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The crystal basis of a tensor product and the two-case operator rule

Statement

Let $M_{1}, M_{2}$ be integrable chosen $P$ -graded type-one quantum weight modules with crystal bases $(L_{1}, B_{1})$ , $(L_{2}, B_{2})$ (Crystal lattices, crystal bases and their crystals), and let $M_{1} \otimes M_{2}$ be the integrable module of Tensor products of integrable weight modules and of their crystal lattices.

Then $(L_{1} \otimes_{A_{0}} L_{2}, B_{1} \otimes B_{2})$ is a crystal basis of $M_{1} \otimes M_{2}$ , and its crystal is the tensor product crystal $B_{1} \otimes B_{2}$ of The tensor product crystal $B_{1} \otimes B_{2}$ and its two-case convention: for all $b_{1} \in B_{1}$ , $b_{2} \in B_{2}$ and $i \in I$ ,

${\tilde{f}}_{i} (b_{1} \otimes b_{2}) = {\begin{cases} {\tilde{f}}_{i} b_{1} \otimes b_{2}, & φ_{i} (b_{1}) > ε_{i} (b_{2}), \\ b_{1} \otimes {\tilde{f}}_{i} b_{2}, & φ_{i} (b_{1}) \leq ε_{i} (b_{2}), \end{cases} {\tilde{e}}_{i} (b_{1} \otimes b_{2}) = {\begin{cases} {\tilde{e}}_{i} b_{1} \otimes b_{2}, & φ_{i} (b_{1}) \geq ε_{i} (b_{2}), \\ b_{1} \otimes {\tilde{e}}_{i} b_{2}, & φ_{i} (b_{1}) < ε_{i} (b_{2}), \end{cases}$

with value $0$ as soon as the selected factor maps to $0$ . In particular ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ map $B_{1} \otimes B_{2}$ into $(B_{1} \otimes B_{2}) \cup {0}$ , and for all $b, b^{'} \in B_{1} \otimes B_{2}$ one has ${\tilde{f}}_{i} b = b^{'}$ if and only if $b = {\tilde{e}}_{i} b^{'}$ , so the partial-inverse axiom holds. The equality case $φ_{i} (b_{1}) = ε_{i} (b_{2})$ is resolved as in the definition: ${\tilde{f}}_{i}$ acts on the second factor and ${\tilde{e}}_{i}$ on the first.

No finite dimensionality, bounded support or uniform nilpotence is assumed, and no choice principle is used.

Facts & Assumptions

Given: Integrable chosen $P$ -graded type-one quantum weight modules $M_{1}, M_{2}$ with crystal bases $(L_{1}, B_{1})$ , $(L_{2}, B_{2})$ (Crystal lattices, crystal bases and their crystals), the tensor product module $M_{1} \otimes M_{2}$ of Tensor products of integrable weight modules and of their crystal lattices with its Kashiwara operators ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ , and its $A_{0}$ -tensor lattice $L_{1} \otimes L_{2} = L_{1} \otimes_{A_{0}} L_{2}$ with residue basis $B_{1} \otimes B_{2}$ .

[F1]

Crystal lattice and crystal basis axioms (Crystal lattices, crystal bases and their crystals): $L_{j}$ is free over $A_{0}$ , weight-graded, generates $M_{j}$ over $Q (q)$ and is stable under ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ ; $B_{j} = ⨆_{μ} (B_{j})_{μ}$ is a $Q$ -basis of $L_{j} / q L_{j}$ with ${\tilde{e}}_{i} B_{j} \subseteq B_{j} \cup {0}$ , ${\tilde{f}}_{i} B_{j} \subseteq B_{j} \cup {0}$ and ${\tilde{f}}_{i} b = b^{'}$ if and only if $b = {\tilde{e}}_{i} b^{'}$ .

[F2]

String components of a lattice vector (The $i$ -string components of a lattice vector and the single surviving component modulo $q$ ): for $u \in L_{λ}$ with admissible decomposition $u = \sum_{n} F_{i}^{(n)} u_{n}$ , every $u_{n}$ lies in $L$ ; the lattice decomposes as $L = ⨁_{n \geq 0} {\tilde{f}}_{i}^{n} (L \cap \ker E_{i})$ ; if $u + q L \in B_{λ}$ then exactly one component $u_{n_{0}}$ is outside $q L$ , with $u_{n_{0}} + q L = {\tilde{e}}_{i}^{n_{0}} (u + q L) \in B$ and $u \equiv F_{i}^{(n_{0})} u_{n_{0}} (m o d q L)$ ; and the residues of ${\tilde{e}}_{i} u, {\tilde{f}}_{i} u$ are those of $F_{i}^{(n_{0} - 1)} u_{n_{0}}$ , $F_{i}^{(n_{0} + 1)} u_{n_{0}}$ .

[F3]

The rank-one tensor strings (Rank-one tensor strings and their crystal limit rule): for rank-one simples of nonnegative integral highest weights $a_{1}, a_{2}$ with divided bases $v_{j} = F_{i}^{(j)} v_{0}$ , $w_{k} = F_{i}^{(k)} w_{0}$ and their $A_{0}$ -tensor lattice, that lattice is stable under ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ and modulo $q$ one has ${\tilde{f}}_{i} (v_{j} \otimes w_{k}) = v_{j + 1} \otimes w_{k}$ if $a_{1} - j > k$ and $v_{j} \otimes w_{k + 1}$ otherwise, while ${\tilde{e}}_{i}$ acts on the first factor if $a_{1} - j \geq k$ and on the second otherwise, with zero outside the divided-basis ranges; this is a finite, choice-free calculation independent of the grand loop.

[F4]

Kashiwara operators (The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module): on the unique admissible string decomposition, ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ shift the divided string positions down and up by one; they are linear, preserve the given factor lattices $L_{j}$ and commute with module homomorphisms.

[F5]

Uniqueness of admissible decompositions (Uniqueness of the $i$ -string decomposition of an integrable module): every vector of an integrable weight module has exactly one finite admissible $i$ -string decomposition; on $\ker E_{i} \cap M_{μ}$ the maps $F_{i}^{(n)}$ are injective for $n \leq ⟨ h_{i}, μ ⟩$ and zero for $n > ⟨ h_{i}, μ ⟩$ , and $\ker E_{i} \cap M_{μ} = 0$ for $⟨ h_{i}, μ ⟩ < 0$ .

[F6]

Crystal graph invariants (The crystal graph, $ε_{i}$ , $φ_{i}$ and the weight function): $ε_{i} (b) = \max {n : {\tilde{e}}_{i}^{n} b \neq 0}$ and $φ_{i} (b) = \max {n : {\tilde{f}}_{i}^{n} b \neq 0}$ are finite and nonnegative, $φ_{i} (b) = ε_{i} (b) + ⟨ h_{i}, w t (b) ⟩$ , and on an existing ${\tilde{f}}_{i}$ -edge $φ_{i}$ decreases by one and $ε_{i}$ increases by one.

[F7]

Tensor integrability lemma (Tensor products of integrable weight modules and of their crystal lattices): $M_{1} \otimes M_{2}$ is an integrable chosen $P$ -graded type-one weight module for the direct sum grading $(M_{1} \otimes M_{2})_{μ} = ⨁_{μ_{1} + μ_{2} = μ} (M_{1})_{μ_{1}} \otimes (M_{2})_{μ_{2}}$ ; $L_{1} \otimes L_{2}$ is free and generates $M_{1} \otimes M_{2}$ over $Q (q)$ , is weight-graded accordingly, and $B_{1} \otimes B_{2}$ is a weight-graded $Q$ -basis of $(L_{1} \otimes L_{2}) / q (L_{1} \otimes L_{2})$ (stability under the crystal operators is not assumed; it is proved here).

[F8]

The tensor product crystal and the target rules (The tensor product crystal $B_{1} \otimes B_{2}$ and its two-case convention).

[F9]

Weight modules and integrability (Quantum weight modules, weights in $P$ , and integrability): the tensor module is a chosen $P$ -graded type-one weight module in the sense stated there, and integrability is local nilpotence of all $E_{i}, F_{i}$ .

Proof

technique · direct

1.1F2givenalgebra

Reduction to single string components. Fix $b_{1} \otimes b_{2} \in B_{1} \otimes B_{2}$ and lifts $u_{j} \in L_{j}$ with $u_{j} + q L_{j} = b_{j}$ , and put $n_{j} : = ε_{i} (b_{j})$ . By [F2] applied to $u_{j}$ there are top components $w_{j} \in L_{j} \cap \ker E_{i}$ with $w_{j} + q L_{j} = {\tilde{e}}_{i}^{n_{j}} b_{j} \in B_{j}$ and $u_{j} \equiv F_{i}^{(n_{j})} w_{j} (m o d q L_{j})$ ; hence $u_{1} \otimes u_{2} \equiv h : = F_{i}^{(n_{1})} w_{1} \otimes F_{i}^{(n_{2})} w_{2} (m o d q (L_{1} \otimes L_{2}))$ (expand the difference as in the bilinearity of the tensor product), and $h + q (L_{1} \otimes L_{2}) = b_{1} \otimes b_{2}$ because the residue of $F_{i}^{(n_{j})} w_{j}$ is $b_{j}$ by [F2].

1.2F2F4F5F6

The rank-one block. The top $w_{j}$ lies in $\ker E_{i} \cap (L_{j})_{μ_{j}}$ with $a_{j} : = ⟨ h_{i}, μ_{j} ⟩ \geq 0$ by [F5]; the divided powers $F_{i}^{(k)} w_{j}$ , $0 \leq k \leq a_{j}$ , form the string basis of the finite-dimensional $U_{i}$ -submodule $N_{j} : = U_{i} w_{j}$ , all of them lie in $L_{j}$ by [F2], and on this string ${\tilde{f}}_{i}$ acts as the divided-power shift and ${\tilde{e}}_{i}$ as its partial inverse by [F4]. The tensor product of the inclusions $N_{1} \otimes N_{2} \to M_{1} \otimes M_{2}$ is injective and $U_{i}$ -linear, and for a vector of $N_{1} \otimes N_{2}$ its admissible decomposition computed inside $N_{1} \otimes N_{2}$ is also admissible in $M_{1} \otimes M_{2}$ , so by uniqueness [F5] it is the admissible decomposition there; hence the operators ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ of $M_{1} \otimes M_{2}$ restrict on the image to the string operators computed from $N_{1}, N_{2}$ .

2.1step 1.2F3

The rank-one computation. Apply [F3] to the pair $(w_{1}, w_{2})$ with divided bases $v_{j} = F_{i}^{(j)} w_{1}$ , $w_{k} = F_{i}^{(k)} w_{2}$ and their $A_{0}$ -tensor lattice $Λ : = \sum_{j, k} A_{0} v_{j} \otimes w_{k} \subseteq L_{1} \otimes L_{2}$ , which is preserved by both string operators by [F3] and by step 1.2; modulo $q Λ$ one has ${\tilde{f}}_{i} (v_{n_{1}} \otimes w_{n_{2}}) = v_{n_{1} + 1} \otimes w_{n_{2}}$ if $a_{1} - n_{1} > n_{2}$ and $v_{n_{1}} \otimes w_{n_{2} + 1}$ otherwise, while ${\tilde{e}}_{i}$ acts on the first factor if $a_{1} - n_{1} \geq n_{2}$ and on the second otherwise, with $0$ outside the divided-basis ranges.

3.1step 1.1step 2.1F2F6algebra

Transport to the residues. Since $h \in Λ$ , step 2.1 computes ${\tilde{f}}_{i} h$ and ${\tilde{e}}_{i} h$ modulo $q Λ \subseteq q (L_{1} \otimes L_{2})$ , and the residues of $v_{n_{1} + 1} = F_{i}^{(n_{1} + 1)} w_{1}$ and $v_{n_{1}} = F_{i}^{(n_{1})} w_{1}$ are ${\tilde{f}}_{i} b_{1}$ and $b_{1}$ ; likewise on the second factor by [F2]. Because $a_{1} - n_{1} = φ_{i} (b_{1})$ by [F6] and $n_{2} = ε_{i} (b_{2})$ , the two cases of step 2.1 are exactly $φ_{i} (b_{1}) > ε_{i} (b_{2})$ and $φ_{i} (b_{1}) \leq ε_{i} (b_{2})$ for ${\tilde{f}}_{i}$ , and $φ_{i} (b_{1}) \geq ε_{i} (b_{2})$ or $φ_{i} (b_{1}) < ε_{i} (b_{2})$ for ${\tilde{e}}_{i}$ , giving the displayed two-case rules for the residue $b_{1} \otimes b_{2}$ of $h$ ; a selected factor with ${\tilde{f}}_{i}$ or ${\tilde{e}}_{i}$ equal to $0$ produces the value $0$ , consistently on both sides of the congruence.

4.1step 1.2step 2.1step 3.1F2F3F7

Stability of the tensor lattice. For arbitrary $u_{j} \in L_{j}$ use the full string decomposition of [F2]: $u_{j} = \sum_{n} F_{i}^{(n)} w_{j, n}$ with finitely many nonzero $w_{j, n} \in L_{j} \cap \ker E_{i}$ , and each pair $(w_{1, n}, w_{2, n^{'}})$ spans a finite rank-one block as in step 1.2 whose $A_{0}$ -tensor lattice is preserved by both string operators by [F3]; since the inclusions commute with the operators (step 1.2), every term of the finite expansion of $u_{1} \otimes u_{2}$ has its images under ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ inside $L_{1} \otimes L_{2}$ , and linearity gives ${\tilde{e}}_{i} (L_{1} \otimes L_{2}) \subseteq L_{1} \otimes L_{2}$ and ${\tilde{f}}_{i} (L_{1} \otimes L_{2}) \subseteq L_{1} \otimes L_{2}$ . In particular ${\tilde{f}}_{i} (q (L_{1} \otimes L_{2})) \subseteq q (L_{1} \otimes L_{2})$ and likewise for ${\tilde{e}}_{i}$ , so the congruence $u_{1} \otimes u_{2} \equiv h$ of step 1.1 may be transported by the operators and the two-case rules of step 3.1 hold for the residues of arbitrary lifts, not only for $h$ .

5.1step 3.1step 4.1F2F6

Operator rules and residues. Combining steps 3.1 and 4.1: for all $b_{1} \in B_{1}, b_{2} \in B_{2}$ the classes ${\tilde{f}}_{i} (b_{1} \otimes b_{2})$ and ${\tilde{e}}_{i} (b_{1} \otimes b_{2})$ are computed by the displayed two-case rules, and they lie in $(B_{1} \otimes B_{2}) \cup {0}$ because each case is the residue of an element of the form $v \otimes w$ with $v \in B_{1} \cup {0}$ , $w \in B_{2} \cup {0}$ ; the equality case $φ_{i} (b_{1}) = ε_{i} (b_{2})$ sends ${\tilde{f}}_{i}$ to the second factor, since the first case requires the strict inequality, and ${\tilde{e}}_{i}$ to the first factor, since its first case is non-strict.

6.1step 5.1F1F6

Partial inverses. Suppose first $φ_{i} (b_{1}) > ε_{i} (b_{2})$ and let $b^{'} : = {\tilde{f}}_{i} (b_{1} \otimes b_{2}) = {\tilde{f}}_{i} b_{1} \otimes b_{2} \neq 0$ ; then $φ_{i} ({\tilde{f}}_{i} b_{1}) = φ_{i} (b_{1}) - 1 \geq ε_{i} (b_{2})$ by [F6], so the ${\tilde{e}}_{i}$ -rule of step 5.1 selects the first factor at $b^{'}$ and ${\tilde{e}}_{i} b^{'} = {\tilde{e}}_{i} {\tilde{f}}_{i} b_{1} \otimes b_{2} = b_{1} \otimes b_{2}$ by the factor partial inverse [F1]. Suppose next $φ_{i} (b_{1}) \leq ε_{i} (b_{2})$ and $b^{'} : = b_{1} \otimes {\tilde{f}}_{i} b_{2} \neq 0$ ; then ${\tilde{f}}_{i} b_{2} \neq 0$ and $ε_{i} ({\tilde{f}}_{i} b_{2}) = ε_{i} (b_{2}) + 1 > φ_{i} (b_{1})$ by [F6], so the ${\tilde{e}}_{i}$ -rule selects the second factor and ${\tilde{e}}_{i} b^{'} = b_{1} \otimes {\tilde{e}}_{i} {\tilde{f}}_{i} b_{2} = b_{1} \otimes b_{2}$ . The converse implication is the same computation read backwards, and at the tie $φ_{i} (b_{1}) = ε_{i} (b_{2})$ both consistency conditions hold simultaneously: this is exactly why ${\tilde{e}}_{i}$ is assigned the non-strict and ${\tilde{f}}_{i}$ the strict comparison.

7.1step 5.1step 6.1F7F8F9given∎

Conclusion. By [F7] the pair $(L_{1} \otimes L_{2}, B_{1} \otimes B_{2})$ has the lattice axioms of generation over $Q (q)$ , weight grading and residue-basis properties; step 4.1 supplies operator stability and step 5.1 the operator axioms on $B_{1} \otimes B_{2}$ ; step 6.1 supplies the partial-inverse axiom. Hence it is a crystal basis of $M_{1} \otimes M_{2}$ whose crystal is the tensor product crystal $B_{1} \otimes B_{2}$ of [F8] with the displayed rules. Every step used finite string data of two vectors and the one-factor crystal bases; no finite dimensionality, bounded support, uniform nilpotence or choice principle is assumed.

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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 Math. J. 63 (1991), no. 2, 465-516 (standard reference, not scraped)
Richard Borcherds, Mark Haiman, Theo Johnson-Freyd, Nicolai Reshetikhin and Vera Serganova, Berkeley Lectures on Lie Groups and Quantum Groups (book-length lecture notes, last updated 18 January 2024) (standard reference, not scraped)