root of unity quantum cluster algebras and …

ROOT OF UNITY QUANTUM CLUSTER ALGEBRAS AND

DISCRIMINANTS

BACH NGUYEN, KURT TRAMPEL, AND MILEN YAKIMOV

Abstract. We describe a connection between the subjects of cluster algebras and dis-criminants. For this, we define the notion of root of unity quantum cluster algebras andprove that they are polynomial identity algebras. Inside each such algebra we construct a(large) canonical central subalgebra, which can be viewed as a far reaching generalizationof the central subalgebras of big quantum groups constructed by De Concini, Kac andProcesi and used in representation theory. Each such central subalgebra is proved to beisomorphic to the underlying classical cluster algebra of geometric type. When the rootof unity quantum cluster algebra is free over its central subalgebra, we prove that thediscriminant of the pair is a product of powers of the frozen variables times an integer.An extension of this result is also proved for the discriminants of all subalgebras generatedby the cluster variables of nerves in the exchange graph. These results can be used forthe effective computation of discriminants. As an application we prove an explicit formulafor the discriminant of the integral form over Z[ε] of each quantum uniponet cells of DeConcini, Kac and Procesi for arbitrary symmetrizable Kac–Moody algebras.

1. Introduction

1.1. Cluster algebras and discriminants. Cluster algebras were introduced by Fominand Zelevinsky in [12] and since then, they have played a fundamental role in a number ofdiverse areas such as representation theory, combinatorics, Poisson and algebraic geometry,mathematical physics, and others [15, 32].

Discriminants of number fields were defined by Dedekind in the 1870s. They have provenan invaluable tool in number theory, algebraic geometry, combinatorics, and orders in centralsimple algebras [21, 34, 35]. In more recent years, new applications of discriminants havebeen found in the noncommutative setting. Bell, Ceken, Palmieri, Wang and Zhang used thediscriminant as an invariant in determining the automorphism groups of certain polynomialidentity algebras [6, 7] and to address the the Zariski cancellation problem (when A[t] ' B[t]implies A ' B) [1]. Discriminant ideals are also intrinsically related to the representationtheory of the corresponding noncommutative algebra [5].

In this paper we connect the subjects of cluster algebras and discriminants. We define thenotion of root of unity quantum cluster algebra, show that these algebras are polynomialidentity (PI) algebras, and construct a canonical large central subalgebra in each of themwhich is shown to be isomorphic to the underlying classical cluster algebra. These specialcentral subalgebras can be viewed as far reaching generalizations of the De Concini–Kac–Procesi central subalgebras of big quantum groups [10, 11]. We prove a theorem giving anexplicit formula for the discriminant of a root of unity quantum cluster algebra, and applyit to compute the discriminants of the big quantum unipotent cells for all symmetrizableKac–Moody algebras at roots of unity.

2010 Mathematics Subject Classification. Primary: 13F60, Secondary: 16G30, 17B37, 14A22.Key words and phrases. Quantum cluster algebras at roots of unity, algebras with trace, discriminants,

Kac–Moody algebras, quantum unipotent cells.The research of B.N. was supported by an AMS-Simons Travel Grant. The research of M.Y. was supported

by NSF grant DMS-1901830 and Bulgarian Science Fund grant DN02/05.1

2 B. NGUYEN, K. TRAMPEL, AND M. YAKIMOV

1.2. Root of unity quantum cluster algebras. Let ε1/2 be a primitive `-th root of unityfor a positive integer `. We define a root of unity quantum cluster algebra by constructingmutations in the skew field of fraction of the based quantum torus over Z[ε1/2] with basis{Xf | f ∈ ZN} and relations

XfXg = εΛ(f,g)/2Xf+g, ∀f, g ∈ ZN

for a skew-symmetric bilinear form Λ : ZN × ZN → Z` := Z/`Z. Quantum frames Mε areintroduced in this setting as in the quantum setting of Berenstein and Zelevinsky [4], butweaker compatibility assumptions between the bilinear form Λ and the exchange matrix

B are imposed (Definition 3.2). In particular, B need no longer have a full rank as thequantum case in [4]. A subset inv of the exchange indices ex is allowed to be inverted and

the corresponding Z[ε1/2]-algebra generated by all cluster variables and the inverses of the

frozen ones in inv is denoted by Aε(Mε, B, inv) (Definition 3.7).In the special case of ` = 1 this construction exactly recovers the definition of a classical

cluster algebra of geometric type. Quantum Weyl algebras and quantum unipotent cellsat roots of unity for all symmetrizable Kac–Moody algebras are examples of root of unityquantum cluster algebras (Sect. 5.6 and 8.3). In addition to the standard properties ofclassical and quantum cluster algebras, such as the Laurent phenomenon, we prove the

following key results for the algebras Aε(Mε, B, inv):

Theorem A. Let ε1/2 be a primitive `-th root of unity for a positive integer `.

(1) All root of unity quantum cluster algebras Aε(Mε, B, inv) are polynomial identityalgebras (see [31, Ch. 13]).

(2) The subring of Aε(Mε, B, inv) generated by the `-th powers of all cluster variablesand the inverses of the `-th powers of the frozen ones in inv is in the center of

Aε(Mε, B, inv). If ` is odd and coprime to the entries of the symmetrizing diagonal

matrix for the principal part of B, this subring is isomorphic to the corresponding

classical cluster algebra A(B, inv).

(3) Under the assumption in part (2) the exchange graphs of Aε(Mε, B, inv) and A(B, inv)are canonically isomorphic.

Denote by

(1.1) Cε(Mε, B, inv)

the Z[ε1/2]-extension of the subring of Aε(Mε, B, inv) in part (2) of the theorem. In concrete

important situations Aε(Mε, B, inv) is module finite over Cε(Mε, B, inv) (Sect. 6.4 and 8.3).For quantum unipotent cells at roots of unity, the latter is proved to be precisely the specialDe Concini–Kac-Procesi subalgebra [11]. The punchline of part (2) of the theorem is thatit not only constructs a large central subalgebra in vast generality, but it also gives a fullcontrol on it via cluster theory. As an upshot, the representation theory of the algebras in[11] can be studied within the framework of root of unity and classical cluster algebras.

The proof of part (3) uses a different strategy from the Berenstein–Zelevinsky [4] resultfor the isomorphism between classical and quantum exchange graphs. It is based on thespecial central subalgebras from part (2).

Root of unity quantum cluster algebras do not necessarily arise as specializations ofquantum cluster algebras. For instance, in the case ` = 1 we recover all cluster algebrasof geometric type. For these reasons we introduce a subclass of strict root of unityquantum cluster algebras, defined as those for which the skew-symmetric bilinear formΛ : ZN × ZN → Z` comes from a skew-symmetric bilinear form ZN × ZN → Z which is

compatible with the exchange matrix B in the sense of [4]. In the case ` = 1, that notion

ROOT OF UNITY QUANTUM CLUSTER ALGEBRAS AND DISCRIMINANTS 3

is the same as the notion of a classical cluster algebra with a compatible Poisson structure

in the sense of Gekhtman–Shapiro–Vainshtein [22]. If the quantum cluster algebra for Bequals the corresponding upper quantum cluster algebra, then we prove that the root of

unity Aε(Mε, B, inv) arises as a specialization from a quantum cluster algebra (Sect. 5).This gives an effective tool for the construction of root of unity quantum cluster algebras(Sect. 5.6 and 8.3).

1.3. Discriminants. Knowing the explicit form of the discriminant of a noncommutativealgebra has a number of important applications, but its calculation is very difficult. Onlya few results are known to date and they concern concrete classes of algebras. Skew-polynomial algebras were treated in [6, 7], their Veronese subrings in [9], low dimensionArtin–Schelter regular algebras in [1, 37, 38], Ore extensions without skew-derivations andskew group extensions in [17], quantized Weyl algebras in [8, 29], Taft algebra smash prod-ucts in [18] and others. A Poisson geometric method for computing discriminants viadeformation theory was given in [33].

We prove the following general results for the computation of the discriminants of all rootof unity quantum cluster algebras over their special central subalgebras (1.1) arising fromTheorem A(2):

Theorem B. Let ε1/2 be a primitive `-th root of unity and Aε(Mε, B, inv) be a root ofunity quantum cluster algebra such that ` is odd and coprime to the entries of the skew-

symmetrizing diagonal matrix for the principal part of B. Let Θ be any collection of seedsthat is a nerve (in the sense of [16] and Definition 6.4) and Aε(Θ, inv), A(Θ, inv) (resp.

Cε(Θ, inv)) be the subalgebras of Aε(Mε, B, inv), A(B, inv) (resp. Cε(Mε, B, inv)) gener-ated by the cluster variables from the seeds in Θ (resp. their `-th powers).

(1) If Aε(Θ, inv) is a free module over Cε(Θ, inv), then Aε(Θ, inv) is a finite rankCε(Θ, inv)-module of rank `N and its discriminant with respect to the regular trace functionis given by

d (Aε(Θ, inv)/Cε(Θ, inv)) =Cε(Θ,inv)× `N`N

∏i∈[1,N ]\(extinv)

X`aii for some ai ∈ N,

where Xi denote the frozen variables of Aε(Θ, inv) and N := {0, 1, . . .}.(2) If Aε(Θ, inv) is a free module over A(Θ, inv), then Aε(Θ, inv) is a finite rank

A(Θ, inv)-module of rank `Nϕ(`) and its discriminant with respect to the regular trace func-tion is given by

d (Aε(Θ, inv)/A(Θ, inv)) =A(Θ,inv)×

( `(N+1)ϕ(`)∏p|` p

ϕ(`)/(p−1)

)`N ∏i∈[1,N ]\(extinv)

X`cii for some ci ∈ N,

where ϕ(.) denotes Euler’s ϕ-function.

In the theorem one can choose Θ to be the set of all seeds, which gives a formula for

the discriminant of Aε(Mε, B, inv) over Cε(Mε, B, inv). The choice of any nerve Θ inthe collection of all seeds allows for the extra flexibility in computing discriminants ofsubalgebras of root of unity quantum cluster algebras that do not have cluster structures ontheir own. The very specific form of the discriminant in the theorem makes the computationof the integers ai easy by degree and filtration arguments (see e.g. Sect. 8.5).

1.4. The De Concini–Kac–Procesi quantum unipotent cells. Many PI algebras aresecretly root of unity quantum cluster algebras or, more generally, algebras of the formAε(Θ, inv). Let g be an arbitrary symmetrizable Kac–Moody algebra and w a Weyl groupelement. In Theorem 8.4 we prove that this is the case for the integral forms over Z[ε] of


all big quantum unipotent cells Aε(n+(w))Z[ε] of De Concini–Kac–Procesi [11], namely that

Aε(n+(w))Z[ε] ⊗Z[ε] Z[ε1/2] admits the structure of a root of unity quantum cluster algebra.Using Theorem B, we prove:

Theorem C. For all symmetrizable Kac–Moody algebras g, Weyl group elements w andprimitive `-th roots of unity ε such that ` is odd and coprime to the symmetrizing inte-gers of the Cartan matrix of g, the discriminant of the integral form of the correspondingquantum unipotent cell Aε(n+(w))Z[ε] over its De Concini–Kac–Procesi central subalgebraCε(n+(w))Z[ε] with respect to the regular trace is given by

d(Aε(n+(w))Z[ε]/Cε(n+(w))Z[ε]

)=Z[ε]× `

(N`N )∏

i∈S(w)

D`N (`−1)$i,w$i ,

where S(w) is the support of w and D$i,w$i are the standard unipotent quantum minors inAε(n+(w))Z[ε] associated to the fundamental weights $i.

A special and weaker case of this theorem was proved in [33]. It only dealt with the case offinite dimensional simple Lie algebras g, due to the use of Poisson geometric results from[10, 11]. Furthermore, [33] only applied to the case of discriminants of algebras over C(ε)and not over Z[ε], because of the use of Poisson geometric techniques.

Notation, We will use the following notation throughout the paper. For a pair of integersj ≤ k, denote [j, k] := {j, j + 1, . . . , k}. For a pair of positive integers m,n, denote 0m×nthe zero matrix of size m× n.

Acknowledgements. We are grateful to Greg Muller for helpful correspondences and forcommunicating the proof of Lemma 6.5 to us.

2. Preliminaries on classical and quantum cluster algebras

In this section we gather background material on cluster algebras of geometric type andquantum cluster algebras which will be used in the rest of the paper.

2.1. Cluster algebras of geometric type. Cluster algebras were defined by Fomin andZelevinsky in [12]. Let N be a positive integer, ex be a subset of [1, N ], and F be a purely

transcendental extension of Q of transcendence degree N . A pair (x, B) is called a seed if

(1) x = {x1, . . . , xN} is a transcendence basis of F over Q which generates F ;

(2) B ∈MN×ex(Z) and its ex×ex submatrix B (called the principal part of B) is skew-symmetrizable; that is DB is skew-symmetric for a matrix D = diag(dj , j ∈ ex)with dj ∈ Z+.

We call B the exchange matrix of the seed, x the cluster of the seed, xi the cluster variables.

The subset ex ⊆ [1, N ] is called set of exchangeable indices. The columns of B are indexed

by this set. The mutation of B in direction k ∈ ex is given by

µk(B) = (b′ij) :=

{−bij if i = k or j = k

bij +|bik|bkj+bik|bkj |

2 otherwise.

Equivalently, µk(B) = EsBFs where s = ± is a sign and the matrices Es ∈ MN (Z),Fs ∈Mex(Z) are defined by

Es := (eij) =

δij if j 6= k

−1 if i = j = k

max(0,−sbik) if i 6= j = k,

Fs := (fij) =

δij if i 6= k

−1 if i = j = k

max(0, sbkj) if j 6= i = k.


The principal part of µk(B) is the mutation µk(B) of the principal part B of B and thematrix µk(B) is skew-symmetrizable with respect to the same diagonal matrix D that skew-

symmetrizes B, [12]. Mutation µk of the seed (x, B) in the direction of k ∈ ex is given by

µk(x, B) := (x′, µk(B)) where the mutation of x is given by

(2.1) x′ = {x′k} ∪ x\{xk} and xkx′k :=

∏bik>0

xbiki +∏bik<0

x−biki .

Mutation is an involution, µ2k = id, [12]. We say that two seeds (x′, B′), (x′′, B′′) are

mutation-equivalent if (x′′, B′′) can be obtained from (x′, B′) via a finite sequence of muta-

tions. Denote this by (x′, B′) ∼ (x′′, B′′). All seeds that are mutation-equivalent to (x, B)contain the cluster variables c := {xi | i ∈ [1, N ]\ex}, called the frozen variables.

The cluster algebra A(B) is defined as the Z[c±1]-subalgebra of F generated by all cluster

variables in the seeds (x′, B′) ∼ (x, B). For the purposes of applications to coordinate rings,instead of inverting all frozen variables, we often need to pick a subset inv ⊆ [1, N ]\ex to

invert. Then A(B, inv), denotes the Z[c, x−1k , k ∈ inv]-subalgebra generated by all cluster

variables in the seeds (x′, B′) ∼ (x, B). In particular, A(B) = A(B, [1, N ]\ex).

The upper cluster algebra U(B, inv) is the intersection of all mixed polynomial/Laurentpolynomial subrings

Z[x′1, . . . , x′N ][(x′i)

−1, i ∈ inv]

of F for the seeds ((x′1, . . . , x′N ), B′) ∼ (x, B). The Laurent phenomenon of Fomin–Zelevinsky

[13] established that A(B, inv) ⊆ U(B, inv).

2.2. Quantum cluster algebras. Quantum cluster algebras were defined by Berensteinand Zelevinsky in [4]. Let Λ : ZN × ZN → Z be a skew-symmetric bilinear form. By abuseof notation we will denote its matrix in the standard basis e1, . . . , eN of ZN by the samesymbol Λ = (Λ(ei, ej)), and we will use interchangeably both notions. The bilinear form is

uniquely reconstructed from this matrix. Using a formal variable q1/2, we work with theLaurent polynomial ring

(2.2) A1/2q := Z[q±1/2].

Definition 2.1. The based quantum torus Tq(Λ) associated with Λ is defined as the A1/2q -

algebra with a A1/2q -basis {Xf |f ∈ ZN } and multiplication given by

XfXg = qΛ(f,g)/2Xf+g, where f, g ∈ ZN .

The bilinear form Λ can be recovered from the commutation relations of the generatorsXe1 , . . . , XeN of Tq(Λ), because XfXg = qΛ(f,g)XgXf . We denote by F the skew-field

of fractions of Tq(Λ), which is a Q(q1/2)-algebra. Each σ ∈ GLN (Z) gives rise to thebased quantum torus Tq(Λ′) associated to the form Λ′(f, g) = Λ(σf, σg). Note that if we

consider Λ′ as a matrix, then Λ′ = σ>Λσ. Also, we have an A1/2q -algebra isomorphism

Ψσ : Tq(Λ)→ Tq(Λ′) given by Xf 7→ Xσ−1f .

Definition 2.2. Let Fq be a division algebra over Q(q1/2). A toric frame Mq for Fq isdefined as a map Mq : ZN → Fq for which there exists a skew-symmetric matrix Λ ∈MN (Z)satisfying:

(1) There is an A1/2q -algebra embedding φ : Tq(Λ) ↪→ Fq with φ(Xf ) = Mq(f) for all

f ∈ ZN .(2) Fq = Fract (φ(Tq(Λ))).


The skew-symmetric matrix associated to a toric frame Mq will be denoted by ΛMq . Forany σ ∈ GLN (Z), ρ ∈ Aut(Fq), and toric frame Mq, the map ρMqσ is a toric frame with

ΛρMqσ = σ>Λσ. The embedding φ for Mq gives rise to an embedding φ′ : Tq(ΛρMqσ) ↪→ Fqby φ′ = ρ ◦ φ ◦Ψσ−1 , which satisfies the two properties above for ρMqσ.

For a toric frame Mq, we indicate the based quantum torus that lies in Fq with basis{Mq(f) |f ∈ ZN } by Tq(Mq). We have the canonical isomorphism Tq(Mq) ' Tq(ΛMq).

As in the previous subsection fix ex ⊆ [1, N ]. View Λ = (λij) as a skew-symmetric matrix

and let B be an N × ex matrix. We call the pair (Λ, B) compatible if

(2.3)N∑k=1

bkjλki = δijdj for all i ∈ [1, N ], j ∈ ex

for some dj ∈ Z+. Equivalently B>Λ = D where djj = dj for j ∈ ex and otherwise dij = 0.

Denote by D := diag(dj , j ∈ ex) the principal part of D. If (Λ, B) is a compatible pair,

then B has full rank and its principle part B is skew-symmetrized by D, [4]

A pair (Λ, B) is mutated in the direction of k ∈ ex, by setting µk(Λ, B) := (Λ′, B′) where

B′ = EsBFs as in the classical case and Λ′ := E>s ΛEs, which is independent on the choiceof sign s, [4]. As in the classical case µk is an involution, [4].

We call a pair (Mq, B) (consisting of a toric frame Mq for a division algebra Fq and

a matrix B ∈ MN×ex(Z)) a quantum seed if the pair (ΛMq , B) is compatible. We call

{Mq(ej) | j ∈ [1, N ]} the cluster variables of the seed (Mq, B). The subset of clustervariables {Mq(ej) | j 6∈ ex} are called frozen variables.

Proposition 2.3. Suppose Mq is a toric frame, k ∈ [1, N ] and g = (n1, . . . , nN ) ∈ ZNis such that ΛMq(g, ej) = 0 for j 6= k and nk = 0. Then for each s = ±, there is an

automorphism ρg,s = ρMqg,s of Fq, such that

ρg,s(Mq(ej)) =

{Mq(ek) +Mq(ek + sg) if j = k

Mq(ej) if j 6= k.

This is a variation of [4, Proposition 4.2], proved in [23, Lemma 2.8], which will be moresuitable for our root of unity treatment and its relation to the quantum picture via thehomomorphism (5.3).

Mutation µk(Mq, B) of a quantum seed in the direction of k ∈ ex is defined as

(µk(Mq), µk(B)) := (ρMq

bk,sMqEs, EsBFs),

which is independent on the choice of sign, and Λµk(Mq) = µk(ΛMq), [4]. Explicitly, mutationof toric frames is given by

µk(Mq)(ej) = Mq(ej) for j 6= k,

µk(Mq)(ek) = Mq(−ek + [bk]+) +Mq(−ek − [bk]−),(2.4)

[4]. Here, for b = (b1, . . . , bN ) ∈ ZN , set [b]± := (c1, . . . , cN ) ∈ ZN where ci := bi if ±bi ≥ 0and ci := 0 otherwise.

We fix a subset inv ⊆ [1, N ]\ex corresponding to frozen variables that will be inverted.

Definition 2.4. The quantum cluster algebra Aq(Mq, B, inv) is the A1/2q -subalgebra of Fq

generated by all cluster variables M ′q(ej), j ∈ [1, N ] of quantum seeds (M ′q, B′) mutation

equivalent to (Mq, B) and by the inverses Mq(ej)−1 for j ∈ inv.


The upper quantum cluster algebra Uq(Mq, B, inv) is defined as the intersection over

quantum seeds (M ′q, B′) ∼ (Mq, B) of all A1/2

q -subalgebras of Fq of the form

A1/2q 〈M ′q(ei),M ′q(ej)−1 | i ∈ [1, N ], j ∈ ex t inv〉.

These subalgebras of Fq are called mixed quantum tori.

The quantum Laurent phenomenon states that

Aq(Mq, B, inv) ⊆ Uq(Mq, B, inv).

Berenstein and Zelevinsky [4] proved this in the case when all frozen variables are inverted,i.e., when inv = [1, N ]\ex. The general case was proved in [23, Theorem 2.5], where the

result is stated over C(q±1/2) but the proof works over A1/2q .

The exchange graphs of a cluster algebra A(x, B) and a quantum cluster algebra Aq(Mq, B)

are the labelled graphs with vertices corresponding to seeds mutation-equivalent to (x, B),

respectively (Mq, B), and edges given by seed mutation and labelled by the corresponding

mutation number. Those graphs will be denoted by E(B) and Eq(ΛMq , B). A map betweentwo labelled graphs is a graph map that preserves labels of edges. Berenstein and Zelevinsky

[4] proved that there is a (unique) isomorphism between the exchange graphs Eq(ΛMq , B)

and E(B) obtained by sending the vertex corresponding to seed (x, B) to that of (Mq, B).Obviously, the exchange graphs do not depend on the choice of inverted set inv.

3. Root of unity quantum cluster algebras and elementary properties

In this section we define root of unity quantum cluster algebras and describe their ele-mentary properties that are similar to those for quantum cluster algebras. We furthermoreprove that all of them are PI algebras.

3.1. Construction. Let ` be a positive integer. Set

Z` := Z/`Z.

For a matrix C ∈Mn×m(Z) denote its image in Mn×m(Z`) by C. Let ε1/2 ∈ C be a primitive`-th root of unity and set

(3.1) A1/2ε := Z[ε1/2].

Note that in the case of ` odd, ε is also a primitive `-th root of unity and Z[ε1/2] = Z[ε].By abuse of notation, for a skew-symmetric bilinear form Λ : ZN × ZN → Z` we will

denote by the same letter its matrix (Λ(ei, ej)) ∈ MN (Z`). For such a bilinear form define

the root of unity based quantum torus Tε(Λ) to be the A1/2ε -algebra with an A1/2

ε -basis{Xf | f ∈ ZN} and multiplication given by

XfXg = εΛ(f,g)/2Xf+g where f, g ∈ ZN .Hence XfXg = εΛ(f,g)XgXf . The bilinear form Λ can be recovered from the based quantumtorus by

εΛ(f,g)/2 = XfXgX−f−g, ∀f, g ∈ ZN

by using the assumption that ε is a primitive `-th root of unity.

Definition 3.1. A root of unity toric frame Mε of a division algebra Fε over Q(ε1/2) is amap Mε : ZN → Fε such that there is a skew-symmetric matrix Λ ∈MN (Z`) satisfying thefollowing conditions:

(1) There is an A1/2ε -algebra embedding φ : Tε(Λ) ↪→ Fε with φ(Xf ) = Mε(f) for all

f ∈ ZN .


(2) Fε ' Fract (Tε(Λ)).

The matrix Λ ∈MN (Z`) is uniquely reconstructed from the root of unity toric frame Mε.It will be called matrix of the frame Mε and we will denote ΛMε := Λ.

Fix a subset of ex ⊆ [1, N ].

Definition 3.2. Let B ∈ MN×ex(Z) and Λ = (λij) ∈ MN (Z`) be skew-symmetric. The

pair (Λ, B) will be called `-compatible if there exists a diagonal matrix D := diag(dj , j ∈ ex)with dj ∈ Z+ such that

(1) The principal part B of B is skew-symmetrized by D; that is DB is skew-symmetric.

(2)∑N

k=1 bkjλki = δijdj (mod `) for all i ∈ [1, N ], j ∈ ex; that is Λ>B =

[D0

], where

0 denotes the zero matrix of size ([1, N ]\ex)× ex.

We will not require any conditions on dj , so the matrix B need not have full rank like inthe case of quantum cluster algebras.

Similar to the generic case, we define the mutation in direction k ∈ ex of `-compatiblepairs to be

µk(Λ, B) := (E>s ΛEs, EsBFs) for a choice of sign s.

The proof of the following proposition is analogous to [4, Propositions 3.4 and 3.6].

Proposition 3.3. The pair µk(Λ, B) is independent of the choice of sign s. If the pair

(Λ, B) is `-compatible with respect to a diagonal matrix D, then the pair µk(Λ, B) is also`-compatible with respect to the same diagonal matrix D. Mutation µk of `-compatible pairsis an involution.

Definition 3.4. We will call a pair (Mε, B) a root of unity quantum seed if

(1) Mε is a root of unity toric frame of Fε,(2) B ∈MN×ex(Z) and (ΛMε , B) is an `-compatible pair.

Proposition 3.5. Suppose Mε is a root of unity toric frame, k ∈ [1, N ], and g = (n1, . . . , nN ) ∈ZN is such that ΛMε(g, ej) ≡ 0 (mod `) for j 6= k and nk = 0. Then for each s = ±, thereis a unique automorphism ρMε

g,s of Fε, such that

(3.2) ρMεg,s (Mε(ej)) =

{Mε(ek) +Mε(ek + sg) if j = k

Mε(ej) if j 6= k.

Our argument is similar to [23, Lemma 2.8] but we spell out the details because they willbe needed later.

Proof. Denote Fract(Tε(Mε)) by Fε. We have a homomorphism ρg,s : Tε(Mε)→ Fε since

(Mε(ek) +Mε(ek + sg))Mε(ej) = εΛ(ek,ej)Mε(ej) (Mε(ek) +Mε(ek + sg))

for j 6= k. On the A1/2ε -basis {Mε(f)}, one calculates that

ρg,s(Mε(f)) =

{PMε,mkg,s,+ Mε(f) if mk ≥ 0

(PMε,−mkg,s,− )−1Mε(f) if mk < 0

for f = (m1, . . . ,mN ) ∈ ZN , where

PMε,mkg,s,± :=

mk∏p=1

(1 + ε∓s(2p−1)ΛMε (g,ek)/2Mε(sg)

)for mk ≥ 0.


Let G := A1/2ε [Mε(sg)]\{0} ⊂ Tε(Mε). Note that G ·Mε(f) = Mε(f) · G for any f ∈

ZN , and hence G is an Ore set. Moreover, as im(ρg,s) ⊂ Tε(Mε)G−1, we may consider

ρg,s : Tε(Mε) → Tε(Mε)G−1. Since g has nk = 0, the map ρg,s acts by the identity on G.

We can clearly extend the map to an endomorphism ρg,s : Tε(Mε)G−1 → Tε(Mε)G

−1.We can similarly construct an algebra endomorhpism ρ′g,s : Tε(Mε)G

−1 → Tε(Mε)G−1

defined by

ρ′g,s(Mε(ej)) =

{(PMε,1

g,s,+)−1Mε(f) if j = k

Mε(ej) if j 6= k.

Clearly ρg,s and ρ′g,s are inverse to each other and are automorphisms of Tε(Mε)G−1. In

particular, they are injective and can be extended to automorphisms of Fε. Uniquenessfollows since Mε(ej) are skew-field generators of Fε. �

Similar to the generic case, we define mutation of a root of unity quantum seed (Mε, B)in the direction of k ∈ ex by

(3.3) µk(Mε, B) := (ρMεEsbk,s

MεEs, EsBFs).

The proof of the following proposition is analogous to [4, Propositions 4.7 and 4.10].

Proposition 3.6. Given a root of unity quantum seed (Mε, B), the following hold:

(1) For k ∈ ex and either sign s = ±:

ρMεEsbk,s

MεEs(ej) = Mε(ej) for j 6= k,

ρMεEsbk,s

MεEs(ek) = Mε(−ek + [bk]+) +Mε(−ek − [bk]−).

In particular, mutation does not depend on the sign used.

(2) µk(Mε, B) is also a root of unity quantum seed.

Moreover, mutation is an involution.

We consider the equivalence classes under finite sequences of mutations of root of unityquantum seeds. Fix a subset inv ⊆ [1, N ]\ex corresponding to frozen variables that willset as invertible.

Definition 3.7. Given a root of unity quantum seed (Mε, B), we define the quantum

cluster algebra at a root of unity Aε(Mε, B, inv) as the A1/2ε -subalgebra of Fε generated by

all cluster variables of quantum seeds (M ′ε, B′) mutation equivalent to (Mε, B) and by the

inverses of the frozen variables corresponding to inv,

Aε(Mε, B, inv) := A1/2ε 〈M ′ε(ei),Mε(ej)

−1 | i ∈ [1, N ], j ∈ inv, (M ′ε, B′) ∼ (Mε, B)〉.

We have associated to each skew-symmetric bilinear form Λ a based quantum torus.Given subsets ex and inv, we can also associate an algebra in between the correspondingskew-polynomial algebra and the quantum torus,

(3.4) Tε(Λ)≥ := A1/2ε 〈 Xi, X

−1j | i ∈ [1, N ], j ∈ ex t inv〉 ⊂ Tε(Λ).

We call this a mixed based quantum torus. Equivalently, it is the algebra

{⊕A1/2ε Xf | f ∈ ZN≥} with the product XfXg = εΛ(f,g)/2Xf+g, ∀f, g ∈ ZN≥ ,

where

(3.5) ZN≥ := {f = (f1, . . . , fN ) ∈ ZN | fi ≥ 0, ∀i /∈ ex t inv}.We similarly define

Tε(Mε)≥ := 〈 Mε(ei), Mε(ej)−1 | i ∈ [1, N ], j ∈ ex t inv〉 ⊂ Tε(Mε).


Definition 3.8. Given a root of unity quantum seed (Mε, B) and specified subsets ex and

inv, we define the upper quantum cluster algebra at a root of unity Uε(Mε, B, inv) as theintersection of mixed quantum tori corresponding to quantum seeds mutation equivalent to

(Mε, B),

Uε(Mε, B, inv) :=⋂

(Mε,B)∼(M ′ε,B′)

Tε(M ′ε)≥.

Remark 3.9. In the case when ε1/2 = 1 (i.e. ` = 1), a root of unity quantum clusteralgebra can be identified with a classical cluster algebra (of geometric type)

A1(M1, B, inv) = A((M1(e1), . . . ,M1(eN )), B, inv),

and similarly a root of unity upper quantum cluster algebra with an upper cluster algebra

U1(M1, B, inv) = U((M1(e1), . . . ,M1(eN )), B, inv).

3.2. The quantum Laurent phenomenon at roots of unity.

Theorem 3.10. For any root of unity quantum cluster algebra Aε(Mε, B, inv)

Aε(Mε, B, inv) ⊆ Uε(Mε, B, inv).

Proof. The case when B has full rank is proved analogously to [23, Theorem 2.15]. Wededuce the general case of the theorem from the full rank one as follows.

For simplicity of notation, assume that ex = [1, n] for some integer n ≤ N . Consider theaugmented skew-symmetric bilinear form with matrix

Λaug :=

[Λ 0N×n

0n×N 0n×n

],

where 0i×j denotes the zero matrix of size i× j. Denote the augmented exchange matrix

Baug :=

[BIn

]whose principal part is the same as B. The pair (Λaug, Baug) is `-compatible with respectto the same diagonal matrix D because

B>augΛaug =

[D 0

].

Denote by Fε the skew-field Fract(Tε(Λaug)) and consider the toric frame (Mε)aug with

matrix Λaug such that (Mε)aug(ek) := Xk for all k ∈ [N+1, N+n]. Clearly, ((Mε)aug, Baug)

is a root of unity quantum seed. We have a canonical surjectiveA1/2ε -algebra homomorphism

π : Tε((Mε)aug)≥ → Tε(Mε)≥ given by π((Mε)aug(ek)) :=

{Mε(ek), 1 ≤ k ≤ N1, N < k ≤ N + n

because the elements (Mε)aug(ek) are in the center of Tε((Mε)aug)≥ for N < k ≤ N + n.By induction on m ≥ 0 one easily shows that

π(µi1 . . . µim((Mε)aug)(ek)

)= µi1 . . . µim(Mε)(ek)

for all k ∈ [1, N ]. Since the matrix Baug has full rank, by the validity of the root of unityquantum Laurent phenomenon in the full rank case we have

µi1 . . . µim((Mε)aug)(ek) ∈ Tε((Mε)aug)≥.

Hence, µi1 . . . µim(Mε)(ek) ∈ Tε(Mε)≥ for all k ∈ [1, N ], which completes the proof of thetheorem in the general case. �


3.3. PI properties of root of unity quantum cluster algebras.

Theorem 3.11. All root of unity quantum cluster algebras Aε(Mε, B, inv) and root of unity

upper quantum cluster algebras Uε(Mε, B, inv) are PI domains.

Proof. By Theorem 3.10, for every toric frameMε of Aε(Mε, B, inv) we have the embeddings

Aε(Mε, B, inv) ⊆ Uε(Mε, B, inv) ⊆ Tε(Mε) ∼= Tε(ΛMε).

Since each root of unity quantum torus Tε(Λ) is a PI domain, the same is true for the firsttwo algebras in the chain. �

4. Canonical central subrings of root of unity quantum cluster algebras

The main results of this section are the construction of a canonical central subring of a

root of unity quantum cluster algebra Aε(Mε, B, inv) and a theorem that it is isomorphic

to the classical cluster algebra A(B, inv).

4.1. Central embedding of commutative cluster algebras.

Lemma 4.1. If (M ′ε, B′) is mutation-equivalent to (Mε, B), then the element M ′ε(ej)

l ∈Aε(Mε, B, inv) is central for any j ∈ [1, N ].

Proof. We only need show that Mε(ej)l ∈ Z(Aε(Mε, B)) for j ∈ [1, N ], since Aε(Mε, B, inv)

= Aε(M ′ε, B′, inv). Now Mε(ej)

l is central in Tε(Mε) as

Mε(ej)lMε(f) = Mε(lej)Mε(f) = εΛ(lej ,f)Mε(f)Mε(lej) = Mε(f)Mε(ej)

l.

Thus, it is central in Fract(Tε(Mε)) and in Aε(Mε, B, inv). �

For a root of unity quantum seed (Mε, B) and for j ∈ ex, consider the commutation ofelements Mε(−ej + [bj ]+) and Mε(−ek − [bj ]−). The relation in the quantum torus is

Mε(−ej − [bj ]−)Mε(−ej + [bj ]+) = εΛ(−ej−[bj ]−,−ej+[bj ]+)Mε(−ej + [bj ]+)Mε(−ej − [bj ]−).

Set tj := Λ(−ej − [bj ]−,−ej + [bj ]+) for brevity.

Lemma 4.2. Let (Mε, B) be a root of unity quantum seed, so (ΛMε , B) is an `-compatiblepair with respect to a diagonal matrix D = diag(dj , j ∈ ex) with dj ∈ Z+. Then for j ∈ ex,

tj = dj.

Proof. We have that

tj = Λ(−ej − [bj ]−,−ej + [bj ]+)

= Λ(−ej ,−ej) + Λ(−ej , [bj ]+) + Λ(−[bj ]−,−ej) + Λ(−[bj ]−, [bj ]+)

= Λ(bj , ej) + Λ([bj ]+, [bj ]−) = dj + Λ([bj ]+, [b

j ]−).

To evaluate Λ([bj ]+, [bj ]−), we note that bj − [bj ]+ = [bj ]−, so

Λ([bj ]+, [bj ]−) = Λ([bj ]+, b

j)− Λ([bj ]+, [bj ]+) = Λ([bj ]+, b

j).

Since bjj = 0,

Λ([bj ]+, [bj ]−) = Λ([bj ]+, b

j) =∑bij>0

bijΛ(ei, bj) =

∑bij>0

−bijδi,jdj = 0.

Thus tj = dj . �


We will often require the following condition on our root of unity quantum seed (Mε, B):

(Coprime)` is an odd integer coprime to dk for k ∈ ex, where D = diag(dj , j ∈ ex)

is the matrix that skew-symmetrizes the principal part B of B.

The condition (Coprime) only concerns the `-th root of unity ε and the compatible pair

(ΛMε , B), and not the root of unity toric frame Mε.

Remark 4.3. The diagonal matrix D that skew-symmetrizes exchange matrices is notchanged under mutation. Therefore, if a root of unity quantum seed satisfies condition(Coprime), then any mutation equivalent seed does so as well. So, (Coprime) is acondition on a root of unity quantum cluster algebra and not on individual seeds.

The main use of Lemma 4.2 is the following result. The formula appearing should becompared to the mutation relation of (2.1).

Proposition 4.4. Let (Mε, B) be a root of unity quantum seed satisfying the condition(Coprime). Then for k ∈ ex,

Mε(ek)` (µkMε(ek))

` =∏bik>0

(Mε(ei)`)bik +

∏bik<0

(Mε(ei)`)−bik .

Proof. Denote

Y := Mε(−ek + [bk]+), Z := Mε(−ek − [bk]−) ∈ Tε(Mε).

Since ZY = εdkY Z (by Lemma 4.2) and εdk is an `-th primitive root of unity,

(Y + Z)` = Y ` + Z`.

Thus,

(µkMε(ek))` =

(Mε(−ek + [bk]+) +Mε(−ek − [bk]−)

)`= (Y + Z)`

= Mε(−ek + [bk]+)` +Mε(−ek − [bk]−)`

= Mε(−èk + `[bk]+) +Mε(−èk − `[bk]−)

= Mε(−èk)∏bik>0

Mε(`bikei) +Mε(−èk)∏bik<0

Mε(−`bikei).

�

Example 4.5. The previous proposition does not hold if the condition that ` is coprime tothe integers dk is dropped. Consider the following example when ` = 9. Let

ε1/2 = e2πi/9, Λ =

[0 1−1 0

], B =

[0 1−3 0

].

Let Fε := Fract (Tε(Λ)) and Mε : Z2 → Fε be the toric frame related to Λ such that

Mε(f) = Xf and ΛMε := Λ. Clearly, (Mε, B) is a root of unity quantum seed. Here wehave

B>Λ =

[3 00 1

].

In particular, d1 = 3 is not coprime to ` = 9. For Y := Mε(−e1 + [b1]+) = Mε(−e1) andZ := Mε(−e1 − [b1]−) = Mε(−e1 + 3e2), by a direct computation one obtains

(Y + Z)9 = Y 9 + 3Y 6Z3 + 3Y 3Z6 + Z9 6= Y 9 + Z9,

so the conclusion of Proposition 4.4 fails.


In a similar way, dropping the odd root of unity condition will result in a failure of the

statement of Proposition 4.4. Consider the same choice for Λ and B, but with ε1/2 = i, aprimitive fourth root of unity. Then ε = −1 and

(Y + Z)4 = Y 4 + (1 + ε+ 2ε2 + ε3 + ε4)Y 2Z2 + Z4

= Y 4 + 2Y 2Z2 + Z4 6= Y 4 + Z4

leading once again to a failure of the conclusion of Proposition 4.4. The issue in the evencase is that ε is a primitive (`/2)-th root of unity, not a primitive `-th root of unity.

Define the Z-subring

C(Mε, B, inv) := Z〈M ′ε(ei)`, M ′ε(ej)−` | (M ′ε, B′) ∼ (Mε, B), i ∈ [1, N ], j ∈ inv〉

of Aε(Mε, B, inv).

Theorem 4.6. Suppose that (Mε, B) satisfies condition (Coprime). Then the subring

C(Mε, B, inv) of Aε(Mε, B, inv) is isomorphic to A(B, inv).

Proof. Since A({x1, . . . , xN}, B,∅) is constructed as a subalgebra of Q(x1, . . . , xN ), considerthe isomorphism φ : Q(x1, . . . , xN ) → Fract

(Z[Mε(e1)`, . . . ,Mε(eN )` ]

)given by xj 7→

Mε(ej)`. Proposition 4.4 gives us that φ(µi(xj)) = (µiMε(ej))

` for all i ∈ ex, j ∈ [1, N ]. By

induction on the length of the mutation sequence, φ(µik . . . µi1(xj)) = (µik . . . µi1Mε(ej))`.

Since the generators of Z〈M ′ε(ei)` | (M ′ε, B′)∼ (Mε, B), i ∈ [1, N ]〉 are the images of the

generators of A({x1, . . . , xN}, B,∅) under the isomorphism φ, then we have an isomor-phism of Z-algebras. The more general case, when inv 6= ∅, is obtained by adjoining theappropriate inverses of frozen variables. �

Corollary 4.7. If (Mε, B) satisfies condition (Coprime), then the A1/2ε -subalgebra

Cε(Mε, B, inv) :=

A1/2ε 〈M ′ε(ei)`, M ′ε(ej)−` | (M ′ε, B

′) ∼ (Mε, B), i ∈ [1, N ], j ∈ inv〉

of Aε(Mε, B, inv) is isomorphic to A1/2ε ⊗Z A(B, inv).

4.2. Exchange graphs of root of unity quantum cluster algebras. For a root of unity

quantum cluster algebra Aε(Mε, B), define its exchange graph Eε(Mε, B) to be the labelledgraph with vertices corresponding to root of unity quantum seeds mutation-equivalent to

(Mε, B) and with edges given by seed mutation labelled by the corresponding letters.

Theorem 4.8. Let (Mε, B) be a root of unity quantum seed satisfying condition (Coprime).

There is a unique isomorphism of labelled graphs from the exchange graph Eε(Mε, B) to the

exchange graph E(B) which sends the vertex corresponding to the seed (Mε, B) to the vertex

corresponding to the seed (x, B), where x = (Mε(e1)`, . . . ,Mε(eN )`).

We will need the following two propositions for the proof of the theorem which are of

independent interest. Recall that an exchange matrix B is indecomposable if it cannot berepresented in a block diagonal form with blocks of strictly smaller size.

Proposition 4.9. Assume that (Mε, B) and (M ′ε, B′) are two seeds of a root of unity

quantum cluster algebra, where B is indecomposable and B 6= 0. Then for every k ∈ [1, N ]there exists a functional θ : ZN → Z such that

M ′ε(ek) = Mε(f) +∑i

aiMε(fi)


for some ai ∈ A1/2ε and f, fi ∈ ZN such that θ(f) > θ(fi) for all i.

The statment fails when B = 0, because in that case µ1(Mε)(e1) = 2Mε(−e1).

Proof. We prove the proposition by induction on the distance between the vertices in the

exchange graph corresponding to the seeds (Mε, B) and (M ′ε, B′). The case when the

distance equals 1 is trivial because the condition that B is indecomposable and B 6= 0implies that µj(Mε)(ej) = Mε(f1) +Mε(f2) for some f1 6= f2 ∈ ZN .

Assume the validity of the statement when the distance equals m. Consider two seeds

(Mε, B) and (M ′ε, B′) whose vertices are at distance m + 1 in the exchange graph. Then

there exists a seed (M ′′ε , B′′) such that (M ′′ε , B

′′) = µj(Mε, B) for some j ∈ [1, N ] and the

distance between the vertices of the exchange graph corresponding to the seeds (M ′′ε , B′′)

and (M ′ε, B′) equals m. The exchange matrices B′ and B′′ are necessarily indecomposable.

We have

M ′′ε (el) = Mε(el) for l 6= j,

M ′′ε (ej) = Mε(−ej + [bj ]+) +Mε(−ej − [bj ]−).

By the induction hypothesis there exists a functional θ′′ : ZN → Z such that

(4.1) M ′ε(ek) = M ′′ε (g) +∑i

a′′iM′′ε (gi)

for some a′′i ∈ A1/2ε and g, gi ∈ ZN such that θ′′(g) > θ′′(gi) for all i.

Denote by s the sign ± for which θ′′([bj ]+) or −θ′′([bj ]−) is minimal. Define the functionalθ : ZN → Z by

θ(ej) = θ′′(−ej + s[bj ]s), θ(el) = θ′′(el) for l 6= j.

Let Tε(Mε) be the completion of the quantum torus Tε(Mε) spanned by formal sums ofthe form

∞∑m=0

cmMε(h−msbj)

for h ∈ ZN and cm ∈ A1/2ε . It is an A1/2

ε -algebra on its own. We have −s[bj ]−s = s[bj ]s−sbj .Since Mε(−ej + [bj ]+) and Mε(−ej − [bj ]−) skew-commute up to a power of ε, for all n ∈ Z,

M ′′ε (nej) =(Mε(−ej + s[bj ]s) +Mε(−ej − s[bj ]−s)

)n(4.2)

= Mε

(n(−ej + s[bj ]s)

)+

∞∑m=1

cmMε

(n(−ej + s[bj ]s)−msbj

)for some cm ∈ A1/2

ε . Denote ZN−1 := ⊕l 6=jZej ⊂ ZN . For all h ∈ ZN−1 we have

Λ′′(ej , h) = Λ(−ej + [bj ]+, h) = Λ(−ej − [bj ]−, h)

and thus, by using (4.2) and the definition of root of unity toric frames,

(4.3) M ′′ε (nej + h) = Mε

(n(−ej + s[bj ]s) + h

)+

∞∑m=1

cmMε

(n(−ej + s[bj ]s) + h−msbj

)


for some cm ∈ A1/2ε . Write the elements g, gi ∈ ZN in (4.1) in the form g = nej + h,

gi = niej + hi, for n, ni ∈ Z, h, hi ∈ ZN−1 and apply (4.3) to obtain,

M ′ε(ek) = Mε

(n(−ej + s[bj ]s) + h

)+∞∑m=1

cmMε

(n(−ej + s[bj ]s) + h−msbj

)+∑i

a′′iMε

(ni(−ej + s[bj ]s) + hi

)+

∞∑m=1

ci,mMε

(ni(−ej + s[bj ]s) + hi −msbj

)for some ci,m ∈ A1/2

ε . By the root of unity quantum Laurent phenomenon (Theorem 3.10),the sum in the right hand side belongs to Tε(Mε). Furthermore, the definition of thefunctional θ implies that

θ(n(−ej + s[bj ]s) + h) = θ′′(g) > θ′′(gi) = θ(ni(−ej + s[bj ]s) + hi),

θ(sbj) = θ(s[bj ]s) + θ(s[bj ]−s) < 0.

Hence, the above expansion of M ′ε(ek) in Tε(Mε) has the desired properties with respect tothe functional θ. �

Remark 4.10. The proof of Proposition 4.9 directly translates to the case quantum clusteralgebras to yield the validity of the obvious analog of it in that situation.

Proposition 4.11. Assume that (Mε, B) and (M ′ε, B′) are two seeds of a root of unity quan-

tum cluster algebra. Then(M ′ε(e1), . . . ,M ′ε(eN )

)is a permutation of

(Mε(e1), . . . ,Mε(eN )

)if and only if

(M ′ε(e1)`, . . . ,M ′ε(eN )`

)is a permutation of

(Mε(e1)`, . . . ,Mε(eN )`

).

Proof. The forward direction is obvious. For the reverse direction it is sufficient to consider

the case when B is indecomposable. If B = 0, the statement is clear. In the remaining part

we assume that B is indecomposable and B 6= 0. Suppose that

(4.4) M ′ε(ek)` = Mε(eσ(k))

` for some σ ∈ SN .

Consider a root of unity quantum torus Tε(Λ) with generators X±11 , . . . , X±1

N . By using the

standard basis of Tε(Λ), one easily sees that the only solutions of the equation y` = X`k

for y ∈ Tε(Λ) and 1 ≤ k ≤ N are y = (ε1/2)mXk for m ∈ [0, `). By Theorem 3.10,M ′ε(ek) ∈ Tε(Mε), and (4.4) implies that for all 1 ≤ k ≤ N

M ′ε(ek) = (ε1/2)mkMε(eσ(k)) for some mk ∈ [0, `).

Proposition 4.9 implies that mk = 0 for all 1 ≤ k ≤ N , so

M ′ε(ek) = Mε(eσ(k)), ∀1 ≤ k ≤ N.

�

Proof of Theorem 4.8. Any map of labelled graphs from Eε(Mε, B) to E(B) that sends

the vertex corresponding to the seed (Mε, B) to the vertex corresponding to the seed

((Mε(e1)`, . . . ,Mε(eN )`), B) necessarily sends the vertex µi1 . . . µim(Mε, B) to the vertex

µi1 . . . µim((Mε(e1)`, . . . ,Mε(eN )`), B) for all sequences i1, . . . , im in ex. Proposition 4.11implies that this map is well defined. It is obviously surjective. It is injective by Proposition4.11. �


4.3. The full centers of roots of unity quantum cluster algebras. In Corollary

4.7 we constructed a central subalgebra Cε(Mε, B, inv) of each root of unity quantum

cluster algebra Aε(Mε, B, inv) satisfying condition (Coprime). We will call it the specialcentral subalgebra. One can provide a characterization of the full center of the algebra

Aε(Mε, B, inv), as follows. For a skew-symmetric bilinear form Λ : ZN × ZN → Z` denote

Ker Λ := {f ∈ ZN | Λ(f, g) = 0, ∀g ∈ ZN}.

Proposition 4.12. Let Aε(Mε, B, inv) be a root of unity quantum cluster algebra. For

every seed (M ′ε, B′) ∼ (Mε, B), the center of Aε(Mε, B, inv) is given by

Z(Aε(Mε, B, inv)) = Aε(Mε, B, inv) ∩ A1/2ε −Span{M ′ε(f) | f ∈ Ker ΛM ′ε}.

Proof. Using the standard basis of a root of unity quantum torus, one easily shows that

(4.5) Z(Tε(M ′ε)) = A1/2ε −Span{M ′ε(f) | f ∈ Ker ΛM ′ε}

The root of unity quantum Laurent phenomenon (Theorem 3.10) implies that

Aε(Mε, B, inv) ⊆ Tε(M ′ε). Since Fract(Aε(Mε, B, inv)) = Fract(Tε(M ′ε)),

Z(Aε(Mε, B, inv)) = Aε(Mε, B, inv) ∩ Z(Tε(M ′ε))and the proposition follows from (4.5). �

Remark 4.13. Using the full form of the root of unity quantum Laurent phenomenon (The-orem 3.10), one analogously proves the following stronger (but more technical) description

of the center of Aε(Mε, B, inv):

Z(Aε(Mε,B, inv)) = Aε(Mε, B, inv)∩

A1/2ε −Span{Mε(f) | f = (f1, . . . , fN ) ∈ Ker ΛMε , fi ≥ 0,∀i /∈ ex t inv}.

5. Strict root of unity quantum cluster algebras and specializations

In this section we introduce the notion of strict root of unity quantum cluster algebrasand show that, under certain general assumptions, they arise as specializations.

5.1. Construction.

Definition 5.1. Consider a root of unity quantum seed (Mε, B), so that (ΛMε , B) is `-compatible with respect to a diagonal matrix D. We say that this seed is strict if thereexists a skew-symmetric integer matrix Λ ∈MN (Z) such that

(1) Λ = ΛMε and

(2) the pair (Λ, B) is compatible with respect to the diagonal matrix D.

Clearly, condition (2) is stronger than requiring that (ΛMε , B) be `-compatible withrespect to D. The choice of matrix Λ is not unique.

Proposition 5.2. If (Mε, B) is a strict root of unity quantum seed with respect to skew-

symmetric integer matrix Λ ∈MN (Z), then µk(Mε, B) is also a strict root of unity quantumseed with respect to the skew-symmetric integer matrix

Λ′ = E>s ΛEs

Proof. The pair (E>s ΛEs, EsBFs) is the mutation of the compatible pair of matrices (Λ, B).By [4, Proposition 3.4] the first pair is compatible with respect to the matrix D. We have

Λµk(Mε) = E>s ΛMεEs = E

>s ΛEs = Λ

′.

�


Definition 5.3. We call a root of unity quantum cluster algebra strict if one, and thusevery of its seeds, is strict.

Remark 5.4. The class of strict root of unity quantum cluster algebras is a proper subset ofthe class of root of unity quantum cluster algebras. For example, by Remark 3.9, for ` = 1,a root of unity quantum cluster algebra is the same object as a classical cluster algebra. Atthe same time, it is easy to see that a strict root of unity quantum cluster algebra for ` = 1is the same object as a classical cluster algebra with a compatible Poisson structure in thesense of Gekhtman–Shapiro–Vainshtein [22].

5.2. Specialization of quantum tori. Denote the `-th cyclotomic polynomial by

(5.1) Φ`(t) ∈ Z[t].

We have the isomorphism A1/2q /(Φ`(q

1/2)) ' A1/2ε given by q1/2 7→ ε1/2. This makes

Tq(Λ)/(Φ`(q1/2)) an A1/2

ε -algebra.

Lemma 5.5. There is an isomorphism of A1/2ε -algebras Tq(Λ)/(Φ`(q

1/2)) ' Tε(Λ).

Proof of Lemma 5.5. It follows that Tq(Λ)/(Φ`(q1/2)) ' Tε(Λ) since the free A1/2

q -module

Tq(Λ) and the free A1/2ε -module Tε(Λ) both have the basis {Xf | f ∈ ZN}. �

Denote the specialization map

(5.2) κε : Tq(Λ) � Tq(Λ)/(Φ`(q1/2)) ' Tε(Λ).

It is a homomorphism of A1/2q -algebras, where A1/2

q acts on Tε(Λ) via the map q 7→ ε.

Construction 5.6. Let (Mε, B) be a strict root of unity quantum seed associated to a skew-

symmetric integer matrix Λ ∈MN (Z). To it we associate the unique quantum seed (Mq, B)

of Fq := Fract(Tq(Λ)) such that ΛMq = Λ. The compatibility of the pair (Λ, B) with respect

to the matrix D implies that (Mq, B) is indeed a quantum seed.The isomorphisms of quantum tori Tq(Mq) ' Tq(Λ) and Tε(Mε) ' Tε(Λ) and the special-

ization map (5.2) give rise to the specialization map (an A1/2q -algebra homomorphism)

(5.3) κε : Tq(Mq) � Tε(Mε)

with kernel (Φ`(q1/2)). It is given by κε(Mq(f)) := Mε(f) for f ∈ ZN .

The next theorem provides a general realization of a root of unity quantum cluster algebrain terms of the specialization maps (5.3) for toric frames.

Theorem 5.7. Let (Mε, B) be a root of unity quantum toric frame associated to a skew-

symmetric integer matrix Λ ∈ MN (Z) and (Mq, B) be the corresponding quantum toric

frame from Construction 5.6. We have the isomorphism of A1/2ε -algebras

(5.4) κε(Aq(Mq, B, inv)) ' Aε(Mε, B, inv).

In the special case ε1/2 = 1 (i.e. ` = 1) the theorem provides a realization of classicalcluster algebras with a compatible Poisson structure (in the sense of [22]) in terms of toricframe specializations of quantum cluster algebras, recall Remark 3.9.

Proof. Since the elements Mq(ek), 1 ≤ k ≤ N generate Tq(Mq) and κε : Tq(Mq) � Tε(Mε)is a surjective ring homomorphism,

(5.5) Fract(κε(Aq(Mq, B, inv))

)' Fract

(Tε(Mε)

).

We claim that the following hold for all quantum seeds (M ′q, B′) of Aq(Mq, B, inv):


(i) (κεM′q, B

′) is a root of unity quantum seed of κε(Aq(Mq, B, inv)).

(ii) µk(κεM

′q, B

′) = (κε(M′′q ), µk(B

′)) whereM ′′q is the toric frame of the seed µk(M′q, B

′).

Property (i): κεM′q is a root of unity quantum toric frame of Tε(Mε) because of (5.5) and the

fact that κε is a homomorphism of A1/2q -algebras. The compatibility of the pair (ΛMq , B

′)

implies that the matrix of the frame κεM′q and the exchange matrix B′ are `-compatible.

Property (ii) follows from the mutation formulas in Eq. (2.4) and Proposition 3.6(1), and

once again the fact that fact that κε is an A1/2q -algebra homomorphism.

The properties (i)–(ii), the fact that the image κε(Aq(Mq, B, inv)) is generated by the

elements κεM′q(ej) for quantum seeds (M ′q, B

′) ∼ (Mq, B), 1 ≤ j ≤ N and by the inversesof these elements for j ∈ inv imply the isomorphism in (5.4). �

5.3. Generalized specialization. For a commutative ring A and an ideal I of it, denotethe factor ring A′ := A/I.

Lemma 5.8. (1) For every A-module V we have the short exact sequence of A-modules

0→ IV → V → A′ ⊗A V → 0,

where the third map is v 7→ 1⊗ v for v ∈ V and A′⊗A V is made into an A-modulevia the surjection A� A′.

(2) For an A-submodule W ⊆ V , the following are equivalent:(a) the induced map A′ ⊗A − : A′ ⊗AW → A′ ⊗A V is injective,(b) W ∩ IV = IW .

Proof. The first part is well known, see e.g. [20, Lemma 3.1].(2) Consider the commutative diagram

0 IW W A′ ⊗AW 0

0 IV V A′ ⊗A V 0

ιW ηW

ιV ηV

θ θ′

where the horizontal maps are the ones from part (1) and the vertical ones are induced fromthe embedding θ : W ↪→ V .

(a) ⇒ (b) Let v0 ∈ IV and w ∈ W be such that ιV (v0) = θ(w). Then θ′ηW (w) =ηV θ(w) = ηV ιV (v0) = 0. Since (a) holds, ηW (w) = 0, and so w ∈ Im ιW .

(b) ⇒ (a) Let w′ ∈ A′ ⊗A W be such that θ′(w′) = 0. Choose w ∈ W such thatw′ = ηW (w). Because ηV θ(w) = θ′ηW (w) = 0, θ(w) ∈ Im ιV . Since (b) holds, w = ιW (w0)for some w0 ∈ IW , and thus w′ = ηW ιW (w0) = 0.

A special case of the second part of the lemma for principal ideal domains A, prime idealsI and free modules V is stated in [20, Lemma 2.1]. �

The A′-module V/IV ' A′ ⊗A V is called the (generalized) specialization of V at I(traditionally specialization deals with the special case when I is a principal idea). Thecanonical projection map

ηV : V � V/IV ' A′ ⊗A Vis called the specialization map. It is a homomorphism of A-modules. If an A-submoduleW ⊆ V satisfies the equivalent conditions in Lemma 5.8(2), then

W/IW ' ηV (W ) and ηW = ηV |W .


5.4. A general specialization result for quantum cluster algebras. Recall from (5.1)

that Φ`(t) denotes the `-th cyclotomic polynomial and A1/2q /(Φ`(q

1/2)) ' A1/2ε . For a

quantum cluster algebra Aq(Mq, B, inv) denote the corresponding specialization map

ηε : Aq(Mq, B, inv) � Aq(Mq, B, inv)/(Φ`(q1/2)) ' A1/2

ε ⊗A1/2q

Aq(Mq, B, inv),

which is a surjective homomorphism of A1/2q -modules.

Similarly to (3.4), for a quantum seed (M ′q, B′) ∼ (Mq, B) denote the subalgebra

(5.6) Tq(M ′q)≥ := A1/2q 〈M ′q(ei),M ′q(ej)−1 | i ∈ [1, N ], j ∈ ex t inv〉

of the quantum torus Tq(M ′q). It is isomorphic to the mixed (based) quantum torus/skewpolynomial algebra

A1/2ε −Span{Xf | f ∈ ZN≥} with the product XfXg = qΛ′(f,g)/2Xf+g, ∀f, g ∈ ZN≥

for ZN≥ as in (3.5). The specialization map κε : Tq(Mq) � Tε(Mε) ∼= Tq(Mq)/(Φ`(q1/2))

form (5.3) restricts to the specialization map

(5.7) κε : Tq(Mq)≥ � Tε(Mε)≥ ∼= Tq(Mq)≥/(Φ`(q1/2)),

which, by abuse of notation, will be denoted by the same symbol.The following result gives a general way of constructing root of unity quantum cluster

algebras as specializations from quantum cluster algebras.

Theorem 5.9. Let (Mε,Λ, B) be a root of unity quantum toric frame and (Mq, B) be thecorresponding quantum toric frame from Construction 5.6. If

(5.8) Aq(Mq, B, inv) ∩(Φ`(q

1/2)Tq(Mq)≥)

= Φ`(q1/2)Aq(Mq, B, inv),

then the root of unity quantum cluster algebra Aε(Mε,Λ, B, inv) is a specialization of the

quantum cluster algebra Aq(Mq, B, inv):

Aq(Mq, B, inv)/(Φ`(q1/2)) ' Aε(Mε,Λ, B, inv)

and the specialization map ηε is a restriction of the specialization map κε : Tq(Mq)≥ �Tε(Mε)≥ to Aq(Mq, B, inv).

Proof. In light of Lemma 5.8(2), the assumption (5.8) implies that

Aq(Mq, B, inv)/(Φ`(q1/2)) ' κε(Aq(Mq, B, inv)) and ηε = κε|Aq(Mq ,B,inv)

.

Thus we have the commutative diagram

Aq(Mq, B, inv) Tq(Mq)≥ Fract(Tq(Mq))

Tε(Mε)≥ Fract(Tε(Mε))

ηεκε

The theorem now follows from Theorem 5.7. �


5.5. Specialization results for quantum cluster algebras. The following is an exten-sion of [20, Proposition 3.5]:

Proposition 5.10. For each prime element p ∈ A1/2q and k ∈ ex,

Tq(Mq)≥ ∩ (pTq(µkMq)≥) = (pTq(Mq)≥) ∩ Tq(µkMq)≥.

Proof. We follow the line of argument of [20, Proposition 3.5] but include the proof because

the original result in [20] is stated over the base ring k[q±1/2], where k is a field, and for aconcrete choice of p.

Denote by Tq(Mq)◦≥ the subalgebra Tq(Mq)≥ with those generators as in (5.6) such that

i, j 6= k. Let Xk := Mq(ek) and X ′k := µk(Mq)(ek). Tq(Mq)≥ is a free (left and right)

Tq(Mq)◦≥-module with basis {Xj

k | j ∈ Z}:

(5.9) Tq(Mq)≥ =⊕j∈Z

XjkTq(Mq)

◦≥.

For j ∈ Z denote

Qj = qjΛ(ek,[bk]+)/2Mq([b

k]+) + q−jΛ(ek,[bk]−)/2Mq(−[bk]−) ∈ Tq(Mq)

◦≥.

We haveQ1 = XkX

′k and QjXk = XkQ

j−2, ∀j ∈ Z.If y ∈ Tq(Mq)≥ ∩ (pTq(µkMq)≥), then

y =∑j∈Z

Xjkcj =

∑j∈Z

(X ′k)jdj ,

where both sums are finite and cj ∈ Tq(Mq)◦≥, dj ∈ pTq(Mq)

◦≥ for all j ∈ Z. The free module

structure (5.9) implies that

c0 = d0

cj = Q−2j−1 . . . Q3Q1d−j for j < 0,

Q−1Q−3 . . . Q−2j+1cj = d−j for j > 0.

Therefore cj ∈ pTq(Mq)◦≥ for all j ≤ 0. For the case j > 0, first note that A1/2

q /(p) is

an integral domain because p ∈ A1/2q is prime. As a consequence, Tq(Mq)

◦≥/pTq(Mq)

◦≥

is a domain since it is a subalgebra of a quantum torus with coefficients in A1/2q /(p). If

τ : Tq(Mq)◦≥ � Tq(Mq)

◦≥/pTq(Mq)

◦≥ denotes the canonical projection, then

τ(cj)τ(Q2j−1) . . . τ(Q3)τ(Q1) = τ(d−j) = 0.

Because Qi /∈ pTq(Mq)◦≥ for all i ∈ Z and Tq(Mq)

◦≥/pTq(Mq)

◦≥ is a domain, τ(cj) = 0 and

thus cj ∈ pTq(Mq)◦≥ for j > 0. Hence, y ∈ pTq(Mq)≥. �

Theorem 5.11. Let (Mε,Λ, B) be a root of unity quantum toric frame and (Mq, B) be thecorresponding quantum toric frame from Construction 5.6. If

Aq(Mq, B, inv) = Uq(Mq, B, inv),

then the root of unity quantum cluster algebra Aε(Mε,Λ, B, inv) is a specialization of the

quantum cluster algebra Aq(Mq, B, inv):

Aq(Mq, B, inv)/(Φ`(q1/2)) ' Aε(Mε,Λ, B, inv)

and the specialization map ηε is a restriction of the specialization map κε : Tq(Mq)≥ �Tε(Mε)≥ from (5.7) to Aq(Mq, B, inv).


Proof. Applying Proposition 5.10, one proves that for all quantum seeds (M ′q, B′) ∼ (Mq, B),

Aq(Mq, B, inv) ∩(Φ`(q

1/2)Tq(Mq)≥)

= Uq(Mq, B, inv) ∩(Φ`(q

1/2)Tq(Mq)≥)

⊆ Φ`(q1/2)Tq(M ′q)≥

by induction on the distance from (Mq, B) to (M ′q, B′) in the exchange graph. Hence

Aq(Mq, B, inv) ∩(Φ`(q

1/2)Tq(Mq)≥)⊆ Φ`(q

1/2)Uq(Mq, B, inv) = Φ`(q1/2)Aq(Mq, B, inv)

and clearly Aq(Mq, B, inv) ∩(Φ`(q

1/2)Tq(Mq)≥)⊇ Φ`(q

1/2)Aq(Mq, B, inv). This verifiesthe condition (5.8) and the theorem now follows from Theorem 5.9. �

5.6. An example: quantized Weyl algebras at roots of unity. LetQ = (aij) ∈Mn(Z)

be a skew-symmetric integer matrix and ε1/2 ∈ C be a primitive `-th root of unity for ` > 1.

Denote by AQn,ε,C the quantized Weyl algebra at the root of unity ε, which is a C-algebra

generated by xi, yi for i ∈ [1, n] with relations

yiyj = εaijyjyi, xixj = ε1+aijxjxi for i < j,

xiyj = ε−aijyjxi for i < j, xiyj = ε1−aijyjxi for i > j,

xjyj = 1 + εyjxj + (ε− 1)

j−1∑r=1

yrxr.

Note that {xi, (ε − 1)yi | 1 ≤ i ≤ n} is another set of generators for this algebra. Denote

by AQn,ε,Z the A1/2ε -subalgebra generated by xi, (ε− 1)yi. It is an A1/2

ε -form of AQn,ε,C. The

algebra AQn,ε,Z is a specialization of the A1/2q -algebra AQn,q,Z with generators and relations as

in [24, Eq. (4.9)]:

AQn,ε,Z∼= AQn,q,Z/Φ`(q

1/2).

This easily follows by using bases for both algebras.

By [24, Example 4.10] AQn,q,Z has a quantum cluster algebra structure and by [24, Theorem

4.8] this quantum cluster algebra equals the corresponding upper quantum cluster algebra.

Proposition 5.10 implies that AQn,ε,Z has a strict root of unity quantum cluster algebrastructure. The root of unity quantum toric frame for its initial seed is given by

Mε(ei) := xi, Mε(ei+n) := [xi, yi] = 1 + (ε− 1)i∑

r=1

xryr for 1 ≤ i ≤ n

and the corresponding matrix is

Λ =

[Q′ −RR 0n×n

]whose blocks are the n× n integer matrices

(Q′)ij =

aij + 1 if i < j

−aji − 1 if i > j

0 if i = j

(R)ij =

1 if i < j

aji if i > j

0 if i = j

.

The set of exchangeable indices is ex = [1, n] and the set of inverted frozen variable is emptyinv = ∅. The exchange matrix of the seed is

B =

[0n×nS

]where the entries of S ∈Mn(Z) are (S)i,n+1−i = 1, (S)i,n−i = −1, (S)ij = 0 otherwise.


6. Discriminants of root of unity quantum cluster algebras

In this section we prove a general result for the computation of discriminants of root ofunity quantum cluster algebras.

6.1. Background on discriminants. For an algebraic number field K, consider its trace

function tr = trK/Q : K → Q obtained from the composition K ↪→ MN (Q)Tr−→ Q, where

the first embedding is obtained from the K-action on K ' QN (for some positive integerN) and the second map is the trace map on matrices. The discriminant of K is defined by

∆K := det(

tr(yiyj))Ni,j=1

,

where {y1, y2, . . . , yN} is a Z-basis of the ring of integers OK of K. The discriminant doesnot depend on the choice of basis. More generally, we consider algebras with trace:

Definition 6.1. An algebra with trace is a ring R with a central subring C and a C-linearmap tr : R→ C such that

tr(xy) = tr(yx), ∀x, y ∈ R.

Such a ring R is naturally a C-algebra.

Example 6.2. Consider a ring R which is free and of finite rank N over a central subringC ⊆ Z(R). Choosing a C-basis of R gives rise to a C-module isomorphism R ' CN , andthe left action of R on itself gives rise to an algebra homomorphism R → MN (C). Theregular trace of R is defined as the composition

trreg : R→MN (C)Tr−→ C ⊆ R,

where the second maps is the trace map on matrices. The trace map trreg is independentof the choice of C-basis used to construct the homomorphism R→MN (C).

For a commutative ring C, denote by C× its group of units (i.e., invertible elements underthe product operation). Two elements c1, c2 ∈ C are called associates (denoted c1 =C× c2)if c1 = uc2 for some u ∈ C×.

Definition 6.3. Assume that R is an algebra with trace tr : R → C such that R is a freeand of finite rank N over the central subring C ⊆ Z(R). The discriminant of R over C isdefined by

(6.1) d(R/C) :=C× det(

tr(yiyj))Ni,j=1

,

where {y1, y2, . . . , yN} is a C-basis of R. For different choices of C-bases of R, the righthand sides of (6.1) are associates of each other.

6.2. Nerves and the algebras Aε(Θ, inv) and Cε(Θ, inv). Let Aε(Mε, B, inv) be a quan-

tum cluster algebra with exchange graph Eε(Mε, B).

For a collection of seeds Θ in Eε(Mε, B), let

(1) Aε(Θ, inv) be A1/2ε -subalgebra of Aε(Mε, B, inv) generated by M ′ε(ej) for j ∈ [1, N ]

and M ′ε(ei)−1 for i ∈ inv, for all (M ′ε, B

′) ∈ Θ, and

(2) Cε(Θ, inv) be A1/2ε -subalgebra of Cε(Mε, B, inv) generated by M ′ε(ej)

` for j ∈ [1, N ]

and M ′ε(ei)−` for i ∈ inv, for all (M ′ε, B

′) ∈ Θ.

Thus Cε(Θ, inv) is in the center of Aε(Θ, inv).

Definition 6.4. A subset of seeds Θ that satisfies the following conditions is called a nerve:

(1) The subgraph in Eε(Mε, B) induced by Θ is connected.


(2) For each mutable direction k ∈ ex, there are at least two seeds in Θ mutationequivalent by µk.

The concept of nerves was introduced in [16] for a practical way of specifying a quasi-homomorphism of a cluster algebra. A basic example of a nerve would be a star neighbor-hood in Eε(Mε, B) of any particular seed.

6.3. The discriminant of Aε(Θ, inv) over Cε(Θ, inv). For the proof of the main theoremon discriminants we will need the following lemma. Its proof was communicated to us byGreg Muller.

Lemma 6.5. If

(6.2) n

N∏i=1

xaii ∈ A(x, B, inv)

for some ai, n ∈ Z, n 6= 0, then ai ≥ 0 for i /∈ inv.

Proof. It is sufficient to prove the statement in the case inv = ∅ because (6.2) implies that

n∏Ni=1 x

aii

∏i∈inv x

cii ∈ A(x, B,∅) for some ci ∈ N. For the rest of the proof we assume

that inv = ∅.If i ∈ ex, then ai ≥ 0 because, if ai < 0, then expressing the Laurent monomial in terms

of the cluster variables of the seed µi(x, B) would contradict the Laurent phenomenon. Ifi /∈ ex, the statement follows from [14, Proposition 3.6]. �

Theorem 6.6. Consider a root of unity quantum cluster algebra Aε(Mε, B, inv) satisfying

the condition (Coprime), where ε1/2 is a primitive `-th root of unity. Let Θ be collectionof seeds which is a nerve.

(1) If Aε(Θ, inv) is a free Cε(Θ, inv)-module, then Aε(Θ, inv) is a finite rank Cε(Θ, inv)-module of rank `N and its discriminant with respect to the regular trace function is given asa product of noninverted frozen variables raised to the `-th power,

d (Aε(Θ, inv)/Cε(Θ, inv)) =Cε(Θ,inv)× `(N`N )

∏i∈[1,N ]\(extinv)

Mε(ei)`ai for some ai ∈ N.

(2) If Aε(Θ, inv) is a free C(Θ, inv)-module, then Aε(Θ, inv) is a finite rank C(Θ, inv)-module of rank `Nϕ(`) and its discriminant with respect to the regular trace function isgiven by

d (Aε(Θ, inv)/C(Θ, inv)) =C(Θ,inv)×

( `(N+1)ϕ(`)∏p|` p

ϕ(`)/(p−1)

)`N ∏i∈[1,N ]\(extinv)

Mε(ei)`ci

for some ci ∈ N.

Proof. Throughout the proof all discriminants are computed with respect to the regulartraces of the algebras that are involved.

(1) For a root of unity quantum frame M ′ε denote the skew polynomial subalgebra ofTε(M ′ε)

Sε(M ′ε) := A1/2ε 〈M ′ε(ei), 1 ≤ i ≤ N〉 ' A1/2

ε 〈X1, . . . , XN 〉/(XiXj − ελ′ijXjXi),

where λ′ij := ΛM ′ε(ei, ej). By [7, Proposition 2.8], the discriminant of Sε(M ′ε) over the

central subalgebra A1/2ε [M ′ε(ei)

`]Ni=1 is given by

d(Sε(M ′ε)/A1/2

ε [M ′ε(ei)`]Ni=1

)=A1/2ε

× `N`N∏

i∈[1,N ]

(M ′ε(ei)

`N (`−1)).


Therefore the discriminant of its localization

Tε(M ′ε) ' Sε(M ′ε)[M ′ε(ei)−`]Ni=1

is given by

d(Tε(M ′ε)/A1/2

ε [M ′ε(ei)±`]Ni=1

)=(A1/2ε [M ′ε(ei)

±`]Ni=1

)× `N`N .For the rest of the proof assume that (M ′ε, B

′) ∈ Θ. Applying Theorem 4.6 (using

the assumption that Aε(Mε, B, inv) satisfies the condition (Coprime)) and the Laurentphenomenon, we obtain that

Cε(Θ, inv)[M ′ε(ei)−`]Ni=1 ' A1/2

ε [M ′ε(ei)±`]Ni=1.

The root of unity quantum Laurent phenomenon (Theorem 3.10) implies

Aε(Θ, inv)[M ′ε(ei)−`]Ni=1 ' Tε(M ′ε).

Therefore, the rank of Aε(Θ, inv) as an Cε(Θ, inv)-module equals the rank of Tε(M ′ε) as

an A1/2ε [M ′ε(ei)

±`]Ni=1-module. Since the latter rank equals `N , Aε(Θ, inv) is a finite rankCε(Θ, inv)-module of rank `N . Furthermore,

(6.3) d(Aε(Θ, inv)[M ′ε(ei)−`]Ni=1/Cε(Θ, inv)[M ′ε(ei)

−`]Ni=1) =Tε(M ′ε)× `N`N .

Theorem 4.6 implies that

(6.4) Cε(Θ, inv) ∩ Tε(M ′ε)× ⊆ {(A1/2ε )×M ′ε(e1)`a1 . . .M ′ε(eN )`aN | ai ∈ Z}.

Combining (6.3) and (6.4) gives that for all seeds (M ′ε, B′) ∈ Θ,

(6.5) d(Aε(Θ, inv)/Cε(Θ, inv)) =Cε(Θ,inv)× `N`N

∏i∈[1,N ]

(M ′ε(ei)

`)ai

for some integers ai (depending on each seed). We will assume that ai = 0 for i ∈ inv sinceMε(ei)

` ∈ Cε(Θ, inv)× for i ∈ inv. Theorem 4.6 and Lemma 6.5 imply that ai ≥ 0 fori /∈ inv.

Fix k ∈ ex. Since Θ is a nerve, there exists (M ′ε, B′) ∈ Θ such that µk(M

′ε, B

′) ∈ Θ.Applying (6.5) to the two seeds gives

d(Aε(Θ, inv)/Cε(Θ, inv)) =Cε(Θ,inv)× `N`N

(M ′ε(ek)

`)ak ∏

i∈[1,N ]\(invt{k})

(M ′ε(ei)

`)ai

=Cε(Θ,inv)× `N`N

(µkM

′ε(ek)

`)ck ∏

i∈[1,N ]\(invt{k})

(M ′ε(ei)

`)ci

for some ai, ci ∈ Z, i ∈ [1, N ]\inv. By Proposition 4.4,

µkM′ε(ek)

` = M ′ε(−ek + [bk]+)` +M ′ε(−ek − [bk]−)`,

which is not a monomial of the M ′ε(ei)’s for i ∈ [1, N ]\(inv t {k}). Hense,

ak = 0 = ck,

ai = ci for i 6= k.

Because of the connectedness assumption in Definition 6.4(1), for all seeds (M ′ε, B′) ∈ Θ

and k ∈ ex t inv, ak = 0 in (6.5).(2) For every root of unity quantum frame M ′ε, Sε(M ′ε) is a free Z[M ′ε(ei)

`]Ni=1-module of

rank `Nϕ(`). The discriminant of the cyclotomic field extension Q(ε1/2) of Q equals

(−1)ϕ(`)/2`ϕ(`)∏p|` p

ϕ(`)/(p−1)·


From this one easily deduces that

d(Sε(M ′ε)/Z[M ′ε(ei)

`]Ni=1

)=Z×

( `(N+1)ϕ(`)∏p|` p

ϕ(`)/(p−1)

)`N ∏i∈[1,N ]

(M ′ε(ei)

`N (`−1)ϕ(`)).

Using this formula, the proof of part (2) is carried out using exactly the same arguments aspart (1). �

Remark 6.7. Since it is unknown whether Cε(Θ, inv) is a free C(Θ, inv), part (2) of thetheorem is not an consequence of part (1) and the formula for the discriminants of cyclotomicfield extensions.

6.4. An example: discriminants of quantized Weyl algebras at roots of unity.By the construction in Sect. 5.6,

AQn,ε,Z∼= Aε(Mε, B,∅)

for the toric frame Mε and exchange matrix B specified there. The underlying clusteralgebra is of finite type A1 t . . . tA1. Let ε1/2 be a primitive `-th root of unity for an odd

integer ` > 1, A1/2ε = Aε. Denote

CQn,ε,Z := Aε[xì , ((ε− 1)yi)`, 1 ≤ k ≤ n].

It is well known and easy to verify that CQn,ε,Z is in the center of AQn,ε,Z. We apply Theorem6.6 for Θ equal to the set of all seeds of the root of unity quantum cluster algebra. It iseasy to see that it has 2n seeds with cluster variables

(t1, . . . , tn,−z1, . . . , (−1)izi, . . . , (−1)nzn) where ti = (−1)iε1/2xi or ti = (ε− 1)yi.

This implies that Cε(Mε, B,∅) = CQn,ε,Z. The algebra AQn,ε,Z is a free CQn,ε,Z-module withbasis

{xj11 . . . xjnn ym11 . . . ymnn | j1, . . . , jn,m1, . . . ,mn ∈ [0, `− 1]}.

Applying Theorem 6.6 gives that

(6.6) d(AQn,ε,Z/CQn,ε,Z) =Aε× `

2n`2nzà11 . . . zànn

for some ak ∈ N (here and below discriminants are computed with respect to the regular

trace). To determine the integers ak, consider the filtration of AQn,ε,Z given by deg xk =

deg yk = k for k ∈ [1, n]. The associated graded is isomorphic to a skew-polynomial algebrawith generators given by the images of xk, (ε− 1)yk for k ∈ [1, n], which will be denoted by

xk, (ε− 1)yk. The discriminants of skew-polynomial algebras are given by [7, Proposition2.8]:

d(grAQn,ε,Z/ grCQn,ε,Z) =Aε× `2n`2n(x1(ε− 1)y1)(`−1)`n . . . (xn(ε− 1)yn)(`−1)`n .

Applying [7, Proposition 4.10] to (6.6) gives that a1 = . . . = an = (`− 1)`n−1, which provesthe following:

Proposition 6.8. For each root of unity ε1/2 of odd order `, the discriminant of the quan-

tized Weyl algebra AQn,ε,Z over its central subalgebra CQn,ε,Z with respect to the regular traceis given by

d(AQn,ε,Z/CQn,ε,Z) =Aε× `

2n`2nz(`−1)`n

1 . . . z(`−1)`n

n .

In the case of n = 1 this fact was first proved in [8, Theorem 0.1] and for general valuesof n in [29, Theorem B(ii)].


7. Quantum groups

In this section we gather material about quantized universal enveloping algebras of sym-metrizable Kac–Moody algebras, their integral forms and specializations to roots of unity.

7.1. Quantized universal enveloping algebras. We will follow the notation of Kashi-wara for quantized universal enveloping algebras of symmetrizable Kac–Moody algebras,[26]. Let I := [1, r] serve as an index set and (A,P,Π, P∨,Π∨) be a Cartan datum com-prised of the following:

(i) A symmetrizable, generalized Cartan matrix A = (aij)i,j∈I . In particular, aii = 2for i ∈ I, aij ∈ Z≤0 for i 6= j, and there exists a diagonal matrix D = (di)i∈Iconsisting of positive, relatively prime integers di such that DA is symmetric.

(ii) A free abelian group P (weight lattice).(iii) A linearly independent subset Π = {αi | i ∈ I} ⊂ P (set of simple roots).(iv) The dual group P∨ = HomZ(P,Z) (coweight lattice).(v) Two linearly independent subsets Π∨ = {hi | i ∈ I} ⊂ P∨ (set of simple coroots),

such that 〈hi, αj〉 = aij for i, j ∈ I, and {$i | i ∈ I} ⊂ P (set of fundamentalweights), such that 〈hi, $j〉 = δij for i, j ∈ I.

Let P+ := {γ ∈ P | 〈hi, γ〉 ∈ Z≥0}. Denote the root lattice Q :=⊕

i∈I Zαi and setQ+ :=

⊕i∈I Z≥0αi. Set h := Q ⊗Z P

∨. There is a Q-valued nondegenerate, symmetricbilinear form (·, ·) on h∗ = Q⊗Z P that satisfies

〈hi, µ〉 =2(αi, µ)

(αi, αi)and (αi, αi) = 2di for all i ∈ I, µ ∈ h∗.

Note that the existence of such a bilinear form is equivalent to the symmetrizability of thegeneralized Cartan matrix A. Denote ‖µ‖ := (µ, µ) for µ ∈ h∗.

Let g be the symmetrizable Kac–Moody algebra over Q associated to this Cartan datum.It is the Lie algebra generated by h, ei, and fi for i ∈ I with Serre relations for h ∈ h andi, j ∈ I,

h is an abelian Lie subalgebra,

[h, ei] = 〈h, αi〉ei, [h, fi] = −〈h, αi〉fi, [ei, fj ] = δijhi,

(ad ei)1−aij (ej) = 0, (ad fi)

1−aij (fj) = 0.

Let W be the Weyl group of g, acting on (h∗, (·, ·)) by isometries. Denote its generatorsby si for i ∈ I. The length function on W will be written as l : W → Z≥0. The Bruhatorder will be denoted by ≥. Let ∆+ ⊂ Q+ be the set of positive roots of g.

Let n+ and n− denote the Lie subalgebras of g generated by {ei | i ∈ I} and {fi | i ∈ I}.So

n± =⊕α∈∆+

g±α,

where gα is the root space in g corresponding to α. The root spaces are one dimensionalfor real roots; that is roots in W{αi | i ∈ I}. For w ∈ W , we denote the nilpotent Liesubalgebras

n±(w) :=⊕

α∈∆+∩w−1(−∆+)

g±α.

If w has a reduced expression w = si1 · · · siN , then n+(w) is generated by the root vectorscorresponding to the real roots αi1 , si1(αi2), . . . , si1 . . . siN−1(αiN ).


Let Uq(g) be the corresponding quantized universal enveloping algebra defined over Q(q),

which is generated by ei, fi, and qh for i ∈ I, h ∈ h subject to the relations

q0 = 1, qhqh′

= qh+h′ , qheiq−h = q〈h,αi〉ei, qhfiq

−h = q−〈h,αi〉fi,

[ei, fi] = δijqdihi − q−dihi

qi − q−1i

,

1−aij∑s=0

(−1)s[1− aijs

]i

e1−aij−si eje

si = 0,

1−aij∑s=0

(−1)s[1− aijs

]i

f1−aij−si fjf

si = 0, i 6= j

for h, h′ ∈ h, i, j ∈ I, where

qi = qdi , [n]i =qni − q

−ni

qni − q−ni

, [n]i! = [n]i . . . [1]i, and

[n

s

]i

=[n]i!

[n− s]i! [s]i!·

The standard Hopf algebra structure on Uq(g) has counit, coproduct, and antipode givenby

ε(qh) = 1, ε(ei) = ε(fi) = 0,

∆(qh) = qh ⊗ qh, ∆(ei) = ei ⊗ 1 + qdihi ⊗ ei, ∆(fi) = fi ⊗ q−dihi + 1⊗ fi,

S(qh) = q−h, S(ei) = −q−dihiei, S(fi) = −fiqdih,where h ∈ P∨ and i ∈ I. The unital subalgebras generated by {ei | i ∈ I}, {qh | h ∈ P∨},and {fi | i ∈ I} will be denoted by Uq(n+), Uq(h), and Uq(n−). The algebras Uq(b±) :=Uq(n±)Uq(h) are Hopf subalgebras of Uq(g).

For a Uq(g)-module V and µ ∈ P , denote the root space Vµ := {v ∈ V | qh · v =

q〈h,µ〉v,∀h ∈ P∨}.Let {Ti | i ∈ I} be the standard generators of the braid group of W . For a reduced

expression si1 . . . siN of w ∈W , let Tw := Ti1 . . . TiN in the braid group of W (this element isindependent on choice of reduced expression). We use the same notation for Lusztig’s braidgroup action [30] on Uq(g) and on integrable Uq(g)-modules (i.e., modules V on which ei andfi act locally nilpotent for i ∈ I and V = ⊕µ∈PVµ). For µ ∈ P+, let V (µ) be the irreduciblehighest weight Uq(g)-module with highest weight µ, and vµ be a highest weight vector of

it. For w ∈ W , denote vwµ = T−1w−1vµ. In (V (µ)wµ)∗, let ξwµ be such that 〈ξwµ, vwµ〉 = 1.

The quantum minors (viewed as functionals on Uq(g)) are defined as the matrix coefficients∆uµ,wµ := cξuµ,vwµ for u,w ∈ W and µ ∈ P+. Note that ∆uµ,wµ∆uν,wν = ∆u(µ+ν),w(µ+ν)

since T−1w−1(vµ ⊗ vν) = T−1

w−1vµ ⊗ T−1w−1vν .

7.2. Hopf pairings and integral forms. Recall that a Hopf pairing between Hopf K-algebras A and H is a bilinear form (·, ·) : A×H → K such that

(1) (ab, h) = (a, h(1))(b, h(2))

(2) (a, gh) = (a(1), g)(a(2), h)

(3) (a, 1) = εA(a) and (1, h) = εH(h)

for all a, b ∈ A and g, h ∈ H in terms of Sweedler notation.Let d ∈ Z>0 be an integer such that (P∨, P∨) ⊆ Z/d. The Rosso-Tanisaki form (·, ·)RT :

Uq(b−)× Uq(b+)→ Q(q1/d) is the Hopf pairing defined by

(fi, ej)RT = δij1

q−1i − qi

, (qh, qh′)RT = q−(h,h′), (fi, q

h)RT = 0 = (qh, ei)RT


for all i ∈ [1, r] and h ∈ P∨. The Rosso-Tanisaki form has the following useful properties,

(7.1)

(xqh, yqh

′)RT

=(x, y)RTq−(h,h′),(

Uq(n−), Uq(n+))RT⊂ Q(q),

and(Uq(n−)−γ , Uq(n+)δ

)RT

= 0

for x ∈ Uq(n−), y ∈ Uq(n+), and γ, δ ∈ Q+ with γ 6= δ, see [25, Ch. 6].Recall (2.2) and denote

Aq := Z[q±1].

The divided power integral forms Uq(n+)Aq and Uq(n−)Aq of Uq(n±) are the Aq-subalgebrasgenerated by

{eki /[k]i! | i ∈ I, k ∈ Z>0} and {fki /[k]i! | i ∈ I, k ∈ Z>0}.

The dual integral form Uq(n−)∨Aq of Uq(n−) is defined as

Uq(n−)∨Aq := {x ∈ Uq(n−) | (x, Uq(n+)Aq)RT ⊂ Aq}.

7.3. Quantum Schubert cells. Fixing a Weyl group element and a reduced expressionw = si1 . . . siN , we denote the following elements of W :

w≤k := si1 . . . sik , w[j,k] := sij . . . sik , w−1≤k := (w≤k)

−1, and w−1[j,k] := (w[j,k])

−1

where 0 ≤ j ≤ k ≤ N . To each root βk := w≤k−1(αik) ∈ Q+ for k ∈ [1, N ], associate theroot vectors

eβk := T−1

w−1≤k−1

(eik) ∈ Uq(n+)Aq and fβk := T−1

w−1≤k−1

(fik) ∈ Uq(n−)Aq .

The quantum Schubert cells Uq(n+(w)) and Uq(n−(w)) are defined to be the unital Q(q)-subalgebras of Uq(n±) generated by eβ1 , . . . , eβN and fβ1 , . . . , fβN , respectively. Theywere defined by De Concini–Kac–Procesi [11] and Lusztig [30], who considered the anti-isomorphic algebras U±q [w] = ∗

(Aq(n±(w))

). It was proved in [2, 28, 36] that

Uq(n±(w)

)= Uq

(n±)∩ T−1

w−1

(Uq(n∓)).

The dual integral form of Uq(n−(w)) is the Aq-algebra

Uq(n−(w))∨Aq := Uq(n−(w)) ∩ Uq(n−)∨Aq .

The dual PBW generators of Uq(n−(w)) are given by

f ′βk :=1

(fβk , eβk)RTfβk = (q−1

ik− qik)fβk ∈ Uq(n−(w))∨Aq

for k ∈ [1, N ]. Kimura proved [28, Proposition 4.26, Theorems 4.25 and 4.27] that

(7.2) Uq(n−(w))∨Aq = ⊕m1,...,mN∈NAq · (f′β1)m1 · · · (f ′βN )mN .

7.4. Quantum unipotent cells. Let Aq(n+), as in [19], denote the subalgebra of the fulldual Uq(b+)∗ of elements f that satisfy the following conditions:

(1) f(yqh) = f(y) for any y ∈ Uq(n+) and h ∈ P∨.(2) There is a finite subset S ⊆ Q+, such that f(x) = 0 for all x ∈ Uq(n+)γ for

γ ∈ Q+\ S.


The map ι : Uq(n−)→ Uq(b+)∗ given by

〈ι(x), y〉 = (x, y)RT for all x ∈ Uq(n−), y ∈ Uq(b+)

is an algebra homomorphism because the Rosso–Tanisaki form is a Hopf pairing. The imageof ι is contained in Aq(n+) by the properties listed in (7.1). Since the Rosso–Tanisaki formis non-degenerate, ι is an isomorphism onto Aq(n+),

ι : Uq(n−)'−−→ Aq(n+).

Following Geiß–Leclerc–Schroer [19], define the quantum unipotent cell Aq(n+(w)) ⊆Aq(n+) as the image of Uq(n−(w)) ⊆ Uq(n−) under ι,

ι : Uq(n−(w))'−−→ Aq(n+(w)) ⊂ Aq(n+).

The images of the elements of Uq(n−(w)) in Aq(n+(w)) will be denoted by the same symbols.We transport the automorphisms Ti via ι to a partial braid group action on Aq(n+(w)).Quantum unipotent cells also inherit a Q+-grading

(7.3) Aq(n+(w))γ := ι(Uq(n−(w))−γ

)for all γ ∈ Q+.

Finally, the dual integral form of Uq(n−(w)) gives rise to an Aq-integral form of thequantum unipotent cell Aq(n+(w)),

Aq(n+(w))Aq := ι(Uq(n−(w))∨Aq

).

The restriction of ι gives rise to the Aq-algebra isomorphism

(7.4) ι : Uq(n−(w))∨Aq'−−→ Aq(n+(w))Aq .

The integral forms Uq(n−)∨Aq , Uq(n−(w))∨Aq and Aq(n+(w))Aq are often defined by using

the Kashiwara [26] and Lusztig [30] bilinear forms on Uq(n−) instead of the Rosso–Tanisakiform. However, the corresponding Aq-algebras are isomorphic [24, Remark 5.3].

Following [19], define the unipotent quantum minors of Aq(n+(w)) for u ∈W , µ ∈ P+ asthe elements of Aq(n+(w))(u−w)µ such that

〈Duµ,wµ, yqh〉 := 〈ξuµ, yvwµ〉

for all y ∈ Uq(n+) and h ∈ P∨. The quantum minors ∆uµ,wµ ∈ Aq(g) only depend on uµand wµ but not on the individual choice of w, u and µ, [4, §9.3]. Since the unipotent minorsDuµ,wµ can be realized as homomorphic images of them [24, §6.3], the same is true for them.The minors Dµ,wµ q-commute with homogeneous elements with respect to the Q+-grading[24, Eq. (6.9)]:

(7.5) Dµ,wµx = q((w+1)µ,γ)xDµ,wµ, ∀µ ∈ P+, x ∈ Aq(n+(w))γ , γ ∈ Q+.

7.5. Specialization to roots of unity. Recall (3.1) and denote

Aε := Z[ε].

For every symmetrizable Kac–Moody algebra g and Weyl group element w ∈W , define the(integral) quantum unipotent cell at root of unity to be the Aε-algebra

Aε(n+(w))Aε := Aq(n+(w))Aq/(q − ε)Aq(n+(w))Aq .

Denote the canonical projection

(7.6) ηε : Aq(n+(w))Aq � Aε(n+(w))Aε

and for j ∈ [1, N ] set

(7.7) f ′′βj := ηει(f′βj

) ∈ Aε(n+(w))Aε .


By Kimura’s result in (7.2) and the isomorphism (7.4), we have

(7.8) Aε(n+(w))Aε = ⊕m1,...,mN∈NAε · (f′′β1)m1 · · · (f ′′βN )mN .

Theorem 7.1. (De Concini–Kac–Procesi [11]) For every symmetrizable Kac–Moody algebrag, Weyl group element w ∈ W , and primitive `-th root of unity ε such that ` is coprime to{di | i ∈ I},

Cε(n+(w))Aε = ⊕m1,...,mN∈NAε · (f′′β1)m1` · · · (f ′′βN )mN `

is a central Aε-subalgebra of Aε(n+(w))Aε.

The theorem was proved in [11] in the case when g is finite dimensional, but the sameproof works for general symmetrizable Kac–Moody algebras. Alternatively, in the case when` is odd, this theorem also follows by combining Proposition 4.4 and Theorem 8.5 (we notethat the proof of Theorem 8.5 does not use Theorem 7.1).

8. Discriminants of quantum unipotent cells at roots of unity

In this section we obtain an explicit formula for the discriminant of each (integral) quan-tum unipotent cell Aε(n+(w))Aε over the central subalgebra Cε(n+(w))Aε for every sym-metrizable Kac–Moody algebra g and Weyl group element w. It is also proved that thealgebras Aε(n+(w))Aε posses a strict root of unity quantum cluster algebra structure. Inthis picture we give an intrinsic interpretation of the central subalgebras Cε(n+(w))Aε incluster algebra terms.

8.1. Theorem on discriminants of quantum unipotent cell. For a Weyl group ele-ment w denote its support S(w) := {i ∈ I | si ≤ w}.

It follows from (7.8) and the definition of Cε(n+(w))Aε that Aε(n+(w))Aε is a free moduleover Cε(n+(w))Aε of rank `N with basis

(8.1) {(f ′′β1)m1 . . . (f ′′βN )mN | m1, . . . ,mN ∈ [0, `− 1]}.

The corresponding discriminant is given by:

Theorem 8.1. Let g be a symmetrizable Kac–Moody algebra, w be a Weyl group elementwith a reduced expression w = si1 . . . siN , and ` > 2 be an odd integer which is coprime todi for all i ∈ S(w). Let ε be a primitive `-th root of unity. Then

d(Aε(n+(w))Aε/Cε(n+(w))Aε

)=A×ε `

(N`N )∏

i∈S(w)

ηε(D$i,w$i)`N (`−1).

Note that, since Cε(n+(w))Aε is a polynomial algebra over Aε, Cε(n+(w))×Aε = A×ε . Thetheorem is proved in §8.5.

8.2. Cluster structures of the integral forms of quantum unipotent cells. For theconstruction of strict root of unity quantum cluster structure on Aε(n+(w))Aε we will useresults from [24, 27] on a quantum cluster algebra structure on

Aq(n+(w))A1/2q

:= Aq(n+(w))Aq ⊗Aq A1/2q .

Fix a reduced expression w = si1 . . . siN . In terms of the support of w, it is given byS(w) = {t ∈ I | t = ik for some k}. Let p : [1, N ] → [1, N − 1] ∪ {−∞} and s : [1, N ] →[2, N ] ∪ {∞} be the predecessor and successor maps given by

p(k) = max{j < k | ij = ik} where max∅ := −∞,s(k) = min{j > k | ij = ik} where min ∅ :=∞.


The mutable directions in the cluster structure will be given by the subset

ex(w) := {k ∈ [1, N ] | ij = ik for j > k}.

It has cardinality |ex(w)| = N − |S(w)| as each t ∈ S(w) in the support will have only one

j ∈ [1, N ] such that ij = t and s(j) =∞. Let Bw be the N × ex(w) matrix with entries

(Bw)j,k =

1, if j = p(k)

−1, if j = s(k)

aijik if j < k < s(j) < s(k)

−aijik if k < j < s(k) < s(j)

0, otherwise.

The principal part Bw is skew-symmetrizable by the matrix D := diag(dij , j ∈ ex(w)).

Moreover, Bw is compatible with the skew-symmetric N ×N matrix

(Λw)j,k := −(

(w≤j + 1)$ij , (w≤k − 1)$ik

), for 1 ≤ j < k ≤ N,

see [24, Proposition 7.2]. By (7.5), the unipotent quantum minors D$ik ,w≤k$ik, with weight

(1− w≤k)$ik , q-commute amongst themselves:

D$ij ,w≤j$ijD$ik ,w≤k$ik

= q(Λw)j,kD$ik ,w≤k$ikD$ij ,w≤j$ij

, 1 ≤ j < k ≤ N.

There is a unique toric frame Mwq : ZN → Fract(Aq(n+(w))A1/2

q) ' Fract(Tq(Λw)), with

corresponding skew-symmetric matrix Λw, given by

Mwq (ek) = qa[1,k]D$ik ,w≤k$ik

for any k ∈ [1, N ]

where

(8.2) a[j, k] = ‖(w[j,k] − 1)$ik‖2/4 ∈ Z/2.

The above facts show that (Λw, Bw) is a compatible pair and that (Mw

q , Bw) is a quantum

seed. The following theorem is proved in [24] and in [27] in the case of symmetric Kac–Moody algebras.

Theorem 8.2. Let g be any symmetrizable Kac–Moody algebra and w ∈W a Weyl elementwith a reduced expression w = si1 . . . siN . Then the integral form of the corresponding

quantum unipotent cells has a cluster structure, Aq(n+(w))A1/2q' Aq(Mw

q , Bw,∅).

Denote by ΞN the subset of the symmetric group SN consisting of permutations σ suchthat σ([1, k]) is an interval for 1 ≤ k ≤ N . We can combinatorially describe this subset interms of one-line notation for the elements of SN : first move 1 as far right as desired, thenmove 2 as far right as desired up to where 1 now is, then moving 3 right possibly up to2, etc. The elements of SN obtained in this way are precisely those of ΞN . The followingdiagram illustrates this with arrows denoting pairs of elements of ΞN obtained from eachother by a transposition:

[1 2 3 4 . . . N ] [2 1 3 4 . . . N ] [2 3 1 4 . . . N ] [2 3 4 1 . . . N ] . . .

[3 2 1 4 . . . N ] [3 2 4 1 . . . N ] . . .

.... . .


For each σ ∈ ΞN , [24, Theorem 7.3(b)] constructs a quantum seed of Aq(Mwq , B

w,∅).Their toric frames (up to a permutation of the basis as below) have cluster variables

(8.3) Mwq,σ(el) = qa[j,k]Dw≤j−1$ik ,w≤k$ik

= qa[j,k]Tw≤j−1D$ik ,w[j,k]$ik

,

where j = min{m ∈ σ([1, l]) | im = iσ(l)}, k = max{m ∈ σ([1, l]) | im = iσ(l)} and a[j, k]are given by (8.2). In particular, Mw

q,id = Mwq . The exchange matrices of these seeds will

not play a role in this paper. By abuse of notation we will denote by ΞN this collection of

quantum seeds of Aq(Mwq , B

w,∅).By [24, Theorem 7.3(c)] this collection of quantum seeds of Aq(n+(w))A1/2

qis linked by

mutations as follows. Let σ, σ′ ∈ ΞN be such that σ′ = (σ(k), σ(k + 1)) ◦ σ = σ ◦ (k, k + 1)for k ∈ [1, N − 1].

(8.4)If iσ(k) 6= iσ(k+1), then Mw

q,σ′ = Mwq,σ · (k, k + 1);

If iσ(k) = iσ(k+1), then Mwq,σ′ = µk(M

wq,σ),

where we use the canonical action of SN on quantum seeds and toric frames by reorderingof basis elements given by Mq · σ(ej) := Mq(eσ(j)) for σ ∈ SN and 1 ≤ j ≤ N .

The following lemma is simple and is left to the reader:

Lemma 8.3. The collection of quantum seeds ΞN of Aq(Mwq , B

w,∅) is a nerve.

8.3. Root of unity quantum cluster structure on integral quantum unipotentcells. Assume that ε1/2 is a primitive `-th root of unity. Denote

Aε(n+(w))A1/2ε

:= Aε(n+(w))Aε ⊗Aε A1/2ε

In the case when ` is odd, ε is also a primitive `-th root of of unity, A1/2ε = Aε, and

Aε(n+(w))A1/2ε

∼= Aε(n+(w))Aε . In the case when ` is even, ε is a primitive (`/2)-th root of

unity. Consider the canonical extension of the specialization (7.6) to a specialization map

ηε : Aq(n+(w))A1/2q

� Aε(n+(w))A1/2ε' Aq(n+(w))A1/2

q/(Φ`(q

1/2))

such that q1/2 7→ ε1/2. By [24, Theorem 7.3(a)]

Aε(Mwε ,Λw, B

w,∅) = Uε(Mwε ,Λw, B

w,∅),

so we are in a position to apply Theorem 5.11. Firstly, this gives that the maps

Mwε,σ := ηε ◦Mw

q,σ : ZN → Aε(n+(w))A1/2ε

are toric frames for all w ∈W and σ ∈ ΞN . Secondly, we obtain that

Aε(n+(w))A1/2ε' Aε(M

wε ,Λw, B

w,∅).

This leads to the following theorem:

Theorem 8.4. For every symmetrizable Kac–Moody algebra g, a Weyl group element wwith a reduced expression w = si1 . . . siN , and a primitive `-th root of unity ε1/2 for ` ∈ Z+,the following hold:

(1) Aε(n+(w))A1/2ε

has the structure of strict root of unity quantum cluster algebra and

is isomorphic to Aε(Mwε ,Λw, B

w,∅).(2) The root of unity quantum cluster algebra in part (1) has seeds indexed by σ ∈ Ξ

with toric frames Mwε,σ.

(3) The collection of seeds in part (2), to be denoted by ΞN , is a nerve and we have themutation formulas (8.4) between them with Mw

q,σ replaced by Mwε,σ.

(4) Under the isomorphism in part (1), Aε(n+(w))A1/2ε

= Aε(ΞN ,∅).


Proof. Parts (1) and (2) are established above.(3) The mutations formulas (8.4) with Mw

q,σ replaced by Mwε,σ follow from the original

formulas (8.4) by applying the ring homomorphism ηε. It follows from Lemma 8.3 that ΞNis a nerve.

(4) It is clear that, under the isomorphism in part (1), Aε(ΞN ,∅) ⊆ Aε(Mwε ,Λw, B

w,∅) =Aε(n+(w))A1/2

ε. For the inverse inclusion, note that for each k ∈ [1, N ], there exists σ ∈ ΞN

such that σ(1) = k. For that σ we have

Mwq,σ(e1) = q

1/2ikι(f ′βk)

by combining [24, Eq. (3.6), (7.2) and Theorem7.1(c)], and thus

(8.5) Mwε,σ(e1) = ε

dik/2

ikf ′′βk .

Hence, under the isomorphism in part (1), Aε(ΞN ,∅) ⊇ Aε(n+(w))A1/2ε

, which completes

the proof of the theorem. �

8.4. Identification of central subalgebras. Let ε be a primitive `-th root of unity suchthat ` is odd and coprime to the symmetrizing integers di for the Kac–Moody algebra gand i ∈ S(w), w ∈W . Choose a square root ε1/2 of ε such that ε1/2 is also a primitive `-th

root of unity. Then Aε = A1/2ε . By Theorem 8.4 we have the identifications

Aε(n+(w))Aε = Aε(n+(w))A1/2ε

= Aε(Mwε ,Λw, B

w,∅) = Aε(ΞN ,∅).

On the one hand, we have the central subalgebra Cε(ΞN ,∅) of Aε(ΞN ,∅) constructed bycluster theoretic methods, see §6.2. On the other hand, we have the De Concini–Kac–Procesicentral subalgebra Cε(n+(w))Aε of Aε(n+(w))Aε , see §7.5.

Theorem 8.5. In the setting of Theorem 8.1, the canonical central subalgebra Cε(ΞN ,∅)of Aε(ΞN ,∅) = Aε(n+(w))Aε coincides with the De Concini–Kac–Procesi central subalgebraCε(n+(w))Aε.

Proof. It follows from (8.5) that Cε(n+(w))Aε ⊆ Cε(ΞN ,∅). To show the reverse inclusion,we need to show that for all σ ∈ ΞN and j ∈ [1, N ], Mw

ε,σ(j)` ∈ Cε(n+(w))Aε . By (8.3), thisis equivalent to

(8.6) ηε(D`w≤j−1$ik ,w≤k$ik

) ∈ Cε(n+(w))Aε , ∀1 ≤ j ≤ k ≤ N with ij = ik.

We prove (8.6) by induction on k − j. The case k − j = 0 is trivial since

ηε(Dw≤k−1$kk ,w≤k$ik) = ε

dik/2

ikf ′′βk

by (8.5). Now assume that k − j = t for some t ∈ Z+ and that the statement holds forpairs 1 ≤ j′ ≤ k′ ≤ N with k′ − j′ < t. Since ij = ik, j ≤ p(k) and s(j) ≤ k. Consider thefollowing elements of ΞN :

σ = [j + 1, . . . , k − 1, j, k, k + 1, . . . , N, 1 . . . , j − 1] and

σ′ = [j + 1, . . . , k − 1, k, j, k + 1, . . . N, 1, . . . , j − 1] = σ(k − j, k − j + 1)

in the two line notation for elements of SN . By (8.4), Mwε,σ′ = µk−jM

wε,σ. From [23, Theorem

6.6] we have that the (k−j)-th column of the exchange matrix of the root of unity quantumseed of Aε(n+(w))A1/2

εcorresponding to σ has the form (b1, . . . , bN )> with

bk−j+1 = −1, bp(k)−j = −1 if j ≤ p(k),

bi ≥ 0 for i < k − j, i 6= p(k)− j,bi = 0 otherwise.


Combining this with Proposition 4.4 gives

Mwε,σ′(ek−j)

` = (µk−jMwε,σ(ek−j))

` = Mwε,σ(ek−j)

−`(Mwε,σ(ek−j+1)`M ` +

∏i<k−j,bi>0

Mwε,σ(ei)

`)

where

M ` :=

{Mwε,σ(ep(k)−j)

`, if j ≤ p(k)

1, otherwise.

It follows from (8.3) that Mwε,σ(ek−j+1)` = ηε(D

`w≤j−1$ik ,w≤k$ik

) and that Mwε,σ′(ek−j)

` and

Mwε,σ(ei)

` for i ≤ k − j are of the form ηε(D`w≤j′−1$ik′

,w≤k′$ik′) for pairs 1 ≤ j′ ≤ k′ ≤ N

with k′ − j′ < k − j. The induction assumption implies that

ηε(D`w≤j−1$ik ,w≤k$ik

) ∈ Fract(Cε(n+(w))Aε) ∩Aε(n+(w))Aε .

It remains to prove that

(8.7) Fract(Cε(n+(w))Aε) ∩Aε(n+(w))Aε = Cε(n+(w))Aε .

Let

P =∑

pm1,...,mN (f ′′β1)m1` . . . (f ′′βN )mN `, Q =∑

qm1,...,mN (f ′′β1)m1` . . . (f ′′βN )mN `

and

R =∑

rn1,...,nN (f ′′β1)n1 . . . (f ′′βN )nN ,

be such that P = RQ. Since f ′′βi are in the center of Aε(n+(w))Aε ,

QR =∑

rn1,...,nN qm1,...,mN (f ′′β1)n1+m1` . . . (f ′′βN )nm+mN `

In light of the PBW basis (7.8), the identity P = QR implies that rn1,...,nN = 0 unlessn1, . . . , nN are divisible by `. This proves (8.7). �

8.5. Proof of Theorem 8.1. As in the previous subsection we chose a square root ε1/2

of ε such that ε1/2 is also a primitive `-th root of unity. In particular, Aε = A1/2ε . By

Theorems 8.4 and 8.5 we have the identifications

Aε(n+(w))Aε = Aε(n+(w))A1/2ε

= Aε(Mwε ,Λw, B

w,∅) = Aε(ΞN ,∅) and

Cε(ΞN ,∅) = Cε(n+(w))Aε .

Since we are requiring that ` is coprime to all dik for 1 ≤ k ≤ N , the root of unity quantum

seeds of Aε(Mwε ,Λw, B

w,∅) satisfy condition (Coprime). Its frozen variables are

Mwε (ek) = εa[1,k]ηε(D$ik ,w≤k$ik

) = εa[1,k]ηε(D$ik ,w$ik) for k ∈ [1, N ]\ex,

where the last equality holds because w≤k$ik = w$ik for k ∈ [1, N ]\ex. By the definitionsof the sets ex and S(w), up to terms in A×ε , the frozen variables are

ηε(D$i,w$i) for i ∈ S(w).

Theorem 6.6 implies that

(8.8) d(Aε(n+(w))Aε/Cε(n+(w))Aε

)=A×ε `

(N`N )∏

i∈S(w)

ηε(D$i,w$i)ni


for some ni ∈ N. Eq (6.1) and the fact that (8.1) is a basis of Aε(n+(w))Aε over Cε(n+(w))Aεimply that with respect to the Q+-grading (7.3) of Aε(n+(w))Aε ,

deg d(Aε(n+(w))Aε/Cε(n+(w))Aε

)= 2

∑0≤mk≤`−1

deg((f ′′β1)m1 . . . (f ′′βN )mN

)(8.9)

= `N (`− 1)(β1 + · · ·+ βN ).

For k ∈ [1, N ]\ex, let rk is the maximal integer such that prk(k) 6= −∞. Iterating theidentity w≤j$ij = w≤j−1($ij − αij ) = w≤p(j)$ij − βj , ∀j ∈ [1, N ] gives

βpmk (k) + · · ·+ βk = (1− w≤k)$ik = (1− w)$ik .

Therefore,

(8.10) β1 + · · ·+ βN =∑

k∈[1,N ]\ex

(βprk (k) + · · ·+ βk) =∑

i∈S(w)

(1− w)$i.

Combining (8.8)–(8.10) and using that deg ηε(D$i,w$i) = (1− w)$i leads to

(1− w)∑

i∈S(w)

(ni − (`− 1)`N ) = 0.

This implies that ni = (` − 1)`N for all i ∈ S(w) because (1 − w) is nondegenerate onSpan{$i | i ∈ S(w)}. �

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Department of Mathematics, Xavier University of Louisiana, New Orleans, LA 70125E-mail address: [email protected]

Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556E-mail address: [email protected]

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803E-mail address: [email protected]

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