T-systems and Y-systems for quantum affinizations of quantum Kac-Moody algebras

The T-systems and Y-systems are classes of algebraic relations originally associated with quantum affine algebras and Yangians. Recently the T-systems were generalized to quantum affinizations of a wide class of quantum Kac-Moody algebras by Hernandez. In this note we introduce the corresponding Y-systems and establish a relation between T and Y-systems. We also introduce the T and Y-systems associated with a class of cluster algebras, which include the former T and Y-systems of simply laced type as special cases.

The T-systems are generalized by Hernandez [27] to the quantum affinizations of a wide class of quantum Kac-Moody algebras studied in [15,56,34,46,47,26]. In this paper we introduce the corresponding Y-systems and establish a relation between T and Y-systems. We also introduce the T and Y-systems associated with a class of cluster algebras, which include the former T and Y-systems of simply laced type as special cases.
It will be interesting to investigate the relation of the systems discussed here to the birational transformations arising from the Painlevé equations in [50,51], and also to the geometric realization of cluster algebras in [5,23].
The organization of the paper is as follows. In Section 2 basic definitions for quantum Kac-Moody algebras U q (g) and their quantum affinizations U q (ĝ) are recalled. In Section 3 the T-systems associated with the quantum affinizations of a class of quantum Kac-Moody algebras by [27] are presented. Based on the result by [27], the role of the T-system in the Grothendieck ring of U q (ĝ)-modules is given (Corollary 3.8). In Section 4 we introduce the Y-systems corresponding to the T-systems in Section 3, and establish a relation between them (Theorem 4.4). In Section 5 we define the restricted version of T-systems and Y-systems, and establish a relation between them (Theorem 5.3). In Section 6 we introduce the T and Y-systems associated with a class of cluster algebras, which include the restricted T and Y-systems of simply laced type as special cases. In particular, the correspondence between the restricted T and Ysystems of simply laced type for the quantum affinizations and cluster algebras is presented (Corollaries 6.20, 6.21, 6.25, and 6.26).

Quantum Kac-Moody algebras and their quantum affinizations
In this section, we recall basic definitions for quantum Kac-Moody algebras and their quantum affinizations, following [26,27]. The presentation here is a minimal one. See [26,27] for further information and details.
Definition 2.1 ( [14,33]). The quantum Kac-Moody algebra U q (g) associated with C is the C-algebra with generators k h (h ∈ h), x ± i (i ∈ I) and the following relations:

Quantum affinizations
In the following, we use the following formal series (currents): We also use the formal delta function δ(z) = r∈Z z r . 15,34,27]). The quantum affinization (without central elements) of the quantum Kac-Moody algebra U q (g), denoted by U q (ĝ), is the C-algebra with generators x ± i,r (i ∈ I, r ∈ Z), k h (h ∈ h), h i,r (i ∈ I, r ∈ Z \ {0}) and the following relations: When C is of finite type, the above U q (ĝ) is called an (untwisted) quantum affine algebra (without central elements) or quantum loop algebra; it is isomorphic to a subquotient of the quantum Kac-Moody algebra associated with the (untwisted) affine extension of C without derivation and central elements [15,4]. (A little confusingly, the quantum Kac-Moody algebra associated with C of affine type with derivation and central elements is also called a quantum affine algebra and denoted by U q (ĝ).) When C is of affine type, U q (ĝ) is called a quantum toroidal algebra (without central elements).
In general, if C is not of finite type, U q (ĝ) is no longer isomorphic to a subquotient of any quantum Kac-Moody algebra and has no Hopf algebra structure.

The category Mod(U q (ĝ))
Let U q (h) be the subalgebra of U q (ĝ) generated by k h (h ∈ h).
The following theorem is a generalization of the well-known classification of the simple finitedimensional modules of the quantum affine algebras by [7,8]. Theorem 2.5 ([46,47,26]). We have L(λ, Ψ) ∈ Mod(U q (ĝ)) if and only if there is an I-tuple of polynomials (P i (u)) i∈I , We call (P i (u)) i∈I the Drinfeld polynomials of L(λ, Ψ). In the case of quantum affine algebras, λ is also completely determined by the Drinfeld polynomials by the condition λ(α ∨ i ) = deg P i . This is not so in general.
(b) Choose any λ satisfying (λ, α ∨ i ) = 0 (i ∈ I) and also set P i (u) = 1 (i ∈ I). The corresponding module L(λ, Ψ) is written as L(λ). The module L(λ) is one-dimensional; it is trivial in the case of the quantum affine algebras.

Kirillov-Reshetikhin modules
The following is a generalization of the Kirillov-Reshetikhin modules of the quantum affine algebras studied by [37,38,3,8,43,9]. Definition 2.7 ( [27]). For any i ∈ I, m ∈ N, and α ∈ C × , set the polynomials (P j (u)) j∈I as and P j (u) = 1 for any j = i.  [27] in order to make the identification to the forthcoming T-systems a little simpler.

T-systems
Throughout Sections 3-5, we restrict our attention to a symmetrizable generalized Cartan matrix C satisfying the following condition due to Hernandez [27]: where D = diag(d 1 , . . . , d r ) is the diagonal matrix symmetrizing C. In this paper, we say that a generalized Cartan matrix C is tamely laced if it is symmetrizable and satisfies the condition (3.1). As usual, we say that a generalized Cartan matrix C is simply laced if C ij = 0 or −1 for any i = j. If C is simply laced, then it is symmetric, d a = 1 for any a ∈ I, and it is tamely laced.
With a tamely laced generalized Cartan matrix C, we associate a Dynkin diagram in the standard way [35]: For any pair i = j ∈ I with C ij < 0, the vertices i and j are connected by max{|C ij |, |C ji |} lines, and the lines are equipped with an arrow from j to i if C ij < −1. Example 3.1.
(1) Any Cartan matrix of finite or affine type is tamely laced except for types A (1) 1 and A (2) 2ℓ .
(2) The following generalized Cartan matrix C is tamely laced: The corresponding Dynkin diagram is For a tamely laced generalized Cartan matrix C, we set an integer t by t = lcm(d 1 , . . . , d r ).
For a, b ∈ I, we write a ∼ b if C ab < 0, i.e., a and b are adjacent in the corresponding Dynkin diagram.
Let U be either 1 t Z, the complex plane C, or the cylinder C ξ := C/(2π √ −1/ξ)Z for some ξ ∈ C \ 2π √ −1Q, depending on the situation under consideration. The following is a generalization of the T-systems associated with the quantum affine algebras [43].

Definition 3.2 ([27]
). For a tamely laced generalized Cartan matrix C, the unrestricted Tsystem T(C) associated with C is the following system of relations for a family of variables where T and E[x] (x ∈ Q) denotes the largest integer not exceeding x. 1. This is a slightly reduced version of the T-systems in [27, Theorem 6.10]. See Remark 3.7. The same system was also studied by [54] when C is of affine type in view of a generalization of discrete Toda field equations.
2. More explicitly, S For example, for d b = 1, and so on. 3. The second terms in the right hand sides of (3.2) and (3.3) can be written in a unified way as follows [27]:

T-system and Grothendieck ring
Let C continue to be a tamely laced generalized Cartan matrix. The T-system T(C) is a family of relations in the Grothendieck ring of modules of U q (ĝ) as explained below.
1. For a pair of ℓ-highest weight modules V 1 , V 2 ∈ Mod(U q (ĝ)), there is an ℓ-highest weight module V 1 * f V 2 ∈ Mod(U q (ĝ)) called the fusion product. It is defined by using the udeformation of the Drinfeld coproduct and the specialization at u = 1.
Therefore, the Grothendieck ring R(C) of the modules in Mod(U q (ĝ)) having finite compositions is well defined, where the product is given by * f . Let R ′ (C) be the quotient ring of R(C) by the ideal generated by all L(λ, Ψ) − L(λ ′ , Ψ)'s. In other words, we regard modules in R(C) as modules of the subalgebra of U q (ĝ) generated by Proof . It follows from [27,Corollary 4.9] that R ′ (C) is generated by the fundamental modules L(Λ i , Ψ). The q-character morphism χ q defined in [27] induces an injective ring homomorphism We set C t log q := C/(2π √ −1/(t log q))Z, and introduce alternative notation W In terms of the Kirillov-Reshetikhin modules, the structure of R ′ (C) is described as follows: (1) The family W generates the ring R ′ (C).
(3) The proof is the same with that of [32, Theorem 2.8] by generalizing the height of T  As a corollary, we have a generalization of [32, Corollary 2.9] for the quantum affine algebras:

Y-systems 4.1 Y-systems
Definition 4.1. For a tamely laced generalized Cartan matrix C, the unrestricted Y-system Y(C) associated with C is the following system of relations for a family of variables 0 (u) −1 = 0 if they occur in the right hand sides in the relations: Remark 4.2.
1. The Y-systems here are formally in the same form as the ones for the quantum affine algebras [42]. However, p for Z p,m (u) here may be greater than 3.
and so on. There are p 2 factors in Z

Relation between T and Y-systems
Let us write both the relations (3.2) and (3.3) in T(C) in a unified manner where M (a) m (u) is the second term of the right hand side of each relation. Define the transposition Proof . This can be proved by case check for d a > 1 and d a = 1.
For any commutative ring R over Z with identity element, let R × denote the group of all the invertible elements of R.
Theorem 4.4. Let R be any commutative ring over Z with identity element.
(2) We modify the proof in the case of quantum affine algebras [32, Theorem 2.12] so that it is applicable to the present situation. Here, we concentrate on the case U = 1 t Z. The modification of the proof for the other cases U = C and C/(2π √ −1/ξ)Z is straightforward. Case 1. When C is simply laced. Suppose that C is simply laced. Thus, d a = 1 for any a ∈ I and t = 1. For any Y satisfying Y(C), we construct a desired family T in the following three steps: Step 1. Choose arbitrarily T  . (4.9) Repeat it and define T (a) 1 (u) (a ∈ I) for the rest of u ∈ Z by (4.9).
where T (a) 0 (u) = 1. Claim. The family T defined above satisfies the following relations in R for any a ∈ I, m ∈ N, u ∈ Z: . (4.13) Proof of Claim. This ends the proof of Claim. Now, taking the inverse sum of (4.12) and (4.13), we obtain (4.3). Therefore, T satisfies the desired properties.
Case 2. When C is nonsimply laced. Suppose that C is nonsimply laced. Then, in Step 2 above, the factor M  da (u) for a and b with a ∼ b, d a > 1, and d b = 1. Therefore, Step 2 should be modified to define these terms together. For any Y satisfying Y(C), we construct a desired family T in the following three steps: Step 1. Choose arbitrarily T (a) Substep 1. (4.14) where T (a)  (2) There is a (not unique) ring homomorphism There is another variation of Theorem 4.4. Let T × (C) (resp. Y × (C)) be the multiplicative subgroup of all the invertible elements of T(C) (resp. Y(C)). Clearly, T × (C) is generated by T  (1) There is a multiplicative group homomorphism .
(2) There is a (not unique) multiplicative group homomorphism

Restricted T and Y-systems
Here we introduce a series of reductions of the systems T(C) and Y(C) called the restricted T and Y-systems. The restricted T and Y-systems for the quantum affine algebras are important in application to various integrable models. We define integers t a (a ∈ I) by t a = t d a .
Proof . The calculation is formally the same as the one for Theorem 4.4. We have only to take care of the boundary term which formally appears in the right hand sides of (4.1) and (4.2) for m = t a ℓ − 1. Since , d a = 1, the right hand side of (5.1) is 1 under the boundary condition T

T and Y-systems from cluster algebras
In this section we introduce T and Y-systems associated with a class of cluster algebras [17,19] by generalizing some of the results in [19,29,10,36,32,11]. They include the restricted T and Y-systems of simply laced type in Section 5 as special cases.

Systems T(B) and Y ± (B)
We warn the reader that the matrix B in this section is different from the one in Section 2 and should not be confused. Definition 6.1 ([16]). An integer matrix B = (B ij ) i,j∈I is skew-symmetrizable if there is a diagonal matrix D = diag(d i ) i∈I with d i ∈ N such that DB is skew-symmetric. For a skewsymmetrizable matrix B and k ∈ I, another matrix B ′ = µ k (B), called the mutation of B at k, is defined by The matrix µ k (B) is also skew-symmetrizable. The matrix mutation plays a central role in the theory of cluster algebras.
We impose the following conditions on a skew-symmetrizable matrix B: The index set I admits the decomposition I = I + ⊔ I − such that if B ij = 0, then (i, j) ∈ I + × I − or (i, j) ∈ I − × I + . (6.2) Furthermore, for composed mutations µ + = i∈I + µ i and µ − = i∈I − µ i , Note that µ ± (B) does not depend on the order of the product due to (6.2).
Lemma 6.2. Under the condition (6.2), the condition (6.3) is equivalent to the following one: For any i, j ∈ I + , The same holds for i, j ∈ I − .
For Y-systems, it is natural to introduce two kinds of systems.  Theorem 6.6. Let R be any commutative ring over Z with identity element. For any family Then, Y satisfies Y + (B). Similarly, define a family Y by Proof . By Remark 6.5, it is enough to prove the first statement only. Then, Note that for j in the right hand side of (6.5), j ∈ I ∓ by (6.2). By putting (6.6)-(6.8) into (6.5), the right hand side of (6.5) is which is the left hand side of (6.5).

Examples
Let us present some examples of T(B) and Y ± (B).
Definition 6.7. A symmetrizable generalized Cartan matrix C = (C ij ) i,j∈I is said to be bipartite if the index set I admits the decomposition I = I + ⊔ I − such that if C ij < 0, then (i, j) ∈ I + × I − or (i, j) ∈ I − × I + .
Example 6.8 ( [17,19]). Let C be a bipartite symmetrizable generalized Cartan matrix, which is not necessarily tamely laced. Define the matrix B = B(C) by otherwise. (6.9) The rule (6.9) is visualized in the diagram: Then, B is skew-symmetrizable and satisfies the conditions (6.2) and (6.3). The corresponding T(B) and Y − (B) are given by where j ∼ i means C ji < 0. These systems are studied in [17,19]. When C is bipartite and simply laced, they coincide with T 2 (C) and Y 2 (C) (for U = Z) in Section 5. When C is bipartite, tamely laced, but nonsimply laced, they are different from T 2 (C) and Y 2 (C), because the latter include factors depending on u + α (α = 0) in the right hand sides. Example 6.9 (Square product [29,10,36,32,11]). Let C = (C ij ) i,j∈I and C ′ = (C ′ i ′ j ′ ) i ′ ,j ′ ∈I ′ be a pair of bipartite symmetrizable generalized Cartan matrices with I = I + ⊔ I − and I ′ = I ′ + ⊔ I ′ − , which are not necessarily tamely laced. For i = (i, i ′ ) ∈ I × I ′ , let us write i : The rule (6.10) is visualized in the diagram: Since it generalizes the square product of quivers by [36], we call the matrix B the square product B(C) B(C ′ ) of the matrices B(C) and B(C ′ ) of (6.9). Proof . Let diag(d i ) i∈I and diag(d ′ i ) i∈I ′ be the diagonal matrices skew-symmetrizing C and C ′ , respectively, and let D = diag(d i d ′ i ′ ) (i,i ′ )∈I×I ′ . Then, the matrix DB is skew-symmetric. The condition (6.2) is clear from (6.11). To show (6.4), suppose, for example, that i = (i, i ′ ) : (++) and j = (j, j ′ ) : (−−). Then, B ik B kj = 0 only for k = (i, j ′ ) or k = (j, i ′ ); furthermore, B ik , B kj ≥ 0 (resp. ≤ 0) for k = (i, j ′ ) (resp. k = (j, i ′ )), and B ik B kj = C ij C ′ i ′ j ′ for both. Thus, (6.4) holds. The other cases are similar.
The corresponding T(B) and Y + (B) are given by where j ∼ i and j ′ ∼ i ′ means C ji < 0 and C ′ j ′ i ′ < 0, respectively. These systems slightly generalize the ones studied in connection with cluster algebras [29,10,36,32,11]. When C is bipartite and simply laced, and C ′ is the Cartan matrix of type A ℓ−1 with I ′ + = {1, 3, . . . } and I ′ − = {2, 4, . . . }, T(B) and Y + (B) coincide with T ℓ (C) and Y ℓ (C) in Section 5. (The choice of I ′ ± is not essential here.) As in Example 6.8, when C is bipartite, tamely laced, but nonsimply laced, and C ′ is the Cartan matrix of type A ℓ−1 , they are different from T ℓ (C) and Y ℓ (C).
Example 6.11. Let us give an example which does not belong to the classes in Examples 6.8 and 6.9. Let B = (B ij ) i,j∈I with I = {1, . . . , 7} be the skew-symmetric matrix whose positive components are given by The matrix B is represented by the following quiver: With I + = {2, 3} and I − = {1, 4, 5, 6, 7}, the matrix B satisfies the conditions (6.2) and (6.3).

T(B) and Y ± (B) as relations in cluster algebras
The systems T(B) and Y ± (B) arise as relations for cluster variables and coefficients, respectively, in the cluster algebra associated with B. See [19,36] for definitions and information for cluster algebras.

T(B) and cluster algebras
We start from T-systems. Let ε : I → {+, −} be the sign function defined by ε(i) = ε for i ∈ I ε . For (i, u) ∈ I × Z, we set the 'parity conditions' P + and P − by where we identify + and − with 1 and −1, respectively. For ε ∈ {+, −}, define T • (B) ε to be the subring of T • (B) generated by those T i (u) with (i, u) satisfying P ε . Then, we have Let A(B, x) be the cluster algebra with trivial coefficients, where (B, x) is the initial seed [19]. We set x(0) = x and define clusters x(u) = (x i (u)) i∈I (u ∈ Z) by the sequence of mutations (6.12) The ring A T (B, x) is no longer a cluster algebra in general, because it is not closed under mutations.

Y ε (B) and cluster algebras
We present a parallel result for Y-systems.
A semifield (P, +) is an abelian multiplicative group P endowed with a binary operation of addition + which is commutative, associative, and distributive with respect to the multiplication in P [19,31]. (Here we use the symbol + instead of ⊕ in [19] to make the description a little simpler.) Definition 6.16. For ε ∈ {+, −} and a skew-symmetrizable matrix B satisfying the conditions (6.2) and (6.3), letỸ ε (B) be the semifield with generators Y i (u) (i ∈ I, u ∈ Z) and the relations Y ε (B). LetỸ • ε (B) be the multiplicative subgroup ofỸ ε (B) generated by Y i (u) and 1 + Y i (u) (i ∈ I, u ∈ Z). (We use the notationỸ to distinguish it from the ring Y in Definition 4.5.) Let A(B, x, y) be the cluster algebra with coefficients in the universal semifield Q sf (y), where (B, x, y) is the initial seed [19]. To make the setting parallel to T-systems, we introduce the coefficient group G(B, y) associated with A(B, x, y), which is the multiplicative subgroup of the semifield Q sf (y) generated by all the elements y ′ i of coefficient tuples of A(B, x, y) together with 1 + y ′ i . We set x(0) = x, y(0) = y and define clusters x(u) = (x i (u)) i∈I and coefficient tuples y(u) = (y i (u)) i∈I (u ∈ Z) by the sequence of mutations (6.13) Definition 6.17. The Y-subgroup G Y (B, y) of G(B, y) associated with the sequence (6.13) is the multiplicative subgroup of G(B, y) generated by y i (u) and 1 + y i (u) (i ∈ I, u ∈ Z).
Proof . This follows from the exchange relation of a coefficient tuple y by the mutation µ k [19]: Theorem 6.19. The groupỸ • Proof . Let us show thatỸ • + (B) + ≃ G Y (B, y). Let f : Q sf (y) →Ỹ + (B) be the semifield homomorphism defined by Then, due to Lemma 6.18 (2), it can be shown by induction on ±u that we have f : y i (u) → Y i (u) for any (i, u) satisfying P + , and f : y i (u) → Y i (u − 1) −1 for any (i, u) satisfying P − . By the restriction of f , we have a multiplicative group homomorphism f ′ : G Y (B, y) →Ỹ • + (B) + . On the other hand, by Lemma 6.18 (2) again, a semifield homomorphism g :Ỹ + (B) → Q sf (y) is defined by Y i (u) → y i (u) ±1 for (i, u) satisfying P ± . By the restriction of g, we have a multiplicative group homomorphism g ′ :Ỹ • y). Then, f ′ and g ′ are the inverse to each other by Lemma 6.18 (1). Therefore,Ỹ • y). The other cases are similar.

Restricted T and Y-systems and cluster algebras: simply laced case
The restricted T and Y-systems, T ℓ (C) and Y ℓ (C), introduced in Section 5 are special cases of T (B) and Y ± (B), if C is simply laced. Therefore, they are also related to cluster algebras.

Bipartite case
Suppose that C is a simply laced and bipartite generalized Cartan matrix. Then, we have already seen in Examples 6.8 and 6.9 that T ℓ (C) and Y ℓ (C) coincides with T (B) and Y ε (B) for some B and ε. Therefore, we immediately obtain the following results as special cases of Theorems 6.15 and 6.19.   The slight discrepancy of the signs between ℓ = 2 and ℓ ≥ 3 is due to the convention adopted here and not an essential problem.

Nonbipartite case
Let us extend Corollaries 6.20 and 6.21 to a simply laced and nonbipartite generalized Cartan matrix C. The Cartan matrix of type A (1) 2r is such an example. In general, a generalized Cartan matrix C is bipartite if and only if there is no odd cycle in the corresponding Dynkin diagram. Without loss of generality we can assume that C is indecomposable; namely, the corresponding Dynkin diagram is connected. Definition 6.22. Let C = (C ij ) i,j∈I be a simply laced, nonbipartite, and indecomposable generalized Cartan matrix. We introduce an index set I # = I # + ⊔ I # − , where I # + = {i + } i∈I and I # − = {i − } i∈I , and define a matrix C # = (C # αβ ) α,β∈I # by 2, α = β, C ij , (α, β) = (i + , j − ) or (i − , j + ), 0, otherwise.
We call C # the bipartite double of C.
It is clear that C # is a simply laced and indecomposable generalized Cartan matrix; furthermore, it is bipartite with I # = I # + ⊔ I # − . Example 6.23. Let C be the Cartan matrix corresponding to the Dynkin diagram in the left hand side below. Then, C # is the Cartan matrix corresponding to the Dynkin diagram in the right hand side.
Here is another example. Proposition 6.24. Let C = (C ij ) i,j∈I be a simply laced, nonbipartite, and indecomposable generalized Cartan matrix, and C # be its bipartite double.
(2) For ℓ ≥ 3, T • ℓ (C) is isomorphic to A T (B, x) with B = B(C # ) B(C ′ ) by the correspondence T Corollary 6.26. Let C and C # be the same ones as in Proposition 6.24.

Concluding remarks
One can further extend Corollaries 6.20, 6.21, 6.25, and 6.26 to the tamely laced and nonsimply laced case by introducing T and Y-systems associated with another class of cluster algebras 1 . Therefore, we conclude that all the restricted T and Y-systems associated with tamely laced generalized Cartan matrices introduced in Section 5 are identified with the T and Y-systems associated with a certain class of cluster algebras.
The following question is left as an important problem: What are the T and Y-systems associated with nontamely laced symmetrizable generalized Cartan matrices?