pm} \to U_{q} (g)$ send every Serre generator to zero; applying their tensor squares to the formulas gives zero, so the Serre relations are preserved by the prescribed coproduct in the full Drinfeld–Jimbo quotient. Once the other defining relation families are checked, the quotient universal property in [F9] gives the descended coproduct.

Source note

The Berkeley text defines its toral-action-only positive Borel and gives the positive quasiprimitivity statement in Lemma 13.1.3.9, then says the cited Jantzen proof is a tedious q-binomial computation; it does not print the computation or the explicit grouplike factors. Jeong–Kang–Kashiwara display the coproduct convention used here in (1.6) but assert the Hopf structure without proving relation preservation. The coefficient cancellations in steps 4.1–6.1 are supplied locally; the original scaffold's claim that Serre-only ideals in the completely free algebra are coideals was replaced because its toral commutations are not valid before quotienting by the toral-action relations.

Depends on

Used by

Unsymmetrized parameters break the coproduct of the Serre ideal Counterexample
The double-edge Serre relation for the cyclic affine type A₁⁽¹⁾ Example
The quasiprimitive Serre element in type A₂ Example
The Drinfeld–Jimbo formulas define a Hopf algebra Theorem

Dependency tree · two levels

36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)