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Unsymmetrized parameters break the coproduct of the Serre ideal

Statement refuted

The following assertion is false: for every generalized Cartan matrix and every assignment of node parameters $q_{i}$ , the standard Drinfeld–Jimbo coproduct formulas descend to the full node-toral presentation, including its mixed and Serre relations, with $K_{i} E_{j} K_{i}^{- 1} = q_{i}^{a_{i j}} E_{j}$ .

Take $A = (\begin{array}{cc} 2 & - 1 \\ - 2 & 2 \end{array})$ , whose standard symmetrizer is $diag (2, 1)$ , but assign $q_{1} = q_{2} = q$ , with $q$ indeterminate. Precisely, let $B$ be the $Q (q)$ -algebra on $E_{1}, E_{2}, F_{1}, F_{2}, K_{1}^{\pm 1}, K_{2}^{\pm 1}$ , with commuting invertible toral generators, relations $K_{i} E_{j} K_{i}^{- 1} = q^{a_{i j}} E_{j}, K_{i} F_{j} K_{i}^{- 1} = q^{- a_{i j}} F_{j}, [E_{i}, F_{j}] = δ_{i j} \frac{K_{i} - K_{i}^{- 1}}{q - q^{- 1}},$ and both symmetric quantum Serre families with the common parameter $q$ . This explicitly changed parameter assignment is the test presentation, not the symmetrized algebra of The Drinfeld-Jimbo quantized enveloping algebra by generators and relations. Put $S = E_{1}^{2} E_{2} - [2]_{q} E_{1} E_{2} E_{1} + E_{2} E_{1}^{2} = 0$ in $B$ .

The assignments $Δ (E_{i}) = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}$ , $Δ (F_{i}) = F_{i} \otimes 1 + K_{i} \otimes F_{i}$ , $Δ (K_{i}) = K_{i} \otimes K_{i}$ fail to define an algebra map $B \to B \otimes B$ . In an explicit representation of $B$ below, the proposed image of $S$ acts on $v_{0} \otimes v_{1}$ by $q^{- 2} (1 - q) (1 + q^{2}) v_{2} \otimes v_{2} \neq 0.$ Thus the previously detected positive-Borel defect survives in the full quotient. The same proposed formulas already fail the off-diagonal mixed relation $[E_{1}, F_{2}] = 0$ .

Facts & Assumptions

Given: The explicitly stated unsymmetrized full presentation $B$ , with $q$ indeterminate.

[F1]

The symmetric Gaussian coefficients are $[2]_{q} = q + q^{- 1}$ and $[3]_{q} = q^{2} + 1 + q^{- 2}$ (Quantum integers, factorials, Gaussian binomials and divided powers at $q_{i}$ ).

[F2]

The displayed inverse- $K$ positive coproduct and its negative counterpart are the normalized Drinfeld–Jimbo convention; with the correct node parameters their Serre mixed terms cancel (The coproduct preserves the positive and negative quantum Serre ideals, The Drinfeld-Jimbo quantized enveloping algebra by generators and relations).

[F3]

For this matrix the correct symmetry is $q_{1} = q^{2}$ , $q_{2} = q$ ; the assignment tested above violates $q_{1}^{a_{12}} = q_{2}^{a_{21}}$ (Symmetrizable Cartan data for quantum groups).

Counterexample

1.1givenconstruct

On a four-dimensional $Q (q)$ -space with basis $v_{0}, v_{1}, v_{2}, v_{3}$ , let $K_{1}$ have eigenvalues $(1, q^{- 1}, q, 1)$ and $K_{2}$ have eigenvalues $(q^{- 1}, q, q^{- 1}, q)$ . Define $E_{1} v_{1} = v_{2}$ , $F_{1} v_{2} = v_{1}$ , $E_{2} v_{0} = v_{1}$ , $E_{2} v_{2} = v_{3}$ , $F_{2} v_{1} = v_{0}$ , $F_{2} v_{3} = v_{2}$ , and make all other $E / F$ actions zero. Both toral operators are commuting and invertible. An $E_{1}$ arrow changes the pair of toral exponents by $(2, - 2)$ and each $E_{2}$ arrow changes it by $(- 1, 2)$ ; the $F$ arrows reverse these changes. Hence all toral-action relations hold.

2.1step 1.1algebra

The diagonal entries of $[E_{1}, F_{1}]$ are $(0, - 1, 1, 0)$ , and those of $[E_{2}, F_{2}]$ are $(- 1, 1, - 1, 1)$ . Each toral exponent is $0$ or $\pm 1$ , so these are precisely the entries of $(K_{i} - K_{i}^{- 1}) / (q - q^{- 1})$ . For each off-diagonal pair, both operator products $E_{1} F_{2}, F_{2} E_{1}$ and $E_{2} F_{1}, F_{1} E_{2}$ are zero: none of their two-arrow sequences is composable on a basis vector. Thus all four mixed relations hold.

2.2F2step 1.1algebra

On its tensor square put $D_{i} = E_{i} \otimes K_{i}^{- 1} + 1 \otimes E_{i}$ . Direct computation gives $D_{1} (v_{0} \otimes v_{1}) = v_{0} \otimes v_{2}$ and $D_{2} (v_{0} \otimes v_{1}) = q^{- 1} v_{1} \otimes v_{1}$ . Also $D_{1} (v_{1} \otimes v_{1}) = q v_{2} \otimes v_{1} + v_{1} \otimes v_{2}$ , so $D_{1}^{2} D_{2} (v_{0} \otimes v_{1}) = (1 + q^{- 2}) v_{2} \otimes v_{2}$ . Applying $D_{2}$ to $v_{0} \otimes v_{2}$ gives $q v_{1} \otimes v_{2} + v_{0} \otimes v_{3}$ , whose $D_{1}$ image is $v_{2} \otimes v_{2}$ ; thus $D_{1} D_{2} D_{1} (v_{0} \otimes v_{1}) = v_{2} \otimes v_{2}$ . Finally $D_{1}^{2} (v_{0} \otimes v_{1}) = 0$ , so the last Serre term contributes zero.

3.1F1step 1.1step 2.1algebra

All four squares $E_{1}^{2}, E_{2}^{2}, F_{1}^{2}, F_{2}^{2}$ are zero. The compositions $E_{1} E_{2} E_{1}$ and $F_{1} F_{2} F_{1}$ are zero as well: after the first color-1 arrow, the color-2 arrow either vanishes or leads to a vector on which the final color-1 arrow vanishes. These identities give both length-three Serre relations $S_{12}^{\pm} = 0$ . Every term of $S_{21}^{\pm}$ contains a square or cube of the color-2 operator, since its four words are $X_{2}^{3} X_{1}$ , $X_{2}^{2} X_{1} X_{2}$ , $X_{2} X_{1} X_{2}^{2}$ , and $X_{1} X_{2}^{3}$ . Hence both length-four Serre relations vanish too. Steps 1.1 and 2.1 and these checks verify every defining relation of $B$ , so the free-generator assignment factors through a representation of the full quotient.

4.1F1F2F3step 1.1step 3.1step 2.2algebra∎

By [F1] and step 2.2, the proposed coproduct image of $S$ acts as $(1 + q^{- 2} - q - q^{- 1}) v_{2} \otimes v_{2} = q^{- 2} (1 - q) (1 + q^{2}) v_{2} \otimes v_{2}$ , which is nonzero over $Q (q)$ . Since $S = 0$ in the full represented algebra $B$ , this contradicts the relation preservation required of a coproduct algebra map $B \to B \otimes B$ . Independently, expanding the off-diagonal mixed commutator gives $[Δ (E_{1}), Δ (F_{2})] = (q - 1) K_{2} E_{1} \otimes F_{2} K_{1}^{- 1}$ : the cross scalar is $q_{1}^{a_{12}} q_{2}^{- a_{21}} - 1 = q - 1$ . Its action on $v_{1} \otimes v_{1}$ is $(q - 1) v_{2} \otimes v_{0} \neq 0$ . Thus both the full Serre and mixed-relation failures are detected without an assumption of triangular decomposition.

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Sources

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)
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)