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Bar invariance and the congruence modulo $q$ do not imply balancedness

Statement refuted

The proposed implication is: bar-fixedness of an integral vector with a prescribed nonzero crystal residue forces balancedness and unique global lifts. The following two witnesses refute it.

(1) In the two-dimensional module $V = Q (q) v \oplus Q (q) w$ for ${s l}_{2}$ (with $v$ of weight $1$ , $w = F v$ of weight $- 1$ ), take $M_{A} = V_{A} = A v \oplus A w$ and the full free $A_{0}$ -lattice $L^{'} = A_{0} (v + w) \oplus q A_{0} v$ , with bar lattice $ψ (L^{'}) = A_{\infty} (v + w) \oplus q^{- 1} A_{\infty} v$ . Then $E_{Z} = V_{A} \cap L^{'} \cap ψ (L^{'}) = Z (v + w)$ , while $V_{A} \cap L^{'} = Z [q] (v + w) \oplus q Z [q] v$ , so the canonical integral residue map is not surjective and misses the nonzero residue class $q v + q L^{'}$ . The bar-fixed vector $v + w$ has its prescribed residue, but $L^{'}$ is neither weight decomposed nor stable under the Kashiwara operators.

(2) In the one-dimensional trivial module $Q (q) v$ with $ε$ -action, $M_{A} = A v$ , bar fixing $v$ , all Kashiwara operators zero and the full free scaled $A_{0}$ -lattice $L^{''} = q^{- 1} A_{0} v$ with bar lattice $q A_{\infty} v$ , the canonical integral residue map has kernel $Z v \oplus Z q v$ ; the two distinct bar-fixed integral vectors $(q + q^{- 1}) v$ and $(q + q^{- 1}) v + v$ have the same nonzero residue class $q^{- 1} v + q L^{''}$ . This is a genuine nonuniqueness witness even though $L^{''}$ is a valid scaled crystal lattice satisfying a congruence condition, and neither example is the canonical top-normalized Kashiwara triple of a highest weight module.

Facts & Assumptions

Given: The ${s l}_{2}$ -module data of Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules in the library conventions $K v = q v$ , $K w = q^{- 1} w$ , $F v = w$ , $F w = 0$ , $E v = 0$ , $E w = v$ , with $V_{A} = A v \oplus A w$ and $ψ (v) = v$ , $ψ (w) = w$ , $ψ (q) = q^{- 1}$ ; the trivial module $Q (q) v$ with $K v = v$ , $E v = F v = 0$ ; the balanced-triple definitions of Balanced triples over the two charts of $P^{1}$ and the globalization map and The lower global basis of $U_{q}^{-}$ and of a highest weight module; the crystal-lattice axioms of Crystal lattices, crystal bases and their crystals; and the uniqueness theorem Bar invariance, congruence and uniqueness of the lower global basis.

[L1]

Balancedness (Balanced triples over the two charts of $P^{1}$ and the globalization map): the triple $(M_{A}, L_{0}, L_{\infty})$ is balanced exactly when the canonical map $E_{Z} = M_{A} \cap L_{0} \cap L_{\infty} \to (M_{A} \cap L_{0}) / (M_{A} \cap q L_{0})$ is an isomorphism; this is the canonical integral residue map, whose inverse, when it exists, is the globalization map; uniqueness of global lifts with a given residue is the injectivity of this map.

[L2]

Crystal lattices (Crystal lattices, crystal bases and their crystals): a crystal lattice is a full free $A_{0}$ -lattice $L$ with $L = ⨁_{μ} (L \cap M_{μ})$ , each weight piece free, and stable under all Kashiwara operators. A specified weight-compatible residue basis is additional crystal-basis data. In the trivial module there is only the zero weight and all Kashiwara operators vanish, so every full free scaled $A_{0}$ -lattice is a crystal lattice there.

[L3]

Cartan data (Divided-power commutation and the simple $U_{q_{i}} ({s l}_{2})$ string modules): in the two-dimensional module $q$ -powers are computed from $K v = q v$ , $K w = q^{- 1} w$ ; in the trivial module the action is by $ε$ so no $q$ -power appears on $v$ .

[L4]

Local rings (The local ring $A_{0}$ of rational functions regular at $q = 0$ ): $A_{0}$ consists of the elements of $Q (q)$ regular at $q = 0$ , is generated inside $Q (q)$ by $Z [q]$ and inverses of polynomials nonzero at $0$ , and $A_{\infty} = ψ (A_{0})$ ; consequently $A \cap A_{0} = Z [q]$ , $A \cap A_{\infty} = Z [q^{- 1}]$ , $A_{0} \cap A_{\infty} \cap A = Z$ and $A \cap q^{- 1} A_{0} = q^{- 1} Z [q]$ .

[L5]

The canonical triple (Bar invariance, congruence and uniqueness of the lower global basis): the balanced map of the genuine Kashiwara triple is an isomorphism, so any failure or nonuniqueness below is a property of the displayed non-canonical lattices, not a defect of the globalization theorem.

Proof

technique · constructive

1.1L1L4construct

Witness (1): the balanced intersection and the missing residue class. Write $a : = v + w$ and $t : = v$ , so $L^{'} = A_{0} a \oplus q A_{0} t$ and $V_{A} = A v \oplus A w = A a \oplus A t$ , while $V = Q (q) a \oplus Q (q) t$ ; these identities follow from $w = a - t$ and $v = t$ . An element $α a + β t$ with $α \in A_{0}$ , $β \in q A_{0}$ lies in $V_{A} = A v \oplus A w$ exactly when $α \in A_{0} \cap A = Z [q]$ and $β \in q A_{0} \cap A = q Z [q]$ , so $V_{A} \cap L^{'} = Z [q] a \oplus q Z [q] t$ . Similarly $ψ (L^{'}) = A_{\infty} a \oplus q^{- 1} A_{\infty} t$ , and the triple intersection is $E_{Z} = V_{A} \cap L^{'} \cap ψ (L^{'}) = (Z [q] \cap Z [q^{- 1}]) a \oplus (q Z [q] \cap q^{- 1} Z [q^{- 1}]) t = Z a,$ by [L4]. The integral residue of the zero chart is $(V_{A} \cap L^{'}) / (V_{A} \cap q L^{'}) = Z [q] a / q Z [q] a \oplus q Z [q] t / q^{2} Z [q] t$ , which contains the nonzero class $q t + q L^{'} = q v + q L^{'}$ . The balanced map sends $Z a$ onto the classes of integer multiples of $a$ and therefore misses that class, so the triple is not balanced by [L1].

1.2L4

Witness (2): the rank-three balanced intersection. For the trivial module $L^{''} = q^{- 1} A_{0} v$ , so $V_{A} \cap L^{''} = q^{- 1} Z [q] v$ by [L4], and $ψ (L^{''}) = q A_{\infty} v$ , so $V_{A} \cap ψ (L^{''}) = q A_{\infty} v \cap A v$ . An element of $V_{A} \cap L^{''}$ is $q^{- 1} f v$ with $f \in Z [q]$ , and it lies in $q A_{\infty} v$ exactly when $q^{- 2} f \in A_{\infty}$ , i.e. when $f = 0$ or $\deg f \leq 2$ ; hence $E_{Z} = V_{A} \cap L^{''} \cap ψ (L^{''}) = Z q^{- 1} v \oplus Z v \oplus Z q v,$ the three Laurent monomials being exactly the integer-coefficient combinations with exponents between $- 1$ and $1$ .

2.1L2L3step 1.1

Witness (1): the lattice defects. The vector $a = v + w$ is fixed by $ψ$ , since $ψ (v) = v$ , $ψ (w) = w$ and $ψ (q) = q^{- 1}$ only acts on coefficients, and it lies in $L^{'}$ with residue class $a + q L^{'}$ . But $a$ is a sum of two weight vectors of different weights, so $L^{'}$ is not weight decomposed, and $\tilde{e} a = v \notin L^{'}$ because $v = β t$ would require $β = 1 \notin q A_{0}$ ; thus $L^{'}$ fails the crystal-lattice axioms of [L2] and the bar-fixed vector alone provides no parametrization or uniqueness.

2.2L1step 1.2

Witness (2): the kernel and the two lifts. Since $V_{A} \cap q L^{''} = q (q^{- 1} Z [q]) v = Z [q] v$ , the canonical integral residue map $E_{Z} \to (V_{A} \cap L^{''}) / (V_{A} \cap q L^{''}) = q^{- 1} Z [q] v / Z [q] v$ is $(a_{- 1} q^{- 1} + a_{0} + a_{1} q) v \mapsto a_{- 1} [q^{- 1} v]$ , which is onto with kernel $Z v \oplus Z q v \neq 0$ . The elements $g_{1} = (q + q^{- 1}) v$ and $g_{2} = (q + q^{- 1}) v + v = (q + q^{- 1} + 1) v$ both lie in $E_{Z}$ , both are fixed by $ψ$ because $ψ (q + q^{- 1}) = q + q^{- 1}$ and $ψ (v) = v$ , and both have residue class $q^{- 1} v + q L^{''}$ ; they are distinct since $g_{2} - g_{1} = v \neq 0$ . Hence the same nonzero crystal residue has two distinct bar-fixed integral lifts, and bar invariance plus congruence does not give uniqueness without the balanced isomorphism.

3.1L2L5step 2.2

Witness (2) is a genuine crystal lattice. In the trivial module all Kashiwara operators vanish and $L^{''} = q^{- 1} A_{0} v$ is a full free $A_{0}$ -lattice concentrated in the zero weight, with residue generator $q^{- 1} v + q L^{''}$ ; by [L2] it is a crystal lattice, so the failure of uniqueness in step 2.2 is not attributable to an invalid lattice. The genuine Kashiwara triples of [L5] require and possess the actual integral balanced map, which the present lattices do not have; accordingly no claim of the globalization theorem is contradicted.

4.1step 2.1step 3.1discharge-construct∎

Conclusion. Steps 1.1–1.2 exhibit the nonbalanced lattice $L^{'}$ whose bar-fixed vector $v + w$ cannot be globalized onto the residue class $q v + q L^{'}$ ; steps 2.1–3.1 exhibit a valid crystal lattice with two distinct bar-fixed integral lifts of one nonzero crystal residue. Both conclusions are established by explicit coefficient computations in the displayed modules.

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Sources

Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (standard reference, not scraped)
Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Mathematical Journal 63 (1991), 465–516 (standard reference, not scraped)
Math 263C Quantum Groups (Stanford, 2022), lecture slides 'Lusztig's quantum group. Tilting modules. The Fusion Category in Action' (standard reference, not scraped)

Bar invariance and the congruence modulo qq do not imply balancedness