Quantum dilogarithm identities arising from the product formula for universal R-matrix of quantum affine algebras

In arXiv:0912.1346, four quantum dilogarithm identities containing infinitely many factors are proposed as wall-crossing formula for refined BPS invariant. We give algebraic proof of these identities using the formula for universal R-matrix of quantum affine algebra developed by K. Ito, which yields various product presentation of universal R-matrix by choosing various convex orders on affine root system. By the uniqueness of universal R-matrix and appropriate degeneration, we can construct various quantum dilogarithm identities including the ones proposed in arXiv:0912.1346, which turn out to correspond to convex orders of multiple row type.


§1. Introduction
Dimofte, Gukov, and Soibelman proposed four remarkable identities with respect to quantum dilogarithm functions as the wall-crossing formulas for the refined BPS invariants, which they proposed in the study of type II string theory [2]. In [2], it is observed that the refined BPS invariants has very similar wall-crossing behavior to that of motivic Donaldson-Thomas invariants introduced by Kontsevich and Soibelman [10], and it is conjectured that the two invariants coincide under appropriate identification of variables.
Then the identities they found are written down as following [3]. Note that the parameter q in this paper corresponds to q 1/2 in [3]. x n n(1 − q n ) and (1 − q)Li 2,q (x) degenerates to classical dilogarithm by q → 1.
These identities, however, are derived by physical insight, and mathematically rigorous proofs for them have not been given. In this paper, we develop algebraic construction of these identities, which eventually yields mathematical proof of them as equalities of skew formal power series.
In [9], Kashaev and Nakanishi established systematic construction of quantum dilogarithm identities from periods of quantum cluster algebras. Their identities, however, involve only finitely many factors, while the four identities (1.2), (1.3), (1.4), (1.5) contain infinite product. Thus, these identities belong to essentially new class of quantum dilogarithm identities.
On the other hand, K. Ito constructed the product formulas for the (quasi-) universal R-matrix of quantum affine algebra, which correspond to convex orders on affine root system [6]. In the formulas, the factors corresponding to real roots are q-exponential function, which is in fact written as exp q (x) = E((q − q −1 )x). The resemblance between the wall-crossing formulas and product formulas for the universal R-matrix implies existence of connection between wall-crossing formulas and quantum groups.
By this observation, we develop systematic construction of quantum dilogarithm identities containing infinite product, using the product formula for the universal R-matrix. As a result, we show that all four identities Dimofte et al. found can be derived algebraically by our method. In section 2, we review general construction of convex orders on affine root systems, concrete construction of PBW type bases for the positive part U + q of quantum affine algebra U q (g) using convex order, and the explicit product formula for the quasi-universal R-matrix of U q (g).
In section 4, we show how to construct quantum dilogarithm identities using quasi-universal R-matrix Θ of U q (g). By virtue of the uniqueness of Θ, we can equate all the product presentations of Θ associated with convex orders. Thus we have infinite product identities whose parts corresponding to real roots are q-exponential function of root vectors. Next, we construct continuous projection of the completed quantum double algebra U + q ⊗U − q , which contains Θ, onto skew formal power series algebra D Q associated with affine Dynkin quiver Q. By this projection, some root vectors vanish and thus their q-exponential become 1 in the image. If one choose appropriate convex order and Dynkin quiver Q, one can make infinitely many root vectors not to vanish for the convex order, while only finitely many root vectors alive in the image for reversed convex order. Eventually one can obtain various quantum dilogarithm identities of the form "finite product = infinite product".
To obtain concrete identities, we have to compute the root vectors explicitly to determine whether they vanish by the projection. In section 3, we show that every root vector can be written as "q-commutator monomial", which is finite application of q-bracket on the Chevalley generators. We also developed combinatorial algorithm for the computation of root vectors, which enable us to obtain concrete presentations of root vectors as q-commutator monomials. The computation is done as manipulations of binary trees.
We concretely found appropriate convex orders and Dynkin quivers which produce identical identities with (1.2) to (1.5), which will be explicitly presented in section 5. It is remarkable that the factor of Θ correspond to imaginary roots becomes q-exponential function by the projection, in spite of the factor itself is not q-exponential function. We also note that the convex orders correspond to (1.3), (1.4), and (1.5) are of multiple row type, which was newly found by Ito [4]. §2. Product formula for the universal R-matrix of quantum affine algebras First we summarize Ito's works [4] [6] which provide explicit product presentations of the (quasi-) universal R-matrix of quantum affine algebras. §2.1. Quantum algebra U q (g) To begin with, we recall quantum enveloping algebra U q (g) corresponding to symmetrizable Kac-Moody algebra g of rank + 1, where q is an indeterminate (thus we work on generic case). We use following notations as in [8]. We also use the symbol R ± := R ∩ ∆ ± for every R ⊂ ∆.
∆ : U q (g) → U q (g) ⊗ U q (g) and ε : U q (g) → Q(q) are uniquely extended as algebra homomorphisms, and S : U q (g) → U q (g) is also extended as an anti-automorphism. Several subalgebras of U q (g) generated by standard generators are defined as usual: Then we have triangular decomposition of U q (g) [11, 3.2.5].
x (∀λ ∈ P ) be the weight space of weight µ ∈ P . For convenience, let V µ := V ∩ U µ for every subspace V ⊂ U q (g).
Then we also have the weight space decomposition Zα i ⊂ P : root lattice), and U q (g) becomes a Q-graded algebra. Using this gradation, we introduce qbracket [·, ·] q , which is defined on each weight spaces as follows: Convex orders on affine root system Next, we introduce the definition and classification of convex orders on the set of positive roots [4]. We also prepare a number of notations on affine root systems. (1) For any pair of positive real roots β, γ ∈ B∩∆ re + satisfying β < γ and β+γ ∈ B, the order relation β < β + γ < γ holds.
When g is of untwisted affine type, convex orders on ∆ + are already classified by Ito [4]. To describe convex orders in general, we have to introduce a numerous amount of symbols on affine root systems. In the rest of this section, we restrict g to be an untwisted affine Lie algebra of type X (1) , where X is one of A, B, C, D, E, F, G and is a positive integer. We assign indices 0, 1, . . . , for each vertex of the Dynkin diagram corresponds to g as in [7] so that the full subdiagram without the vertex 0 is of finite X type. First, letI := {1, 2, . . . , } be the set of indices other than 0, andg ⊂ g be the Lie subalgebra generated by e i , f i ,α i i ∈I . Theng is isomorphic to the simple Lie algebra of type X due to our assignment of indices, andh := i∈I Cα i ⊂ h is a Cartan subalgebra ofg. Let∆ ⊂h * be the set of all roots ofg, andW = s i i ∈I ⊂ W be the finite Weyl group.
number m is smallest among all the expressions of w as finite product of elements in S J , and the smallest number m is called length of w. Let J (w) denote the length of w. An infinite sequence of elements u 1 , u 2 , . . . in S J is called infinite reduced word when J (u 1 u 2 . . . u m ) = m for all positive integer m. The set of all infinite reduced words is denoted by W ∞ J , and the k-th factor of s ∈ W ∞ J is denoted by s(k) ∈ S J . We also use a function on positive integers φ s : We associate each s ∈ W ∞ J with a infinite set of positive roots Now, we can state the general description of convex orders. To begin with, we pick an element w ∈W . Then we have the decomposition Note that ∆(w, +) = ∆(ww • , −) with the longest element w • ∈W , since w • reverses the sign of every root in∆. Thus the set of positive real roots consists of two sets of the form ∆(w, −). We will construct convex orders on ∆(w, −) and connect them to construct whole order. Convex orders on ∆(w, −) are constructed by the following procedure.
(1) Select a positive integer n and a filtration of indices I = J 0 J 1 J 2 · · · J n = ∅.
· · · ∇(J n , w Jn , y n ) = ∆(w, −), (3) Then every root α ∈ ∆(w, −) can be uniquely written as Using this expression we define a total order ≤ on ∆(w, −) by Then ≤ is well-ordered and its ordinal number is nω, so that this well-order ≤ is called n-low type.
Using this procedure, we construct two convex orders ≤ − , ≤ + on ∆(w, −), ∆(w, +) = ∆(ww • , −) respectively. The parameters used in the procedure can be chosen independently between ≤ − and ≤ + . We also set a total order ≤ 0 on ∆ im + arbitrarily. Then we define a total order ≤ on whole ∆ + as follows.
Notice that ≤ + needs to be reversed, and therefore whole ≤ is not well-ordered.
Theorem 2.5. [4, Theorem 7.9, Corollary 7.10] The total order ≤ on ∆ + constructed above is convex, and any convex order on ∆ + can be constructed by the above procedure. §2.3. Convex bases of U + q constructed by convex orders When g is of finite type, it is known that U + q has canonical bases, which can be described concretely by using braid group action on U q (g) and correspond to each reduced expressions of longest element w • of Weyl group W [11]. In the affine type case, however, a couple of difficulties arise to construct basis of U + q due to the absence of longest element of W and the existence of imaginary roots. These problems are solved by constructing certain elements correspond to imaginary roots, using extended braid group action on U q (g), which is proposed by Beck [1]. Then Ito generalized this construction to general convex orders [6]. In this subsection, we summarize the construction of PBW type bases of U + q from convex orders. We first introduce the notion of convex basis, which is a PBW type basis with convexity property. Definition 2.6. Let U be a Q(q)-algebra, Λ ⊂ U be a subset, and ≤ be a total order on Λ. For every subset Σ ⊂ Λ, the set of increasing monomials consist of the elements in Σ is denoted by We call a subset I ⊂ Λ interval if I = Λ, or I coincide with one of (x, * ), [x, * ), ( * , y), ( * , y], (x, y), [x, y), (x, y], [x, y] for some x, y ∈ Λ, where (x, E < (Λ) is called convex basis of U if it has following properties: (1) E < (Λ) is a basis of U as Q(q)-linear space.
(2) For every interval I ⊂ Λ with respect to given order ≤, let U I denote the Q(q)subalgebra of U generated by I. Then E < (I) is a basis of U I as Q(q)-linear space.
It is known that one can construct convex bases for quantum algebra U q (g) by using the braid group action on U q (g), which is given explicitly by the following fundamental result.
Recall that the braid group B associated with the Weyl group W is defined by generators T i and relation (2.19). It is well-known that B has the following property with respect to reduced expressions in W .
is well-defined and a section of canonical surjection π : B → W ; T i → s i .
Thus we can define the action of w ∈ W on U q (g) by T w := T f (w) , where T b denotes the action of b ∈ B on U q (g).
When g is of finite type, set X q := T i1 T i2 . . . T iq−1 (E iq ) (q = 1, 2, . . . , N ) where w • = s i1 s i2 . . . s i N is a reduced expression of the longest element. Then it is known that increasing monomials X k1 1 X k2 2 . . . X k N N constitutes a convex basis of U + q . X q has weight β q := s i1 s i2 . . . s iq−1 (α iq ) and called a root vector associated to the root β q . Root vectors depend on reduced expression of w • .
Reduced expression of w • = s i1 s i2 . . . s i N induces a convex order β 1 < β 2 < · · · < β N . This is because if α i k + s i k s i k+1 . . . s i l−1 (α i l ) = s i k . . . s im−1 (α im ) and suppose that l < m, then applying s im−1 s im−2 . . . s i k both sides yields α im ∈ ∆ − , which is absurd. Thus the given order has the convexity property. Conversely, let w ∈ W and β 1 < β 2 < · · · < β k be a convex order on Φ(w) := w∆ − ∩ ∆ + . Then β 1 must be a simple root α i1 . To see this, we suppose that β 1 is not simple. Then β 1 can be written as a sum of two positive roots, and at least one of them belong to Φ(w) due to the biconvexity of Φ(w). This contradicts the minimality of β 1 , and now we conclude β 1 = α i1 . Since the action of W preserve the addition of roots, s i1 (β 2 ) < s i1 (β 3 ) < · · · < s i1 (β N ) is a convex order on Φ(s i1 w). By induction on the length of w, we can construct a reduced expression of w from given convex order. Therefore, each convex order on ∆ + generates a reduced expression of w • . These correspondences between convex orders and reduced expression of w • is clearly inverses of each other, the correspondences are one-to-one.
Using the correspondences above, we want to construct bases for U + q from convex orders on ∆ + . To extend the construction for affine case, we have to deal with several problems such as the definition of root vectors when given convex order has multiple lows, existence of imaginary roots, which are unreachable from simple roots by only using the braid group action. These problems have already been solved by Beck [1] and Ito [6].
Before introducing their construction, we need to extend the affine Weyl group properly. We now return to consider the case when g is untwisted affine Lie algebra of type X (1) . The linear map t λ ∈ End h * , called the translation by λ ∈h * , is defined by where δ ∈ ∆ im + is the null root. Let T := t ν(α) α ∈Q be the group of translations, where ν : h → h * is the canonical isometry andQ ⊂ h is the coroot lattice. Then it is well-known that W =W T [8]. Note that in general T does not contain translation t εi by fundamental coweight ε i ∈h * , which is characterized by (ε i , α j ) = δ ij for i, j = 1, . . . , . We extend the affine Weyl group W by appending the translations t εi . Let W denote the subgroup of GL(h * ) generated by W and t εi | h * (i ∈I), where h * := i=0 Cα i ⊂ h * . In fact, the extended Weyl group W coincides with a semidirect product of W and a subgroup of the Dynkin automorphism group. where ρI j ∈ Ω acts on W by ρI j .s i := s ρ(i) .
We define the length of w ∈ W by where we use the decomposition w = ρu (ρ ∈ Ω, u ∈ W ) given by (2.22). Recall that W = WI . We can also consider reduced expressions of w ∈ W . Namely, we call an expression w = t 1 t 2 . . . t m ∈ W , t i ∈ S Ω reduced if the sequence of integer (t 1 ), (t 1 t 2 ), . . . , (w) is increasing. Thus every element of Ω has length 0, and reduced expressions of w ∈ W may have different number of factors but the number of factors which belong to S must coincide with the length of w. Dynkin automorphism ρ acts on the subalgebra U q (g) := E i , F i , K ±1 αi ⊂ U q (g) as algebra automorphism by permuting indices: Thus we have an action of the extended braid group B := Ω B on U q (g) by extending the braid group action of Theorem 2.7. Proposition 2.8 also holds for W and B, and therefore every w ∈ W has an action T w on U q (g).
We will define the root vectors associated to real roots by lifting the expression (2.14) to quantum algebra U q (g), in which process simple reflection s i is replaced by T i and simple root α i is replaced by E i . In this lifting process, we also have to specify appropriate alternatives for δ − θ Jc ∈ Π Jc and s δ−θ Jc ∈ S Jc , where J c ⊂I is a connected subdiagram. The simple root vector E δ−θ Jc is in fact uniquely determined due to the following lemma.
The appropriate alternative for s δ−θ Jc is given by somewhat technical manner.
Definition 2.11. [6, Definition 3.4] First, we fix an index j c ∈ J c satisfying (ε jc , θ Jc ) = 1 for every nonempty connected subdiagram J c ⊂I. Then we define a map · : S J → W by wherej c ∈I is the unique index which satisfies w • (αj c ) = −α jc . We also define the extended map · : W J → W simply by w := t 1 t 2 . . . t m when w = t 1 t 2 . . . t m is a reduced expression in W J . Now, we can describe the construction of root vectors for affine case.
14) of a positive real root α determined by ≤. Then root vector E ≤,α ∈ U α associated to α is defined by , where E si := E i , and Ψ : The root vectors for imaginary roots is constructed using the action of extended braid group, which contains coweight lattice [1]. Since each imaginary root has multiplicity in affine Lie algebra g of type X (1) , we will construct as many number of root vectors as the multiplicity. The construction is rather technical and we proceed step-by-step. First, we introduce weight vectors E nδ−αi (i ∈I), which is independent of convex order.
where T εi := T tε i ∈ Aut U q (g) was defined via the extended braid group action and lifting a reduced expression of t εi ∈ W to B by (2.20). Then we set Despite of these ϕ i,n have weight nδ ∈ ∆ im + , ϕ i,n are not yet suitable for imaginary root vectors. The genuine imaginary root vectors are constructed by modifying ϕ i,n through the following technical procedure. For every i ∈I, let be the generating function of ϕ i,n . U + q [[z]] has topological algebra structure by declaring that z is central and ] has z-adic topology. Then imaginary root vectors I i,n ∈ U + q are defined as the coefficient of the function where the logarithm is defined by log(1 + It is shown that the these root vectors constitute convex bases for positive part of quantum affine algebra U + q . Theorem 2.13. [6, Theorem 8.6] Let ≤ be a convex order on positive roots ∆ + of untwisted affine root system, and let w ∈W be the parameter determined by the decompositon (2.11) of ∆ + in accordance with the given convex order ≤. Let denote the set of root vectors constructed above, and we set the order on Λ by using given order ≤ and Then increasing monomials E < (Λ) constitute a convex basis of U + q .
Once a convex basis of U + q is constructed, we also obtain the one for U − q through Chevalley involution Ω : The convex bases for quantum affine algebra enable explicit construction of quasiuniversal R-matrix. By applying Drinfeld's quantum double construction, Ito obtained the product formula for the quasi-universal R-matrix [6]. Since the quasiuniversal R-matrix does not lie in the algebraic tensor product U q (g) ⊗ U q (g), we have to give appropriate topology on U q (g) ⊗ U q (g) and complete it.
that is, we only count the weight of positive part with respect to the triangular decomposition (2.7). Then we set a topology which is generated by the subsets of the form In short, we give U q (g) ⊗ U q (g) linear topology. Let (2) Θ 0 = 1 ⊗ 1, The uniqueness of Θ will be the core of the proof of identites. Finally, we introduce the product formula for quasi-universal R-matrix.
Theorem 2.15. [5] [6] Let ≤ be a convex order on ∆ + of affine root system, and E ≤,α , I i,n denote the root vectors constructed above. For every i, j ∈I and positive integer n, let We also set .
Then the quasi-universal R-matrix Θ has the product presentation where > α∈∆+ X α means that if α < β, the order of multiplication is X β X α . In short, the order of multiplication is reverse to given convex order. §3. Explicit presentation of root vectors using q-bracket We will construct quantum dilogarithm identities by using various presentation (2.37) of quasi-universal R-matrix Θ, taking advantage of the uniqueness of Θ. However, to obtain specific identities, we have to calculate root vectors explicitly, which is described by braid group action (Theorem 2.7). In this section, we show that in general quantum algebra U q (g) of symmetrizable Kac-Moody algebra g, the element T w (E i ) ∈ U + q (w ∈ W ) can be written as "q-commutator monomial", that is, finite application of q-bracket on the generators E i . We also construct concrete algorithm for getting explicit presentation of T w (E i ) as q-commutator monomial, which enables us direct computation.
Let g be a symmetrizable Kac-Moody algebra of rank n.
We define subsets P k ⊂ U q (g) inductively by We call the elements of the form cM ∈ U q (g) for some c ∈ Q(q), M ∈ ∞ k=0 P k q-commutator monomial.
Our claim is that T w (E i ) ∈ U + q is q-commutator monomial for all i = 1, . . . , n and w ∈ W . To prove it, several formulas have to be prepared. First we recall by definition of the braid group action, and since the Weyl group action preserves the invariant bilinear form. The basic process of calculation for . . T im from the tail using (3.1) and (3.2). However, there is a problem that T k (E k ) = −F k K k may be appeared in the process of expansion. To resolve it, we use the following formula.
Lemma 3.2. For every 1 ≤ i = j ≤ n and positive integer m, Proof. By defining relation of U q (g), Thus the commutation relations of F : Now, we begin the proof by induction on m. Suppose that (3.3) holds for some positive integer m. Let Then by the induction hypothesis, Using these commutation relations, we obtain the equation for the case of m + 1.
Thus we obtain the recursion formula It is easy to verify C m := [m] qi [1 − a ij − m] qi satisfies this recurrence relation. Therefore, (3.3) holds for m + 1.
For the case of m = 1, above calculation works if one uses (3.4) in place of induction hypothesis and lets C 0 := 0.
Proof. Since length of reduced expression of the form s i s j s i s j . . . is at most 5 when finite type case, our task is just compute T w (E k ) directly for all cases. Using the formula (3.3), the computation is easily accomplished. For example, when a ij = a ji = −1 we have Thus we have a reduction formula For infinite type case, we use the following formulas, which can be verified by direct computation using (3.3).
Lemma 3.4. For indices i, j (i = j) and nonnegative integers p, k, let i,j;p satisfy the following recurrence relations.
Proof. The proof is by induction on (w). The case (w) = 1 is immediate by (3.1).
Take a reduced expression of w and let s i be its suffix. Then i = j due to the assumption. Let w {i,j} ∈ W be the shortest element satisfying Let F ij (i = j) denotes the set of q-commutator monomials consist of only E i and E j . We are going to prove that if s i,j;p (p ∈ Z ≥1 ) is a reduced expression, then E si,j;p := . . . T j T i (E j ) ∈ F ij by induction on p. The cases when p = 1, 2 are immediate by (3.1). When α i and α j span a finite root system, s i,j;p is reduced only for finitely many p ∈ Z ≥1 . Thus when (a ij , a ji ) = (−1, −1), (−1, −2), (−1, −3), (−2, −1), (−3, −1), we can verify E si,j;p ∈ F ij by direct computation since there exists only finitely many cases. The computation is easily accomplished using the formula (3.3). For example when (a ij , a ji ) = (−1, −1) and p = 3, we have When α i and α j span infinite root system, then a ij a ji ≥ 4. Now we suppose that p ≥ 2 and E si,j;r ∈ F ij for all r ≤ p. First, E si,j;p+1 can be written as follows.
Since E sj,i;p ∈ F ij , which is the induction hypothesis, we are reduced to verify V i,j;1 ∈ F ij using (3.3), and the recurrence formulas (3.10), (3.11) When a ij ≤ −4, a ji = −1, we need to continue the calculation of (3.12) slightly. Using the formula (3.3), we have Thus E si,j;p+1 can be written as By the proof of the theorem 3.5, we can easily construct an algorithm for describing T w (E j ) as a concrete q-commutator monomial once formulas for the elements of the form . . . T i T j T i (E j ) are prepared. In particular for simply laced case, we have a simple graphical algorithm for the calculation of T w (E j ), which we are going to describe below.
First, we introduce a graphical notation of q-bracket, which is convenient to write down q-commutator monomials.
We also abbreviate E i to i in the schematic notation. For example, the q-Serre relation (2.5) can be written as the following binary tree.
Using these notation, we can describe every q-commutator monomial as a binary tree, whose node represents q-bracket and whose leaf denotes a Chevalley generator E i . Now we can describe the algorithm for simply laced case.
Proposition 3.6. Let g be a simply laced Kac-Moody algebra, w ∈ W and j be an index satisfying w(α j ) ∈ ∆ + . Let w = s i1 s i2 . . . s im be a reduced expres-sion. Then the binary tree constructed by the following procedure represents a q-commutator monomial equal to T w (E j ).
(1) In this procedure, we manipulate a binary tree, each of whose leaf holds a pair of a reduced expression s j1 s j2 . . . s j k and an index p such that s j1 s j2 . . . s j k (α p ) ∈ ∆ + .
(2) At the beginning we have a binary tree consists of only the root, whose reduced expression is s i1 s i2 . . . s im and whose index is j. The procedure terminates immediately when m = 0.
(3) For each leaf of the binary tree, the following manipulations are applied recursively. Let s j1 s j2 . . . s j k and p be the reduced expression and the index of the leaf we are working on respectively.
(a) We are done for the leaf if k = 0.
(b) If k ≥ 1, then j k = p. If a j k p = 0, then delete the factor s j k in the reduced expression since T j k (E p ) = E p . Repeat this deletion until a j k p = −1.
(c) If s j1 s j2 . . . s j k−1 (α p ) ∈ ∆ − , then there exists a number l such that due to the exchange condition [8]. By a j k p = a pj k = −1 and (3.5), we have According to this calculation, replace the reduced expression with s j1 s j2 . . . s j l−1 s j l+1 . . . s j k−1 and replace the index with j k . Repeat this replacement until s j1 s j2 . . . s j k−1 (α p ) ∈ ∆ + .
(d) Finally, when s j1 s j2 . . . s j k−1 (α p ) ∈ ∆ + , then a j k p = a pj k = −1 by the manipulations so far. Thus T j k (E p ) = [E j k , E p ] q by (3.1), and s j1 s j2 . . . s j k−1 s j k , s j1 s j2 . . . s j k−1 s p are reduced. Therefore, create new branch at the current leaf and generate two leaves as below, where s := s j1 s j2 . . . s j k−1 . The new two leaves have index j k , p respectively, and both reduced expression is s . Figure 1 shows this branching procedure, where s[p] denotes reduced expression s and index p.
(4) Repeat above procedure until all reduced expressions in the leaves has length 0. This algorithm terminates within finite steps because each manipulation shortens the length of reduced expression of target leaf.
· · · Figure 1. Branching rule §4. Construction of quantum dilogarithm identities In this section, we show how to construct quantum dilogarithm identities using the product formula (2.37) of the quasi-universal R-matrix Θ. First, we introduce certain projections of the algebra U + q ⊗U − q onto skew formal power series algebras determined by Dynkin quivers. Through the projections, most of the elements of the form Θ ≤,α ∈ U + q ⊗U − q (α ∈ ∆ re + ) become the unit of image, while several factors survive and retain their form as q-exponential function, which can also be seen as quantum dilogarithm functions. Moreover, under appropriate setting of parameters, the product of factors in Θ associated with imaginary roots can be written using quantum dilogarithm functions in the image of the projection. Thus the image of Θ will be written as certain product of quantum dilogarithm functions. Choosing various convex orders, one can obtain various product presentations of the image of Θ, which have finitely or infinitely many factors depending on selected order. Eventually, we can construct quantum dilogarithm identities of the form "finite product = infinite product", which will exactly coincide with the identities proposed in [3] after suitable change of variables. §4.1. Projections of U + q ⊗U − q onto skew formal power series algebras Let g be symmetrisable Kac-Moody algebra of rank n and A = (a ij ) n i,j=1 be its Cartan matrix. Let d 1 , d 2 , . . . , d n be coprime positive integers such that c ij := d i a ij = d j a ji (1 ≤ i, j ≤ n). Then C := (c ij ) n i,j=1 is symmetrized matrix of A. We normalize the invariant bilinear form (·, ·) so that (α i , α j ) = c ij for 1 ≤ i, j ≤ n. Choose σ ij ∈ {±1} for each pair of indices i < j such that a ij = 0, and set Then the matrix B = (b ij ) n i,j=1 is skew-symmetric matrix, and this data can be interpreted as the Dynkin quiver which has an arrow from i to j if σ ij = +1. Let Let P B be a Q(q)-algebra defined by the generators and relations below.
P B has natural Q-graded algebra structure if each e i is supposed to have weight α i , hence q-bracket makes sense on P B . Each weight space of P B is one dimensional subspace spanned by a monomial of the form e k1 1 e k2 2 . . . e kn n . The degree of each monomial . . e kn n := k 1 + k 2 + · · · + k n coincides with the height of its weight. Let P B,m be the subspace of P B spanned by monomials of degree m. Also note that Due to the following well-known fact, P B turns out to be a quotient of U + q ⊂ U q (g). q is isomorphic to the Q(q)-algebra whose generators are E 1 , E 2 , . . . , E n and whose relation is given by the quantum Serre relation (2.5). Let P + B := P B and P − B be a copy of P B but whose generators e i are replaced by f i . Recall that U − q is isomorphic to U + q as algebra [11][3.2.6]. Let π + B := π B : are Q-graded algebra surjections given by the Proposition 4.2. Then we have an algebra surjection We want to construct a completion of this surjection to define image of Θ ∈ U + q ⊗U − q . To define the completion, we give a topology on D B := P + B ⊗ P − B so that the surjection π + B ⊗ π − B becomes continuous. For every nonnegative integer m, set Recalling the definition (2.34) of U q ⊗ U q , composition of the surjection π + B ⊗ π − B : U + q ⊗ U − q D B and inclusion ι : D B → D B is continuous with respect to the relative topology on U + q ⊗U − q ⊂ U q ⊗ U q . Hence this map induces an unique continuous map due to the completeness of D B . §4.2. Skew formal power series algebras be Q(q)-subalgebra generated by y 1 , y 2 , . . . , y n , and S B ⊂ D B be its closure. Since the increasing monomials form a topological basis of D B , the increasing monomials y m1 1 y m2 2 . . . y mn n form a topological basis of S B . Therefore S B is isomorphic to formal power series algebra Q(q)[[y 1 , y 2 , . . . , y n ]] as Q(q)-linear space. This isomorphism endows the Q(q)linear space Q(q)[[y 1 , y 2 , . . . , y n ]] with a complete topological Q(q)-algebra structure, whose multiplication is uniquely determined by the commutation relations y i y j = q 2bij y j y i .
In the same way, skew Laurent polynomial algebra L B can be defined. Namely, L B is Laurent polynomial algebra Q(q)[y ±1 1 , y ±2 2 , . . . , y ±1 n ] as Q(q)-linear space, and multiplication in L B is uniquely defined by the commutation relations y i y j = q 2bij y j y i . S B can be naturally considered as a subalgebra of L B .
Let L be the lower triangular part of B. Since B is skew-symmetric, B = L− t L. We define normal ordered product in L B by where y m := y m1 1 y m2 2 . . . y mn n . Let B = (b kl ) n k,l=1 ∈ M n (Z) be another skew-symmetric matrix. We shall consider algebra homomorphism ψ R : 1 , y ±1 2 , . . . , y ±1 n ] which is determined by a n × n-matrix R ∈ M n ,n (Z) and where v i ∈ Z n is the i-th unit vector. ψ R is well-defined if and only if it preserves the commutation relation y i y j = q 2bij y j y i for all i, j = 1, 2, . . . , n, in other words : y Rvi :: y Rvj : = q 2 t viBvj : y Rvj :: y Rvi : (i, j = 1, 2, . . . , n).
On the other hand, Thus ψ R is well-defined if and only if t v i t RB Rv j = t v i Bv j for all i, j. This shows  Moreover, ψ R preserves the normal ordered product.     In this paper we consider these functions just as formal power series. The function (x; q) ∞ is characterized by the recurrence relation Since −Li 2,q (qx) + log(1 − x) = −Li 2,q (x), exp(−Li 2,q (x)) satisfies the recurrence relation. Therefore which coincides with the product presentation of E(x) in the introduction. By this presentation, E(x) is characterized by the recurrence relation (4.20) (1 + qx)E(x) = E(q 2 x).
Recall that q-exponential function was defined by Then it can be directly verified that exp q (x) satisfies and we conclude that In the same way, we can also prove another presentation of (x; q) ∞ [13], which we will use in the computation of imaginary root vectors.
Computation of the image of quasi-universal R-matrix Θ Now, we suppose that g is untwisted affine Lie algebra. We shall compute π + B ⊗π − B (Θ) ∈ D B for various product presentation of Θ (2.37) and equate them to obtain concrete identities.
It is convenient to introduce the bilinear form α, β B := (α, β) − {α, β} B . Then the values of the bilinear form ·, · B are even integers, since The formula (4.25) shows that [X α , X β ] q vanishes if and only if α, β B = 0, otherwise it is nonzero multiple of X α X β . Therefore we have the following vanishing criteria for q-commutator monomials. if and only if there exists an application of q-bracket [E α , E β ] q in M for some E α , E β ∈ U + q of weight α, β ∈ Q satisfying α, β B = 0. If there are no such application of q-bracket in M , π + B (M ) ∈ P + B is a nonzero monomial.
Thus F ≤,α are also q-commutator monomials, and they coincide with E ≤,α except for multiple of ±q k and replacing E i with F i . There exists unique anti-isomorphism of Q-algebra Ω B : P + B → P − B which sends e i to f i and q to q −1 . This is useful to compute π − B (F ≤,α ) because (4.27) Since ht α − 1 times of q-brackets occur in E ≤,α for α ∈ ∆ re + , its image takes the form below.
where C ∈ Q(q) is the coefficient of E ≤,α as q-commutator monomial, α = n i=0 m i α i and u, k i ∈ Z. When simply-laced case, C = 1 because no non-trivial scalar multiple occur in the algorithm of section 3 (Proposition 3.6). We also note that each k i is a value of the bilinear form ·, · B and thus even integer. Using Ω B , the image of F ≤,α is (4.29) Recall that the subalgebra S B ⊂ D B , which is generated by Using these notations, when simply-laced case we have : y m0 0 y m1 1 . . . y mn n :.
While we have the general simple description of the images of real root vectors, the computation of the images of imaginary root vectors requires some ingenuity. Recall that ϕ i,n ∈ U + nδ were defined as and imaginary root vectors I i,n were polynomials consists of ϕ i,n . Since T n εi T −1 i (E i ) can be written as a q-commutator monomial by using the algorithm in section 3, ϕ i,n themselves are q-commutator monomials. But we need to compute T w (I i,n ) (w ∈W ) for general convex order, and we cannot apply the algorithm to T w (ϕ i,n ) when w(α i ) ∈ ∆ − because T w (E i ) no longer lies in U + q . First, we compute T i (ϕ i,n ) using the following fact.  T εj (ϕ i,n ) = ϕ i,n .
We will also use the property that for u, v ∈ W , Since t εi (α i ) = −δ+α i ∈ ∆ − , the length of u := t εi s i in W is (u) = (t εi )−1. Thus T εi = T u T i and we have Now we can compute arbitrary T w (ϕ i,n ) for w ∈W . When w(α i ) ∈ ∆ + , we have and thus simply applying the algorithm to T w T n−1 εi T u (E i ) and T w (E i ) yields explicit presentation of T w (ϕ i,n ) as a q-commutator monomial.
Recall the anti-automorphism of Q(q)-algebra Ψ : U + q → U + q defined by Ψ(E i ) := E i . Since Ψ preserves weights, it reverses q-bracket: It is also easy to verify that Finally we have We can apply the algorithm to Tw v E τ −1 (i) . Since Ψ just reverses the directions of q-brackets, ΨTw v E τ −1 (i) is a q-commutator monomial.
Proposition 4.9. T w (ϕ i,n ) is a q-commutator monomial for every w ∈W , i ∈I, and positive integer n. §5. Examples of quantum dilogarithm identities In this final section, we give specific convex orders and Dynkin quivers, which eventually induce the identities proposed in [3].
Recall that affine positive root system ∆ + is decomposed as and convex orders on ∆ + consists of convex orders on each ∆(w, ±) (the order on ∆ im + is not significant since any total order can be chosen). Convex order on ∆(w, −) was determined by the following parameters with several restrictions (2.12) (2.13): (1) A positive integer n and a filtration of indices I = J 0 J 1 J 2 · · · J n = ∅.
We have to specify not only the parameters for ∆(w, −), but also for ∆(w, +) = ∆(ww • , −) to construct whole convex order on ∆ + . In the examples below, let· denote the parameters for ∆(w, +). For instance,w = ww • .
where (s) ∞ := sss . . . denote infinite repetition of s. Then corresponding convex order ≤ turns out to coincide with (2.10). Next, we compute the root vectors from this convex order. Since A 1 is not simply-laced, we cannot use the algorithm of section 3. But the following formula is sufficient to accomplish the computation.
By definition of root vectors and the chosen order, one can verify for all n = 0, 1, 2, . . . .
Using the reduced expression t ε1 = ρs 1 , where ρ ∈ Ω is the transposition of 0 and 1, one can show that for any positive integer m, We set the projection of section 4.1 by σ 01 := +1. Then corresponding skewsymmetric matrix is B = 0 −2 2 0 , and matrix presentation of the bilinear form ·, · B is ( α i , α j B ) 1 i,j=0 = 2 0 −4 2 . Since α 0 , α 1 B = 0, the projection π + B : U + q → P + B annihilates [E 0 , E 1 ] q . Thus all the root vectors vanish in P + B except for simple root vectors E ≤,αi = E i (i = 0, 1). Therefore, the image of quasi-universal Rmatrix Θ is Beware that the order of product is reverse to given convex order (2.37). Now we consider the reversed order ≤ , which is in fact obtained by just swapping ∆(w, −) for ∆(w, +). The corresponding parameters are also just swapping every parameter · for·. Thus E ≤ ,α = ΨE ≤,α for every real root α ∈ ∆ re + . One can verify that all the real root vectors for ≤ satisfy the condition of Proposition 4.7 and thus do not vanish. Sincew = s 1 , T 1 (I 1,m ) (m ≥ 1) are used as imaginary root vectors. Using (5.3) we have Let D := (q − q −1 ) 2 e 0 e 1 . Then the image of generating function and therefore Let D := Ω B (D) = (q −1 − q) 2 f 1 f 0 . By virtue of (4.27) and ΩT i = T i Ω (i = 0, 1, . . . , ), we can compute as follows.
By (4.24), we obtain the image of Θ im := ∞ m=1 Θ mδ . Comparing with (5.4), we eventually obtain the following quantum dilogarithm identity, which was first found by Terasaki [13]. where : y m0 0 y m1 1 : = q 2m0m1 y m0 0 y m1 1 .  Remark. We shall call a group homomorphism Z : Q → C central charge. When Z is injective and Z(∆ + ) lies in the (closure of) upper half plane H := { z ∈ C | Im z ≥ 0 }, defines a convex order on positive real roots, where we choose principal value of argument so that 0 ≤ arg z < 2π.
Using the algorithm of Proposition 3.6 and notation (3.13), real root vectors in the first row of ∆(w, −) are computed as follows. This can be directly proven by induction on m, noting that , which means that applying T 0 T 1 T 2 on [E 0 , E 1 ] q adds 3 branches from right.
Proof. Firstly, we have to check the equality. It is enough to compare the action of both sides on simple roots since W ⊂ GL (h * ). Moreover, since every element of W fixes the null root δ = α 0 + α 1 + · · · + α , it is enough to check the action on α j for j = 1, 2, . . . , . On the one hand, t εi (α j ) = α j − δ ij δ. On the other hand, let R i := ρ −1 s 1 s 2 . . . s i . When = 1, R 1 (α 1 ) = ρ −1 (−α 1 ) = −α 0 = α 1 − δ and hence (5.17) holds. Next, we assume ≥ 2. Recall that for 1 ≤ i, j ≤ , in the root system of type A . By direct calculation we have When j = i, the case i = is the above formula. When i < , notice that R i (α i + α i+1 ) = α i . Using this inductively, To verify that R +1−i i is reduced expression, it is enough to show that the length of t εi ∈ W is i( + 1 − i). The length of t εi coincides with the number of positive roots which t εi sends to negative roots. Recall that in finite root system of type A . t εi translates the roots containing ±α i by ∓δ. Thus, if α = mδ +ε ∈ ∆ + (m ∈ Z ≥0 , ε ∈∆) satisfies t εi (α) ∈ ∆ − , then m = 0 and ε must be a positive root containing α i . Such ε takes the form α j + α j+1 + · · · + α k (j ≤ i ≤ k), and the number of such (j, k) is i( − i + 1). This shows that the length of t εi is i( + 1 − i). Now we can compute real root vectors in the second row. Since w J1 = s J1 1 = 1 and E δ−α1 = T 0 (E 1 ) = T ρ T 2 (E 1 ), Observing Moreover, one can deform this presentation to the form Ψ(T u ) m T v (E i ) by realizing . In the extended braid group B, Therefore we have As a result, we obtain The advantage of this presentation is that inductive computation becomes easy. In fact, and thus applying T 2 T ρ 2 T 1 to [E 2 , E 0 ] q adds 2 branches from right. By virtue of [E 2 , E 1 ] q and E 0 being invariant by T 2 T ρ 2 T 1 , finally we have (5.29) E ≤,mδ−α1 = 0 1 2 0 1 2 · · · 0 1 2 0 1 2 0 2 Next, we compute imaginary root vectors. Since w = s 1 , T 1 (I i,m ) (i = 1, 2; m ∈ Z ≥1 ) are used as imaginary root vectors. We use (4.34) to compute T 1 (ϕ 1,m ). Since t ε1 = ρs 2 s 1 = us 1 , u = ρs 2 . Thus The computation of T 1 (ϕ 2,m ) is more easy. By definition, and in the computation of E ≤,mδ−α1 we have already computed We also require Tw(ϕ i,m ) = T 2 T 1 (ϕ i,m ) for the reversed order ≤ . Fortunately, in this case we can simply apply T 2 on every leaf of the presentation of T 1 (ϕ 1,m ).
and [E 1 , E 2 ] q lie in the kernel of π + B . Examining the computed presentations of root vectors above, all the root vectors except for simple root vectors vanish by π + B . Therefore, the image of quasi-universal R-matrix Θ is Now we reverse the given order. Recall that the real root vectors for the reversed order ≤ are obtained just by reversing all the direction of q-bracket. Then one can verify that the real root vectors in the first row, namely, E ≤ ,mδ−α1 = Ψ(E ≤,mδ−α1 ), E ≤ ,mδ−α1−α2 = Ψ(E ≤,mδ−α1−α2 ) (m ≥ 1) satisfy the condition of Proposition 4.7.
Next, we have to compute the image of imaginary root vectors T 2 T 1 (I i,m ). Using (4.25) and the presentations (5.32) (5.33), we have The second equality is due to the following calculation. Recall that the commutation relations in P + B become e 1 e 0 = qe 0 e 1 , e 2 e 0 = qe 0 e 2 , e 2 e 1 = qe 1 e 2 . Thus Thus the image of imaginary root vectors are At last, we compute the image of S m := (T 2 T 1 ⊗ T 2 T 1 )(S m ) (m ≥ 1) in (2.36). By definition, and thus S m is written down as Therefore the image of S m is computed as follows.
This result coincides with the case of type A 1 . Therefore Comparing (5.34), finally we attain the following identity.
where the null root δ = α 0 + α 1 + α 2 + α 3 . Set (corresponding quiver: The matrix presentation of the bilinear form ·, · B is In the same way in the examples so far, one can verify that all the root vectors except for simple root vectors vanish by the projection π + B . On the other hand, real root vectors for the reversed order ≤ , namely, E ≤ ,mδ+α for α = ±α 1 , ±α 2 , ±α 3 ,  Although the computation of imaginary root vectors also has resemblance to the previous examples and in fact we will obtain identical presentation of π + B ⊗π − B (Θ im ), the process of computation is far from obvious. After somewhat lengthy computation (we used T −1 ε2 (E i ) = E i for i = 1, 3 in the process), one will obtain Thus the images of imaginary root vectors are π + B Tw(I 1,m ) = π + B Tw(I 3,m ) = where D := (q − q −1 ) 4 e 0 e 2 e 1 e 3 . Our last task is to compute the image of S m := (Tw ⊗ Tw)(S m ) (m ≥ 1). The definition (2.35) reads where M k := [km] q for k = 1, 2. Thus the image of S m is computed as follows. Then the corresponding convex order ≤ is as follows.