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Crystal lattices, crystal bases and their crystals

Definition

Let $A_{0}$ be the local ring of The local ring $A_{0}$ of rational functions regular at $q = 0$ and let $M$ be an integrable weight module with its Kashiwara operators ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ (The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module, Quantum weight modules, weights in $P$ , and integrability).

A crystal lattice of $M$ is a free $A_{0}$ -submodule $L \subseteq M$ such that

(i) $L$ generates $M$ over $Q (q)$ , i.e. the canonical map $Q (q) \otimes_{A_{0}} L \to M$ is an isomorphism;

(ii) $L = ⨁_{μ \in P} L_{μ}$ with $L_{μ} : = L \cap M_{μ}$ , and each $L_{μ}$ is again a free $A_{0}$ -module;

(iii) ${\tilde{e}}_{i} L \subseteq L$ and ${\tilde{f}}_{i} L \subseteq L$ for all $i \in I$ .

Because $A_{0}$ is a principal ideal domain, each graded piece $L_{μ}$ has a well-defined rank ${rank}_{A_{0}} L_{μ}$ , and (ii) says that ${rank}_{A_{0}} L$ is the sum of the ranks of the $L_{μ}$ .

A crystal basis of $M$ is a pair $(L, B)$ consisting of a crystal lattice $L$ and a $Q$ -basis $B$ of the crystal limit $L / q L ≅ Q \otimes_{A_{0}} L$ (The local ring $A_{0}$ of rational functions regular at $q = 0$ ) such that

(iv) $B = ⨆_{μ \in P} B_{μ}$ , where $B_{μ} : = B \cap (L_{μ} / q L_{μ})$ ;

(v) ${\tilde{e}}_{i} B \subseteq B \cup {0}$ and ${\tilde{f}}_{i} B \subseteq B \cup {0}$ for all $i \in I$ ;

(vi) for all $b, b^{'} \in B$ and all $i$ : ${\tilde{f}}_{i} b = b^{'}$ if and only if $b = {\tilde{e}}_{i} b^{'}$ ; here ${\tilde{e}}_{i}, {\tilde{f}}_{i}$ denote the operators induced on $L / q L$ .

The crystal graph of $(L, B)$ is the coloured directed graph with vertex set $B$ and an arrow $b \overset{i}{\to} b^{'}$ whenever ${\tilde{f}}_{i} b = b^{'}$ (by (vi) this is equivalent to $b = {\tilde{e}}_{i} b^{'}$ ). The crystal of $M$ is this graph, together with the weight function $b \mapsto μ$ for $b \in B_{μ}$ . For every $μ \in P$ one has $\dim_{Q (q)} M_{μ} = {rank}_{A_{0}} L_{μ} = # B_{μ}$ .

A crystal basis of the negative half $U_{q}^{-}$ is defined by the same data with $U_{q}^{-}$ in place of $M$ : a free $A_{0}$ -submodule $L \subseteq U_{q}^{-}$ with $U_{q}^{-} ≅ Q (q) \otimes_{A_{0}} L$ and ${\tilde{e}}_{i} L, {\tilde{f}}_{i} L \subseteq L$ (The $i$ -string decomposition of the negative half), together with a $Q$ -basis $B$ of $L / q L$ satisfying (v) and (vi) for the operators on the negative half. Here the root grading replaces the weight grading, and the lattice and residue basis are required to respect that root grading. In the negative half, ${\tilde{e}}_{i} {\tilde{f}}_{i} = i d$ implies ${\tilde{f}}_{i} B \subseteq B$ ; this refinement is specific to its infinite strings. No choice principle is used in these definitions.

Facts & Assumptions

Given: The local ring $A_{0} = Q [q]_{(q)}$ with residue field $Q$ , an integrable chosen $P$ -graded type-1 module $M$ with Kashiwara operators, and a crystal lattice $L \subseteq M$ with $L = ⨁_{μ} L_{μ}$ as in (ii).

[F1]

${\tilde{e}}_{i}, {\tilde{f}}_{i}$ are $Q (q)$ -linear endomorphisms of $M$ shifting the label weight by $+ α_{i}$ and $- α_{i}$ ; on a string they move $F_{i}^{(n)} u_{n}$ to $F_{i}^{(n - 1)} u_{n}$ or $F_{i}^{(n + 1)} u_{n}$ , with the top of an ${\tilde{e}}_{i}$ -string and the bottom of an ${\tilde{f}}_{i}$ -string mapped to $0$ (The Kashiwara operators ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ of an integrable module).

[F2]

$M = ⨁_{μ \in P} M_{μ}$ with the toral action and gradings of Quantum weight modules, weights in $P$ , and integrability.

[F3]

For a free $A_{0}$ -module $N$ the canonical map $Q \otimes_{A_{0}} N \to N / q N$ is an isomorphism, and $Q \otimes_{A_{0}} N$ is a $Q$ -vector space of dimension equal to the rank of $N$ ; $Q (q) \otimes_{A_{0}} N$ has dimension equal to the rank of $N$ over $Q (q)$ (The local ring $A_{0}$ of rational functions regular at $q = 0$ ).

[F4]

On the negative half, ${\tilde{e}}_{i} {\tilde{f}}_{i} = {id}_{U_{q}^{-}}$ , and the operators shift the root degree by $\mp α_{i}$ (The $i$ -string decomposition of the negative half, part (iii)).

Verification

technique · direct. The induced operators, the rank and residue counts and the negative-half refinement are read off the axioms using the crystal-limit isomorphism

1.1F1F3given

Induced operators. By (iii) the operators preserve $L$ and commute with multiplication by $A_{0}$ , so they descend to $Q$ -linear endomorphisms of $L / q L$ , which is canonically $Q \otimes_{A_{0}} L$ by [F3]; by [F1] they shift the label weight by $\pm α_{i}$ , so they map $L_{μ} / q L_{μ}$ to $L_{μ + α_{i}} / q L_{μ + α_{i}}$ and to $L_{μ - α_{i}} / q L_{μ - α_{i}}$ , and axiom (v) says that they map $B$ into $B \cup {0}$ .

1.2F2F3givenalgebra

Rank decomposition. Since $L = ⨁_{μ} L_{μ}$ with every $L_{μ}$ free by (ii), ${rank}_{A_{0}} L = \sum_{μ} {rank}_{A_{0}} L_{μ}$ , and applying $Q (q) \otimes_{A_{0}} -$ gives $Q (q) \otimes_{A_{0}} L = ⨁_{μ} Q (q) \otimes_{A_{0}} L_{μ}$ with each summand of dimension ${rank}_{A_{0}} L_{μ}$ by [F3]. The isomorphism of (i) is weight graded, because $L_{μ} \subseteq M_{μ}$ and the target is the direct sum of the $M_{μ}$ ; hence $\dim_{Q (q)} M_{μ} = {rank}_{A_{0}} L_{μ}$ for every $μ$ .

2.1F3step 1.2givenalgebra

Residue counts. The crystal limit isomorphism of [F3] is compatible with the grading, and $L / q L = ⨁_{μ} L_{μ} / q L_{μ}$ ; by (iv) each $B_{μ}$ is contained in $L_{μ} / q L_{μ}$ and $B = ⨆_{μ} B_{μ}$ is a $Q$ -basis of $L / q L$ , so $B_{μ}$ is a $Q$ -basis of $L_{μ} / q L_{μ}$ . Hence $# B_{μ} = \dim_{Q} L_{μ} / q L_{μ} = {rank}_{A_{0}} L_{μ} = \dim_{Q (q)} M_{μ}$ , which is the displayed equality, and the weight function of the crystal is well defined by (iv).

2.2givenstep 1.1

The graph and the partial-inverse property. By (vi) the arrow relation $b \overset{i}{\to} b^{'}$ is symmetric under exchanging ${\tilde{e}}_{i}$ and ${\tilde{f}}_{i}$ , so the crystal graph has no ambiguity: an arrow exists exactly when ${\tilde{f}}_{i} b = b^{'}$ , equivalently $b = {\tilde{e}}_{i} b^{'}$ . In a finite-dimensional piece a string may end at a bottom, where ${\tilde{f}}_{i} b = 0$ ; by (v) this is allowed and no vertex is added.

2.3F4step 1.1

The negative half. For $U_{q}^{-}$ the same discussion applies with the root grading in place of the weight grading. By [F4] the induced endomorphisms of $L / q L$ satisfy ${\overset{ˉ}{\tilde{e}}}_{i} {\overset{ˉ}{\tilde{f}}}_{i} = i d$ , so ${\overset{ˉ}{\tilde{f}}}_{i}$ is injective and no element of $B$ is mapped to $0$ by ${\tilde{f}}_{i}$ ; with (v) this gives ${\tilde{f}}_{i} B \subseteq B$ , the infinite-string refinement.

3.1given∎

Choice. The definitions select no bases, no complements and no representatives; the lattice, its grading and the residue basis are given data, and all statements above are computations with them, so no choice principle is used.

Depends on

Used by

Dependency tree · two levels

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Sources

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)
Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390 (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)