Quantum cluster algebra at roots of unity and discriminant formula - - PowerPoint PPT Presentation

Quantum cluster algebra at roots of unity and discriminant formula

Bach Nguyen

Louisiana State University

A conference celebrating the 60-th birthday of Vyjayanthi Chari June 04, 2018

Quantum Cluster Algebra

We will be working over Z[q±1/2] for an formal variable q.

• For a skew symmetric matrix Γ ∈ MN(Z) we define a quantum

torus Tq(Γ) over Z[q±1/2].

Quantum Cluster Algebra

We will be working over Z[q±1/2] for an formal variable q.

• For a skew symmetric matrix Γ ∈ MN(Z) we define a quantum

torus Tq(Γ) over Z[q±1/2].

• A map M : ZN −

→ F is a toric frame if there exist Γ such that it defines an embedding Tq(Γ) ֒ → F where F ∼ = Fract(Tq(Γ)).

Quantum Cluster Algebra

We will be working over Z[q±1/2] for an formal variable q.

• For a skew symmetric matrix Γ ∈ MN(Z) we define a quantum

torus Tq(Γ) over Z[q±1/2].

• A map M : ZN −

→ F is a toric frame if there exist Γ such that it defines an embedding Tq(Γ) ֒ → F where F ∼ = Fract(Tq(Γ)).

• Fix n ≤ N, and let ex ⊆ [1, N] such that |ex| = n. An integral

matrix ˜ BN×ex is called exchange matrix if the submatrix Bex is skew symmetrizable.

Quantum Cluster Algebra

We will be working over Z[q±1/2] for an formal variable q.

• For a skew symmetric matrix Γ ∈ MN(Z) we define a quantum

torus Tq(Γ) over Z[q±1/2].

• A map M : ZN −

→ F is a toric frame if there exist Γ such that it defines an embedding Tq(Γ) ֒ → F where F ∼ = Fract(Tq(Γ)).

• Fix n ≤ N, and let ex ⊆ [1, N] such that |ex| = n. An integral

matrix ˜ BN×ex is called exchange matrix if the submatrix Bex is skew symmetrizable.

• A compatible pair (M, ˜

B) is called a quantum seed and its corresponding quantum cluster variables are M(ej) for j ∈ [1, N].

Quantum Cluster Algebra

Torus frame M : ZN − → F Exchange matrix ˜ B Quantum seed (M, ˜ B) Quantum cluster variables M(ej)’s

Quantum Cluster Algebra

• For each k ∈ ex, one has mutation µk which takes quantum

seed to quantum seed. µk((M, ˜ B)) = (µk(M), µk( ˜ B)).

Quantum Cluster Algebra

• For each k ∈ ex, one has mutation µk which takes quantum

seed to quantum seed. µk((M, ˜ B)) = (µk(M), µk( ˜ B)).

• The cluster variables indexed by [1, N] \ ex are called frozen

variables.

Quantum Cluster Algebra

• For each k ∈ ex, one has mutation µk which takes quantum

seed to quantum seed. µk((M, ˜ B)) = (µk(M), µk( ˜ B)).

• The cluster variables indexed by [1, N] \ ex are called frozen

variables.

• Let inv ⊆ [1, N] \ ex.

Quantum Cluster Algebra

• For each k ∈ ex, one has mutation µk which takes quantum

seed to quantum seed. µk((M, ˜ B)) = (µk(M), µk( ˜ B)).

• The cluster variables indexed by [1, N] \ ex are called frozen

variables.

• Let inv ⊆ [1, N] \ ex.
• Mutation-equivalent of quantum seeds

(M, ˜ B)

µ1

− → · · ·

µk

− → (M′, ˜ B′) M(ej) M′(ej)

Quantum Cluster Algebra

• For each k ∈ ex, one has mutation µk which takes quantum

seed to quantum seed. µk((M, ˜ B)) = (µk(M), µk( ˜ B)).

• The cluster variables indexed by [1, N] \ ex are called frozen

variables.

• Let inv ⊆ [1, N] \ ex.
• Mutation-equivalent of quantum seeds

(M, ˜ B)

µ1

− → · · ·

µk

− → (M′, ˜ B′) M(ej) M′(ej)

• The quantum cluster algebra Aq(M, ˜

B, inv) is the algebra generated by all cluster variables M′(ej), j ∈ [1, N] and M′(ek)−1, k ∈ inv for all quantum seeds (M′, ˜ B′) which are mutation-equivalent to (M, ˜ B).

Quantum Cluster Algebra at Roots of Unity

Let ǫ1/2 be a primitive ℓth root of unity and we work over Z[ǫ±1/2].

• The based quantum torus is now Tǫ(Γ)

Quantum Cluster Algebra at Roots of Unity

Let ǫ1/2 be a primitive ℓth root of unity and we work over Z[ǫ±1/2].

• The based quantum torus is now Tǫ(Γ)
• Note that we are not specialize Tq(Γ) at ǫ but simply define a

quantum torus over Z[ǫ±1/2].

Quantum Cluster Algebra at Roots of Unity

Let ǫ1/2 be a primitive ℓth root of unity and we work over Z[ǫ±1/2].

• The based quantum torus is now Tǫ(Γ)
• Note that we are not specialize Tq(Γ) at ǫ but simply define a

quantum torus over Z[ǫ±1/2].

• Define the toric frame M as before and Γ be its skew

symmetric matrix. Similarly, we have the root of unity quantum seed (M, ˜ B, Γ).

Quantum Cluster Algebra at Roots of Unity

Let ǫ1/2 be a primitive ℓth root of unity and we work over Z[ǫ±1/2].

• The based quantum torus is now Tǫ(Γ)
• Note that we are not specialize Tq(Γ) at ǫ but simply define a

quantum torus over Z[ǫ±1/2].

• Define the toric frame M as before and Γ be its skew

symmetric matrix. Similarly, we have the root of unity quantum seed (M, ˜ B, Γ).

• The quantum cluster algebra at root of unity

Aǫ(M, ˜ B, Γ, inv) is a Z[ǫ±1/2]-algebra generated by all cluster variables M′(ej), j ∈ [1, N] and M′(ek)−1, k ∈ inv for all root

• f unity quantum seeds (M′, ˜

B′, Γ′) which are mutation-equivalent to (M, ˜ B, Γ). [N.–Trampel–Yakimov]

Quantum Cluster Algebra at Roots of Unity

Let A( ˜ B) be the cluster algebra associated to the exchange matrix ˜ B.

Quantum Cluster Algebra at Roots of Unity

Let A( ˜ B) be the cluster algebra associated to the exchange matrix ˜ B.

Theorem 1 (N.–Trampel–Yakimov)

The exchange graphs of Aq(M, ˜ B), Aǫ(M, ˜ B, Γ) and A( ˜ B) are all

• isomorphic. Moreover, the root of unity quantum cluster algebra

satisfies the Laurent phenomenon.

Quantum Cluster Algebra at Roots of Unity

Let A( ˜ B) be the cluster algebra associated to the exchange matrix ˜ B.

Theorem 1 (N.–Trampel–Yakimov)

The exchange graphs of Aq(M, ˜ B), Aǫ(M, ˜ B, Γ) and A( ˜ B) are all

• isomorphic. Moreover, the root of unity quantum cluster algebra

satisfies the Laurent phenomenon.

Theorem 2 (N.–Trampel–Yakimov)

The elements M′(ej)ℓ, j ∈ [1, N] and M′(ek)−ℓ, k ∈ inv are central in Aǫ(M, ˜ B, Γ). Moreover, the central subalgebra generated by them is isomorphic to the cluster algebra A( ˜ B).

Discriminant of Algebras

Let A be a noncommutative algebra .

• We call (A, tr) is an algebra with trace if tr : A −

→ A such that for any x, y ∈ A tr(xy) = tr(yx), tr(y)x = xtr(y), tr(xtr(y)) = tr(y)tr(x). Note that these conditions imply im(tr) = C ⊂ Z(A) and tr is C-linear.

Discriminant of Algebras

Let A be a noncommutative algebra .

• We call (A, tr) is an algebra with trace if tr : A −

→ A such that for any x, y ∈ A tr(xy) = tr(yx), tr(y)x = xtr(y), tr(xtr(y)) = tr(y)tr(x). Note that these conditions imply im(tr) = C ⊂ Z(A) and tr is C-linear.

• Let (A, tr) be an algebra with trace and Y = {y1, ..., yn} ⊂ A.

We define discriminant of Y to be d(Y : tr) = det[tr(yiyj)] ∈ C.

Discriminant of Algebras

Let A be a noncommutative algebra .

• We call (A, tr) is an algebra with trace if tr : A −

→ A such that for any x, y ∈ A tr(xy) = tr(yx), tr(y)x = xtr(y), tr(xtr(y)) = tr(y)tr(x). Note that these conditions imply im(tr) = C ⊂ Z(A) and tr is C-linear.

• Let (A, tr) be an algebra with trace and Y = {y1, ..., yn} ⊂ A.

We define discriminant of Y to be d(Y : tr) = det[tr(yiyj)] ∈ C.

• When A is free of rank n over some central subalgebra C, we

use the map tr : A ֒ → Mn(C) − → C. Then discriminant of A

• ver C is d(A/C) =C × d(Y : tr) for a chosen C-basis Y of A.

Discriminant of Quantum Cluster Algebra

Suppose Θ is a finite set of seeds in Aǫ(M, ˜ B, Γ) such that every 2 seeds in Θ are connected by a sequence of mutations in Θ and every nonfrozen vertex is mutated at least one time in Θ.

Discriminant of Quantum Cluster Algebra

Suppose Θ is a finite set of seeds in Aǫ(M, ˜ B, Γ) such that every 2 seeds in Θ are connected by a sequence of mutations in Θ and every nonfrozen vertex is mutated at least one time in Θ.

Proposition 3 (N.–Trampel–Yakimov)

Let Aǫ(Θ) be the subalgebra of Aǫ(M, ˜ B, Γ) generated by the cluster variables in Θ. Let Cǫ(Θ) be the central subalgebra of Aǫ(Θ) generated by the ℓth power of the cluster variables. Then Aǫ(Θ) is finitely generated as a Cǫ(Θ)-module.

Discriminant of Quantum Cluster Algebra

Theorem 4 (N.–Trampel–Yakimov)

Suppose Aǫ(Θ) is free over Cǫ(Θ). Then d(Aǫ(Θ)/Cǫ(Θ)) =

• (noninverted frozen variables)

.

Discriminant of Quantum Cluster Algebra

Theorem 4 (N.–Trampel–Yakimov)

Suppose Aǫ(Θ) is free over Cǫ(Θ). Then d(Aǫ(Θ)/Cǫ(Θ)) =

• (noninverted frozen variables)

. Consider the quantum group Uq(g) for a symmetrizable Kac–Moody algebra g, then the quantum Schubert cell algebra Uq(n+ ∩ w(n−)) is a cluster algebra due to [Geiss–Leclerc–Schroer, Goodearl–Yakimov].

Discriminant of Quantum Cluster Algebra

Theorem 4 (N.–Trampel–Yakimov)

Suppose Aǫ(Θ) is free over Cǫ(Θ). Then d(Aǫ(Θ)/Cǫ(Θ)) =

• (noninverted frozen variables)

. Consider the quantum group Uq(g) for a symmetrizable Kac–Moody algebra g, then the quantum Schubert cell algebra Uq(n+ ∩ w(n−)) is a cluster algebra due to [Geiss–Leclerc–Schroer, Goodearl–Yakimov].

Theorem 5 (N.–Trampel–Yakimov)

For any symmetrizable Kac–Moody algebra g, w ∈ W , and ǫ an

• dd primitive root of unity,

d(Uǫ(n+ ∩ w(n−))/Cǫ) =

• ∆ℓt+1(ℓ−1)

ωi,wωi

where t is the length of w.