Categorical crystals for quantum affine algebras

(I)

The notion of strong (resp. complete) duality datum for 𝒞 𝔤 0 was introduced by investigating root modules. The complete duality datum generalizes the notion of Q-datum one step further. Let 𝒟 = { 𝖫 i } i ∈ J be a strong duality datum associated with a finite ADE type Cartan matrix 𝖢 , and let

be the quantum affine Schur–Weyl duality functor [24], where R 𝖢 is the symmetric quiver Hecke algebra associated with 𝖢 . Let 𝒞 𝒟 be the image of ℱ 𝒟 (see Section 3 for the precise definition). It was proved in [35] that the duality functor ℱ 𝒟 sends simple modules in R 𝖢 ⁢ - gmod to simple modules in 𝒞 𝒟 and it preserves the invariants Λ, Λ ∞ and 𝔡 (see [34] for details on Λ, Λ ∞ and 𝔡 for 𝒞 𝔤 0 ). The invariants for 𝒞 𝔤 0 take a crucial role in the recent developments about block decomposition, the PBW theory, and monoidal categorification for quantum affine algebras (see [37, 35, 36]). The duality functor ℱ 𝒟 tells us that 𝒞 𝒟 has the categorical crystal structure induced from R 𝖢 ⁢ - gmod via ℱ 𝒟 .

(II)

The notion of affine cuspidal modules for 𝒞 𝔤 0 is introduced. Let 𝒟 be a complete duality datum. Let w 0 ¯ be a reduced expression of the longest element w 0 of the Weyl group of 𝔤 fin , and let { 𝖵 k } k = 1 , … , ℓ be the set of the cuspidal modules (see [46] and see also [33]) of the quiver Hecke algebra R 𝖢 associated with w 0 ¯ . The affine cuspidal modules { 𝖲 k } k ∈ ℤ for 𝒞 𝔤 0 are defined by using { 𝖵 k } k = 1 , … , ℓ via the duality functor ℱ 𝒟 (see Section 3 for the precise definition). When 𝒟 arises from a Q-datum, the affine cuspidal modules coincide with the fundamental modules in 𝒞 𝔤 0 . It turned out that all simple modules in 𝒞 𝔤 0 can be obtained uniquely as the simple heads of the ordered tensor products 𝖯 𝒟 , w 0 ¯ ⁢ ( 𝐚 ) of affine cuspidal modules. Moreover, the module 𝖯 𝒟 , w 0 ¯ ⁢ ( 𝐚 ) , called a standard module, has the unitriangularity property. This generalizes the classical simple module construction taking the head of ordered tensor products of fundamental representations ([1, 31, 47, 48, 54]). From the viewpoint of the PBW theory, it is natural that the Grothendieck ring K ⁢ ( 𝒞 𝔤 0 ) can be viewed as a product of infinite copies of K ⁢ ( 𝒞 𝒟 ) . Note that, when 𝒟 arises from a Dynkin quiver of a finite ADE type, it was already observed by Hernandez and Leclerc (see [20]).

The purpose of this paper is to study a new categorical crystal structure of the category 𝒞 𝔤 0 by using the invariants Λ, Λ ∞ , 𝔡 and the PBW theory. Our main results can be summarized as follows:

1. We introduce the notion of extended crystals B ^ 𝔤 ⁢ ( ∞ ) for an arbitrary quantum group U q ⁢ ( 𝔤 ) . The extended crystal B ^ 𝔤 ⁢ ( ∞ ) can be understood as a product of infinite copies of B ⁢ ( ∞ ) and its crystal operators are defined by using the crystal structure of B ⁢ ( ∞ ) . Thus the extended crystal B ^ 𝔤 ⁢ ( ∞ ) may look like an affinization of the crystal B ⁢ ( ∞ ) . As a usual crystal, the extended crystal B ^ 𝔤 ⁢ ( ∞ ) has the colored graph structure induced from its crystal operators.

2. We prove that the category 𝒞 𝔤 0 has a new categorical crystal structure isomorphic to the extended crystal B ^ 𝔤 fin ⁢ ( ∞ ) . Let 𝒟 = { 𝖫 i } i ∈ I fin be a complete duality datum of 𝒞 𝔤 0 and let ℬ 𝒟 ⁢ ( 𝔤 ) be the set of the isomorphism classes of simple modules in 𝒞 𝔤 0 . The operators ℱ ~ i , k , ℰ ~ i , k , ℱ ~ i , k * , and ℰ ~ i , k * defined in (5.9) gives a categorical crystal structure on ℬ 𝒟 ⁢ ( 𝔤 ) . This crystal structure turns out isomorphic to that of B ^ 𝔤 fin ⁢ ( ∞ ) . Thus the simple modules in 𝒞 𝔤 0 are parameterized by the extended crystal B ^ 𝔤 fin ⁢ ( ∞ ) which leads us to a new combinatorial approach to study 𝒞 𝔤 0 .

3. We provide explicit formulas to compute the invariants Λ and 𝔡 between 𝒟 k ⁢ 𝖫 i ( k ∈ ℤ ) and an arbitrary simple module in terms of the extended crystal.

4. We give an explicit combinatorial description of the extended crystal ℬ 𝒟 ⁢ ( 𝔤 ) for affine type A n ( 1 ) in terms of affine highest weights.

Let us explain our results more precisely. Let I be an index set and let 𝔤 be the Kac–Moody algebra associated with a symmetrizable generalized Cartan matrix. The extended crystal is defined as

where B ⁢ ( ∞ ) is the crystal of U q - ⁢ ( 𝔤 ) . Let

For 𝐛 = ( b k ) k ∈ ℤ ∈ B ^ 𝔤 ⁢ ( ∞ ) and ( i , k ) ∈ I ^ , we define the extended crystal operator F ~ ( i , k ) ⁢ ( 𝐛 ) (resp. E ~ ( i , k ) ⁢ ( 𝐛 ) ) in terms of the usual crystal operators f ~ i ⁢ ( b k ) and e ~ i * ⁢ ( b k + 1 ) (resp. e ~ i ⁢ ( b k ) and f ~ i * ⁢ ( b k + 1 ) ) according to the values of ε i ⁢ ( b k ) and ε i * ⁢ ( b k + 1 ) (see (4.1)). We show that there exist natural injections ι k : B ⁢ ( ∞ ) ↣ B ^ 𝔤 ⁢ ( ∞ ) ( k ∈ ℤ ) and interesting bijections ⋆ and D on B ^ 𝔤 ⁢ ( ∞ ) compatible with the extended crystal structure (see Lemma 4.2 and Lemma 4.3). Similarly to the usual crystals, the extended crystal B ^ 𝔤 ⁢ ( ∞ ) has a connected I ^ -colored graph structure induced from the operators F ~ i , k (see Lemma 4.4).

We next deal with the category 𝒞 𝔤 0 from the viewpoint of the extended crystal. Let U q ′ ⁢ ( 𝔤 ) be a quantum affine algebra of arbitrary type and let 𝒟 = { 𝖫 i } i ∈ I fin be a complete duality datum of 𝒞 𝔤 0 . We define ℬ ⁢ ( 𝔤 ) to be the set of the isomorphism classes of simple modules in 𝒞 𝔤 0 . For 𝐛 = ( b k ) k ∈ ℤ ∈ B ^ 𝔤 fin ⁢ ( ∞ ) , we define

where hd ⁡ ( X ) stands for the head of a module X, 𝒟 is the right dual functor and ℒ 𝒟 ⁢ ( b k ) is the simple module in 𝒞 𝒟 corresponding to b k via the duality functor ℱ 𝒟 (see Lemma 3.2 (i)). Proposition 5.5 says that ℒ 𝒟 ⁢ ( 𝐛 ) is simple and the map

is bijective. Moreover, the bijection D on B ^ 𝔤 fin ⁢ ( ∞ ) is compatible with the functor 𝒟 under the map Φ 𝒟 . For M ∈ ℬ ⁢ ( 𝔤 ) and ( i , k ) ∈ I ^ fin , we define

where ∇ ⁡ X Y denotes the head of ⊗ ⁡ X Y . Since the operators ℱ ~ i , k and ℰ ~ i , k depend on the choice of 𝒟 , we write ℬ 𝒟 ⁢ ( 𝔤 ) instead of ℬ ⁢ ( 𝔤 ) when we consider ℬ ⁢ ( 𝔤 ) together with the operators ℱ ~ i , k and ℰ ~ i , k . We prove that ℱ ~ i , k and ℰ ~ i , k are inverse to each other and the bijection Φ 𝒟 has compatibility with the extended crystal operators, i.e.,

(see Theorem 5.10). Therefore ℬ 𝒟 ⁢ ( 𝔤 ) has the same extended crystal structure as B ^ 𝔤 fin ⁢ ( ∞ ) and, for any two quantum affine algebras U q ′ ⁢ ( 𝔤 ) and U q ′ ⁢ ( 𝔤 ′ ) , ℬ 𝒟 ⁢ ( 𝔤 ) ≃ ℬ 𝒟 ′ ⁢ ( 𝔤 ′ ) as a colored graph if and only if 𝔤 fin ≃ ( 𝔤 ′ ) fin as a simple Lie algebra. In particular, the I ^ fin -colored graph structure of ℬ 𝒟 ⁢ ( 𝔤 ) does not depend on the choice of complete duality data 𝒟 (see Corollary 5.11).

Explicit formulas for computing the invariants Λ and 𝔡 between 𝒟 k ⁢ 𝖫 i ( k ∈ ℤ ) and a simple module are given in Theorem 6.3. Let M be a simple module in 𝒞 𝔤 0 and let

By investigating properties of root modules, we prove that

1. Λ ⁢ ( 𝒟 k ⁢ 𝖫 i , M ) = 2 ⁢ max { x , r } + ∑ t ∈ ℤ ( - 1 ) k + δ ( t > k ) ⁢ ( α i , wt t ⁡ ( 𝐛 ) ) ,

2. Λ ⁢ ( M , 𝒟 k ⁢ 𝖫 i ) = 2 ⁢ max { y , s } + ∑ t ∈ ℤ ( - 1 ) k + δ ( t < k ) ⁢ ( α i , wt t ⁡ ( 𝐛 ) ) ,

3. 𝔡 ⁡ ( 𝒟 k ⁢ 𝖫 i , M ) = max { x , r } + max { y , s } + ( - 1 ) k ⁢ ( α i , wt k ⁡ ( 𝐛 ) ) ,

where := ⁡ x ε ( i , k + 1 ) * ⁢ ( 𝐛 ) , := ⁡ r ε ( i , k ) ⁢ ( 𝐛 ) , := ⁡ s ε ( i , k ) * ⁢ ( 𝐛 ) , and := ⁡ y ε ( i , k - 1 ) ⁢ ( 𝐛 ) . We remark that these formulas can be understood as a quantum affine algebra analogue of the ones given in [33, Corollary 3.8].

For affine type A n ( 1 ) , we provide an explicit combinatorial description of the extended crystal ℬ 𝒟 ⁢ ( 𝔤 ) in terms of affine highest weights. Let U q ′ ⁢ ( 𝔤 ) be the quantum affine algebra of type A n ( 1 ) and let 𝒞 𝔤 0 be the Hernandez–Leclerc category corresponding to

Let

where := ⁡ 𝒮 n { ( i , a ) ∈ I 0 × ℤ : a - i ≡ 1 mod 2 } . Note that the set of fundamental modules in 𝒞 𝔤 0 is { V ⁢ ( ϖ i ) ( - q ) a : ( i , a ) ∈ 𝒮 n } . For any λ = ∑ ( i , a ) ∈ 𝒮 n c ( i , a ) ⁢ ( i , a ) ∈ 𝒫 n , we denote by V ⁢ ( λ ) the simple module in 𝒞 𝔤 0 with the affine highest weight ∑ ( i , a ) ∈ 𝒮 n c ( i , a ) ⁢ ( i , ( - q ) a ) (see Theorem 2.3 iv), which gives the bijection

For ( i , k ) ∈ I ^ 0 , we define the crystal operators F ~ i , k and E ~ i , k on 𝒫 n in a combinatorial way (see Section 7.3). We briefly explain the combinatorial rule for F ~ i , k . For a given λ ∈ 𝒫 n , we first choose a suitable subset S i , k = { 𝗏 2 ⁢ n , 𝗏 2 ⁢ n - 1 , … , 𝗏 1 } of 𝒮 n and make a sequence of + and - according to S i , k and the coefficients of λ. We then cancel out all possible ( + , - ) pairs to obtains a sequence of - ’s followed by + ’s. Let 𝗏 t be the element of S i , k corresponding to the leftmost + in the resulting sequence of λ. If such an 𝗏 t exists, then we define

Otherwise, we define

We remark that this combinatorial rule is quite similar to the ( + , - ) -signature rule for the tensor product of crystals. Let := ⁡ 𝒟 { V ⁢ ( ϖ 1 ) ( - q ) 2 ⁢ i - 2 } i ∈ I 0 . Theorem 7.5 tells us that there is a bijection Υ n : B ^ 𝔤 0 ⁢ ( ∞ ) → ∼ 𝒫 n which is compatible with the crystal operators F ~ i , k and E ~ i , k and, for ( i , k ) ∈ I ^ 0 and λ ∈ 𝒫 n ,

Hence the diagram

commutes, and the arrows are I ^ 0 -colored graph isomorphisms. In the course of proofs, the multisegment realization for the crystal B ⁢ ( ∞ ) is used crucially (see Section 7.1).

The paper is organized as follows. In Section 2, we briefly review necessary background on crystals, quantum affine algebras, and the invariants Λ, Λ ∞ and 𝔡 . In Section 3, we recall the PBW theory for 𝒞 𝔤 0 developed in [35]. In Section 4, we introduce the notion of the extended crystals. In Section 5, we prove that the category 𝒞 𝔤 0 has a categorical crystal structure isomorphic to the extended crystal B ^ 𝔤 fin ⁢ ( ∞ ) . In Section 6, we give formulas to compute the invariants Λ and 𝔡 in terms of the extended crystal. In Section 7, we give a combinatorial description of the extended crystal ℬ 𝒟 ⁢ ( 𝔤 ) in terms of affine highest weights.

2.1 Crystals

A quintuple ( 𝖠 , 𝖯 , Π , 𝖯 ∨ , Π ∨ ) is a (symmetrizable) Cartan datum if it consists of

1. a generalized Cartan matrix 𝖠 = ( a i ⁢ j ) i , j ∈ I ,

2. a free abelian group 𝖯 , called the weight lattice,

3. Π = { α i : i ∈ I } ⊂ 𝖯 , called the set of simple roots,

4. 𝖯 ∨ = Hom ℤ ⁡ ( 𝖯 , ℤ ) , called the coweight lattice,

5. Π ∨ = { h i ∈ 𝖯 ∨ : i ∈ I } , called the set of simple coroots,

which satisfy the following properties:

1. 〈 h i , α j 〉 = a i ⁢ j for i , j ∈ I ,

2. Π is linearly independent over ℚ ,

3. for each i ∈ I , there exists Λ i ∈ 𝖯 such that 〈 h j , Λ i 〉 = δ j , i for all j ∈ I .

4. there is a symmetric bilinear form ( ⋅ , ⋅ ) on 𝖯 satisfying ( α i , α i ) ∈ ℚ > 0 and

   〈 h i , λ 〉 = 2 ⁢ ( α i , λ ) ( α i , α i ) .

We denote by := ⁡ 𝖰 + ∑ i ∈ I ℤ ⩾ 0 ⁢ α i the positive cone of the root lattice := ⁡ 𝖰 ⊕ i ∈ I ℤ ⁢ α i , and set 0 ⁢ p ⁢ t ⁢ β := ∑ i ∈ I k i for any β = ∑ i ∈ I k i ⁢ α i ∈ 𝖰 + . Let Δ + denote the set of positive roots for 𝖠 and write Δ - := - Δ + . Let 𝖶 be the Weyl group and let s i ( i ∈ I ) be the simple reflections of 𝖶 . Let U q ⁢ ( 𝔤 ) denote the quantum group corresponding to ( 𝖠 , 𝖯 , 𝖯 ∨ , Π , Π ∨ ) , that is an algebra over ℚ ⁢ ( q ) generated by f i , e i and q h with certain defining relations, where q is an indeterminate (see [21, Chapter 3] for details). Let U q - ⁢ ( 𝔤 ) be the subalgebra of U q ⁢ ( 𝔤 ) generated by f i for all i ∈ I . For the notion of crystals, we refer the reader to [28, 29, 30] and [21, Chapter 4].

We denote by B 𝔤 ⁢ ( ∞ ) the crystal of U q - ⁢ ( 𝔤 ) and let 𝟣 be the highest vector of B 𝔤 ⁢ ( ∞ ) . We simply write B ⁢ ( ∞ ) instead of B 𝔤 ⁢ ( ∞ ) if no confusion arises. The ℚ ⁢ ( q ) -antiautomorphism * of U q ⁢ ( 𝔤 ) defined by

gives an involution on B ⁢ ( ∞ ) . This provides another crystal structure with e ~ i * , f ~ i * , ε i * , φ i * on U q - ⁢ ( 𝔤 ) , which is denoted by B ⁢ ( ∞ ) * . We set

2.2 Quantum affine algebras

Let 𝖠 = ( a i , j ) i , j ∈ I be an affine Cartan matrix. We take a ℚ -valued non-degenerate symmetric bilinear form ( ⋅ , ⋅ ) such that, for any i ∈ I and λ ∈ 𝖯 ,

where δ ∈ 𝖯 is the imaginary root and c ∈ 𝖯 ∨ is the central element. Let 𝔤 be the affine Kac–Moody algebra associated with 𝖠 and set := ⁡ I 0 I ∖ { 0 } . We refer the reader to [35, Section 2.3] for the choice of 0 ∈ I . We define 𝔤 0 to be the subalgebra of 𝔤 generated by e i , f i for i ∈ I 0 .

Let 𝐤 the algebraic closure of the subfield ℂ ⁢ ( q ) in the algebraically closed field

Let U q ′ ⁢ ( 𝔤 ) be the 𝐤 -subalgebra of the quantum group U q ⁢ ( 𝔤 ) generated by e i , f i , := ⁡ K i q i ± h i ( i ∈ I ) , where := ⁡ q i q ( α i , α i ) / 2 . Let 𝒞 𝔤 denote the category of finite-dimensional integrable U q ′ ⁢ ( 𝔤 ) -modules. Recall that a finite-dimensional U q ′ ⁢ ( 𝔤 ) -module M is integrable if it has a weight decomposition

where := ⁡ 𝖯 cl 𝖯 / ( 𝖯 ∩ ℚ ⁢ δ ) . Let ⊗ denote the tensor product of 𝒞 𝔤 . We write

for k ∈ ℤ ⩾ 0 , and denote by ∇ ⁡ M N (resp. Δ ⁡ M N ) the head of ⊗ ⁡ M N (resp. socle of ⊗ ⁡ M N ) for M , N ∈ 𝒞 𝔤 .

We say that M and N commute if ⊗ ⁡ M N ≃ ⊗ ⁡ N M . We say that M and N strongly commute if ⊗ ⁡ M N is simple. A simple U q ′ ⁢ ( 𝔤 ) -module L is real if ⊗ ⁡ L L is simple. Note that, for simple modules M and N, if M and N strongly commute, then they commute. If M and N are simple and one of them is real, then M and N commutes if and only if M and N are strongly commute ([25, Corollary 3.16]). For M ∈ 𝒞 𝔤 , we denote by 𝒟 ⁢ M and 𝒟 - 1 ⁢ M the right and the left dual of M, respectively. We extend this to 𝒟 k ⁢ ( M ) for all k ∈ ℤ (see [15, Section 3 and 4] for dual functors, and see also [34, Section 2]).

Every simple module L has a distinguished element, called a dominant extremal weight vector, and the weight of a dominant extremal weight vector is called a dominant extremal weight. For each i ∈ I 0 , we define

where the central element c is written as ∑ i ∈ I 𝖼 i ⁢ h i . Let V ⁢ ( ϖ i ) denote the i-th fundamental representation (see [31, Section 5.2] for details).

For a module M, let M aff be the affinization of M with the U q ′ ⁢ ( 𝔤 ) -module automorphism z M : M aff → M aff of weight δ. We often write M z for M aff to emphasize z. For any x ∈ 𝐤 × , we set

Here x is called a spectral parameter (see [31, Section 4.2] for details).

We set

where the equivalence relation ∼ is defined by ( i , x ) ∼ ( j , y ) if and only if V ⁢ ( ϖ i ) x ≃ V ⁢ ( ϖ j ) y . As σ ⁢ ( 𝔤 ) has a quiver structure arising from the denominators of R-matrices (see Section 2.3), we take a connected component σ 0 ⁢ ( 𝔤 ) of σ ⁢ ( 𝔤 ) . Note that the connected component σ 0 ⁢ ( 𝔤 ) is determined uniquely up to a parameter shift (see [35, Section 2.5] for details).

The Hernandez–Leclerc category 𝒞 𝔤 0 is the smallest full subcategory of 𝒞 𝔤 such that

1. 𝒞 𝔤 0 contains V ⁢ ( ϖ i ) x for all ( i , x ) ∈ σ 0 ⁢ ( 𝔤 ) ,

2. 𝒞 𝔤 0 is stable by taking subquotients, extensions and tensor products.

2.3 R-matrices and related invariants

In this subsection, we briefly recall the invariants Λ, Λ ∞ and 𝔡 introduced in [34] and several results of [34, 35]. For the notion of (universal) R-matrices and related properties, we refer the reader to [13], [1, Appendices A and B], [31, Section 8] and [18].

For non-zero modules M and N, we denote by

the universal R-matrix. We say that R M , N z univ is rationally renormalizable if there is an element f ⁢ ( z ) ∈ 𝐤 ⁢ ( ( z ) ) × such that

When R M , N z univ is rationally renormalizable, there exists c M , N ⁢ ( z ) ∈ 𝐤 ⁢ ( ( z ) ) × such that

sends M ⊗ N z to N z ⊗ M and its specialization

does not vanish at any z = x ∈ 𝐤 × . Here R M , N z ren is called the renormalized R-matrix and c M , N ⁢ ( z ) is called the renormalizing coefficient. Let 𝐫 M , N be the specialization at z = 1

which is called the R-matrix. Note that 𝐫 M , N never vanishes by the definition.

For simple modules M and N, there is a M , N ⁢ ( z ) ∈ 𝐤 ⁢ [ [ z ] ] × such that

where u and v are dominant extremal weight vectors of M and N, respectively. We have a unique ⊗ ⁡ 𝐤 ⁢ ( z ) U q ′ ⁢ ( 𝔤 ) -module isomorphism

between 𝐤 ⁢ ( z ) ⊗ 𝐤 ⁢ [ z ± 1 ] ( M ⊗ N z ) and 𝐤 ⁢ ( z ) ⊗ 𝐤 ⁢ [ z ± 1 ] ( N z ⊗ M ) . Here a M , N ⁢ ( z ) is called the universal coefficient of M and N, and R M , N z norm is called the normalized R-matrix. Denote by d M , N ⁢ ( z ) a monic polynomial of the smallest degree such that the image d M , N ⁢ ( z ) ⁢ R M , N z norm ⁢ ( ⊗ ⁡ M N z ) is contained in N z ⊗ M . We call it the denominator of R M , N z norm . Then

and

up to a multiple of 𝐤 ⁢ [ z ± 1 ] × (see [34] for details).

In the following theorem, we refer to [31] for the notion of good modules. Note that any good module is real simple and every fundamental module V ⁢ ( ϖ i ) is a good module.