Based Quantum Cluster Algebras Karl Schmidt University of Oregon - - PowerPoint PPT Presentation

Based Quantum Cluster Algebras

Karl Schmidt

University of Oregon

June 4, 2018 Interactions of quantum affine algebras with cluster algebras, current algebras, and categorification Catholic University of America

Karl Schmidt Based Quantum Cluster Algebras

Motivation

Fix d ∈ Z>0, let q

1 2d be an indeterminate, and set K = Q(q 1 2d ). Karl Schmidt Based Quantum Cluster Algebras

Motivation

Fix d ∈ Z>0, let q

1 2d be an indeterminate, and set K = Q(q 1 2d ).

Definition For any integers m, n ≥ 1, Aq[Matm,n] is the K-algebra generated by symbols {xi,j | 1 ≤ i ≤ m and 1 ≤ j ≤ n} and subject to the relations xi,ℓxi,j = qxi,jxi,ℓ (j < ℓ), xk,jxi,j = qxi,jxk,j (i < k), xk,jxi,ℓ = xi,ℓxk,j (i < k, j < ℓ), xk,ℓxi,j = xi,jxk,ℓ + (q − q−1)xi,ℓxk,j (i < k, j < ℓ).

Karl Schmidt Based Quantum Cluster Algebras

Theorem Aq[Matm,n] is isomorphic (as an algebra) to a quantum Schubert cell of Uq(slm+n).

Karl Schmidt Based Quantum Cluster Algebras

Theorem Aq[Matm,n] is isomorphic (as an algebra) to a quantum Schubert cell of Uq(slm+n). Theorem (Geiß-Leclerc-Schr¨

• er ’11, Goodearl-Yakimov ’13)

Every quantum Schubert cell is a quantum cluster algebra.

Karl Schmidt Based Quantum Cluster Algebras

Theorem Aq[Matm,n] is isomorphic (as an algebra) to a quantum Schubert cell of Uq(slm+n). Theorem (Geiß-Leclerc-Schr¨

• er ’11, Goodearl-Yakimov ’13)

Every quantum Schubert cell is a quantum cluster algebra. Theorem (Kang-Kashiwara-Kim-Oh ’15) If g is of type ADE, the quantum cluster monomials of every quantum Schubert cell are elements of the dual canonical basis.

Karl Schmidt Based Quantum Cluster Algebras

If m ≥ 2, then Aq[Matm,n] is a locally finite Uq(slm)-module algebra.

Karl Schmidt Based Quantum Cluster Algebras

If m ≥ 2, then Aq[Matm,n] is a locally finite Uq(slm)-module algebra. Given any two Uq(g)-module algebras A and B satisfying certain conditions, there is a natural way to make their tensor product (as vector spaces) into a Uq(g)-module algebra: the braided tensor product A⊗B.

Karl Schmidt Based Quantum Cluster Algebras

If m ≥ 2, then Aq[Matm,n] is a locally finite Uq(slm)-module algebra. Given any two Uq(g)-module algebras A and B satisfying certain conditions, there is a natural way to make their tensor product (as vector spaces) into a Uq(g)-module algebra: the braided tensor product A⊗B. Proposition Aq[Matm,n1]⊗Aq[Matm,n2] ∼ = Aq[Matm,n1+n2].

Karl Schmidt Based Quantum Cluster Algebras

“Definition” A quantum cluster algebra is a K-algebra with a distinguished set

• f generators, called (quantum) cluster variables. Cluster variables

are gathered into maximal (overlapping) subsets called clusters, the elements of which pairwise commute up to powers of q. Every cluster is reachable from any other cluster by a series of single-variable exchanges, called mutations.

Karl Schmidt Based Quantum Cluster Algebras

“Definition” A quantum cluster algebra is a K-algebra with a distinguished set

• f generators, called (quantum) cluster variables. Cluster variables

are gathered into maximal (overlapping) subsets called clusters, the elements of which pairwise commute up to powers of q. Every cluster is reachable from any other cluster by a series of single-variable exchanges, called mutations. Key features for our discussion: Every quantum cluster algebra has a unique anti-linear algebra anti-involution which fixes the cluster variables (usually denoted by z → z).

Karl Schmidt Based Quantum Cluster Algebras

“Definition” A quantum cluster algebra is a K-algebra with a distinguished set

• f generators, called (quantum) cluster variables. Cluster variables

are gathered into maximal (overlapping) subsets called clusters, the elements of which pairwise commute up to powers of q. Every cluster is reachable from any other cluster by a series of single-variable exchanges, called mutations. Key features for our discussion: Every quantum cluster algebra has a unique anti-linear algebra anti-involution which fixes the cluster variables (usually denoted by z → z). Quantum cluster monomials are monomials in the cluster variables of a single cluster, scaled by the power of q

1 2d so

that they are fixed under the bar.

Karl Schmidt Based Quantum Cluster Algebras

“Definition” Uq(g) is a Hopf algebra over K, generated by symbols {K

± 1

2

i

, Ei, Fi | i ∈ I}, subject to some relations.

Karl Schmidt Based Quantum Cluster Algebras

“Definition” Uq(g) is a Hopf algebra over K, generated by symbols {K

± 1

2

i

, Ei, Fi | i ∈ I}, subject to some relations.We use the comultiplication with (for example) ∆(Ei) = Ei ⊗ K

1 2

i + K − 1

2

i

⊗ Ei.

Karl Schmidt Based Quantum Cluster Algebras

“Definition” Uq(g) is a Hopf algebra over K, generated by symbols {K

± 1

2

i

, Ei, Fi | i ∈ I}, subject to some relations.We use the comultiplication with (for example) ∆(Ei) = Ei ⊗ K

1 2

i + K − 1

2

i

⊗ Ei. Key features for our discussion: Uq(g) has a unique anti-linear algebra involution such that K

± 1

2

i

= K

∓ 1

2

i

, Ei = Ei, and Fi = Fi.

Karl Schmidt Based Quantum Cluster Algebras

“Definition” Uq(g) is a Hopf algebra over K, generated by symbols {K

± 1

2

i

, Ei, Fi | i ∈ I}, subject to some relations.We use the comultiplication with (for example) ∆(Ei) = Ei ⊗ K

1 2

i + K − 1

2

i

⊗ Ei. Key features for our discussion: Uq(g) has a unique anti-linear algebra involution such that K

± 1

2

i

= K

∓ 1

2

i

, Ei = Ei, and Fi = Fi. Uq(g) has a “universal R-matrix”, which braids the category

• f weight modules on which Ei acts locally nilpotently for

each i ∈ I. R satisfies “R = R−1”.

Karl Schmidt Based Quantum Cluster Algebras

Definition A barred module algebra is a pair (A, ¯), where A is a Uq(g)-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u(a) = u(a) for u ∈ Uq(g) and a ∈ A.

Karl Schmidt Based Quantum Cluster Algebras

Definition A barred module algebra is a pair (A, ¯), where A is a Uq(g)-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u(a) = u(a) for u ∈ Uq(g) and a ∈ A. Example (Aq[Matm,n], ¯) is a barred module over Uq(slm), where ¯ : Aq[Matm,n] → Aq[Matm,n] is the unique anti-linear algebra anti-involution such that xi,j = xi,j.

Karl Schmidt Based Quantum Cluster Algebras

Definition A barred module algebra is a pair (A, ¯), where A is a Uq(g)-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u(a) = u(a) for u ∈ Uq(g) and a ∈ A. Example (Aq[Matm,n], ¯) is a barred module over Uq(slm), where ¯ : Aq[Matm,n] → Aq[Matm,n] is the unique anti-linear algebra anti-involution such that xi,j = xi,j. Furthermore, the dual canonical basis is fixed under the bar.

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) Given barred module algebras (A, ¯) and (A′, ¯), there is a unique barred module algebra structure (A⊗A′, ¯) so that a ⊗ 1 = a ⊗ 1 and 1 ⊗ a′ = 1 ⊗ a′ for a ∈ A and a′ ∈ A′.

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) Given barred module algebras (A, ¯) and (A′, ¯), there is a unique barred module algebra structure (A⊗A′, ¯) so that a ⊗ 1 = a ⊗ 1 and 1 ⊗ a′ = 1 ⊗ a′ for a ∈ A and a′ ∈ A′. Caution: It is almost never the case that a ⊗ a′ = a ⊗ a′.

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) Given barred module algebras (A, ¯) and (A′, ¯), there is a unique barred module algebra structure (A⊗A′, ¯) so that a ⊗ 1 = a ⊗ 1 and 1 ⊗ a′ = 1 ⊗ a′ for a ∈ A and a′ ∈ A′. Caution: It is almost never the case that a ⊗ a′ = a ⊗ a′.Namely, a ⊗ a′ = (a ⊗ 1)(1 ⊗ a′) = (1 ⊗ a′)(a ⊗ 1).

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) If B and B′ are bar-invariant bases of barred module algebras (A, ¯) and (A′, ¯) satisfying certain criteria, there is a canonical choice B ⋄ B′ of bar-invariant basis of (A⊗A′, ¯) satisfying the same criteria.

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) If B and B′ are bar-invariant bases of barred module algebras (A, ¯) and (A′, ¯) satisfying certain criteria, there is a canonical choice B ⋄ B′ of bar-invariant basis of (A⊗A′, ¯) satisfying the same criteria. Let Bm,n be the dual canonical basis for Aq[Matm,n].

Karl Schmidt Based Quantum Cluster Algebras

Theorem (S) If B and B′ are bar-invariant bases of barred module algebras (A, ¯) and (A′, ¯) satisfying certain criteria, there is a canonical choice B ⋄ B′ of bar-invariant basis of (A⊗A′, ¯) satisfying the same criteria. Let Bm,n be the dual canonical basis for Aq[Matm,n]. Theorem (S) The Uq(g)-module algebra isomorphism Aq[Matm,n1]⊗Aq[Matm,n2] ∼ = Aq[Matm,n1+n2] carries Bm,n1 ⋄ Bm,n2 to Bm,n1+n2.

Karl Schmidt Based Quantum Cluster Algebras

Example Aq[U], Aq[B], and Aq[G/U] are certain Uq(g)-module algebras, generated (as module algebras) by {xi | i ∈ I}, {v±1

i

| i ∈ I}, and {vi | i ∈ I}, respectively.

Karl Schmidt Based Quantum Cluster Algebras

Example Aq[U], Aq[B], and Aq[G/U] are certain Uq(g)-module algebras, generated (as module algebras) by {xi | i ∈ I}, {v±1

i

| i ∈ I}, and {vi | i ∈ I}, respectively. Each has a unique anti-linear algebra anti-involution fixing these generators and giving it a barred module algebra structure. In fact, these are all quantum cluster algebras with the anti-linear anti-involutions fixing the quantum cluster monomials.

Karl Schmidt Based Quantum Cluster Algebras

Example Aq[U], Aq[B], and Aq[G/U] are certain Uq(g)-module algebras, generated (as module algebras) by {xi | i ∈ I}, {v±1

i

| i ∈ I}, and {vi | i ∈ I}, respectively. Each has a unique anti-linear algebra anti-involution fixing these generators and giving it a barred module algebra structure. In fact, these are all quantum cluster algebras with the anti-linear anti-involutions fixing the quantum cluster monomials. Theorem (Berenstein-S) Aq[U]⊗n ∼ = Aq[U]⊗n and Aq[B]⊗n ∼ = Aq[B]⊗n (as algebras).

Karl Schmidt Based Quantum Cluster Algebras

Example Aq[U], Aq[B], and Aq[G/U] are certain Uq(g)-module algebras, generated (as module algebras) by {xi | i ∈ I}, {v±1

i

| i ∈ I}, and {vi | i ∈ I}, respectively. Each has a unique anti-linear algebra anti-involution fixing these generators and giving it a barred module algebra structure. In fact, these are all quantum cluster algebras with the anti-linear anti-involutions fixing the quantum cluster monomials. Theorem (Berenstein-S) Aq[U]⊗n ∼ = Aq[U]⊗n and Aq[B]⊗n ∼ = Aq[B]⊗n (as algebras). Corollary Aq[U]⊗n and Aq[B]⊗n are quantum cluster algebras.

Karl Schmidt Based Quantum Cluster Algebras

What about Aq[G/U]?

Unlike the others, Aq[G/U]⊗n doesn’t “trivialize”.

Karl Schmidt Based Quantum Cluster Algebras

What about Aq[G/U]?

Unlike the others, Aq[G/U]⊗n doesn’t “trivialize”. Conjecture Aq[G/U]⊗n is a quantum cluster algebra.

Karl Schmidt Based Quantum Cluster Algebras

What about Aq[G/U]?

Unlike the others, Aq[G/U]⊗n doesn’t “trivialize”. Conjecture Aq[G/U]⊗n is a quantum cluster algebra. Example Aq[SL2/U]⊗n ∼ = Aq[Mat2,n] and is therefore a quantum cluster algebra.

Karl Schmidt Based Quantum Cluster Algebras

What about Aq[G/U]?

Unlike the others, Aq[G/U]⊗n doesn’t “trivialize”. Conjecture Aq[G/U]⊗n is a quantum cluster algebra. Example Aq[SL2/U]⊗n ∼ = Aq[Mat2,n] and is therefore a quantum cluster algebra. Example (S) Aq[SL3/U] is a quantum cluster algebra of type A1, so the cluster monomials form a basis (call it B). Aq[SL3/U]⊗Aq[SL3/U] is a quantum cluster algebra of type D4 and B ⋄ B coincides with the set of cluster monomials.

Karl Schmidt Based Quantum Cluster Algebras

Thank You!

Karl Schmidt Based Quantum Cluster Algebras