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 1 1 2 d be an indeterminate, and set K = Q ( q 2 d ). Fix d ∈ Z > 0 , let q Karl Schmidt Based Quantum Cluster Algebras

Motivation 1 1 2 d be an indeterminate, and set K = Q ( q 2 d ). Fix d ∈ Z > 0 , let q Definition For any integers m , n ≥ 1, A q [ Mat m , n ] is the K -algebra generated by symbols { x i , j | 1 ≤ i ≤ m and 1 ≤ j ≤ n } and subject to the relations x i ,ℓ x i , j = qx i , j x i ,ℓ ( j < ℓ ) , x k , j x i , j = qx i , j x k , j ( i < k ) , x k , j x i ,ℓ = x i ,ℓ x k , j ( i < k , j < ℓ ) , x k ,ℓ x i , j = x i , j x k ,ℓ + ( q − q − 1 ) x i ,ℓ x k , j ( i < k , j < ℓ ) . Karl Schmidt Based Quantum Cluster Algebras

Theorem A q [ Mat m , n ] is isomorphic (as an algebra) to a quantum Schubert cell of U q ( sl m + n ) . Karl Schmidt Based Quantum Cluster Algebras

Theorem A q [ Mat m , n ] is isomorphic (as an algebra) to a quantum Schubert cell of U q ( sl m + n ) . Theorem (Geiß-Leclerc-Schr¨ oer ’11, Goodearl-Yakimov ’13) Every quantum Schubert cell is a quantum cluster algebra. Karl Schmidt Based Quantum Cluster Algebras

Theorem A q [ Mat m , n ] is isomorphic (as an algebra) to a quantum Schubert cell of U q ( sl m + n ) . Theorem (Geiß-Leclerc-Schr¨ oer ’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 A q [ Mat m , n ] is a locally finite U q ( sl m )-module algebra. Karl Schmidt Based Quantum Cluster Algebras

If m ≥ 2, then A q [ Mat m , n ] is a locally finite U q ( sl m )-module algebra. Given any two U q ( g )-module algebras A and B satisfying certain conditions, there is a natural way to make their tensor product (as vector spaces) into a U q ( g )-module algebra: the braided tensor product A ⊗ B . Karl Schmidt Based Quantum Cluster Algebras

If m ≥ 2, then A q [ Mat m , n ] is a locally finite U q ( sl m )-module algebra. Given any two U q ( g )-module algebras A and B satisfying certain conditions, there is a natural way to make their tensor product (as vector spaces) into a U q ( g )-module algebra: the braided tensor product A ⊗ B . Proposition A q [ Mat m , n 1 ] ⊗A q [ Mat m , n 2 ] ∼ = A q [ Mat m , n 1 + n 2 ] . Karl Schmidt Based Quantum Cluster Algebras

“Definition” A quantum cluster algebra is a K -algebra with a distinguished set of 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 of 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 of 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 1 2 d so variables of a single cluster, scaled by the power of q that they are fixed under the bar. Karl Schmidt Based Quantum Cluster Algebras

“Definition” U q ( g ) is a Hopf algebra over K , generated by symbols ± 1 { K 2 , E i , F i | i ∈ I } , subject to some relations. i Karl Schmidt Based Quantum Cluster Algebras

“Definition” U q ( g ) is a Hopf algebra over K , generated by symbols ± 1 { K 2 , E i , F i | i ∈ I } , subject to some relations.We use the i comultiplication with (for example) 1 − 1 ∆( E i ) = E i ⊗ K i + K 2 2 ⊗ E i . i Karl Schmidt Based Quantum Cluster Algebras

“Definition” U q ( g ) is a Hopf algebra over K , generated by symbols ± 1 { K 2 , E i , F i | i ∈ I } , subject to some relations.We use the i comultiplication with (for example) 1 − 1 ∆( E i ) = E i ⊗ K i + K 2 2 ⊗ E i . i Key features for our discussion: U q ( g ) has a unique anti-linear algebra involution such that ± 1 ∓ 1 K 2 = K 2 , E i = E i , and F i = F i . i i Karl Schmidt Based Quantum Cluster Algebras

“Definition” U q ( g ) is a Hopf algebra over K , generated by symbols ± 1 { K 2 , E i , F i | i ∈ I } , subject to some relations.We use the i comultiplication with (for example) 1 − 1 ∆( E i ) = E i ⊗ K i + K 2 2 ⊗ E i . i Key features for our discussion: U q ( g ) has a unique anti-linear algebra involution such that ± 1 ∓ 1 K 2 = K 2 , E i = E i , and F i = F i . i i U q ( g ) has a “universal R -matrix”, which braids the category of weight modules on which E i 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 U q ( g )-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u ( a ) = u ( a ) for u ∈ U q ( g ) and a ∈ A . Karl Schmidt Based Quantum Cluster Algebras

Definition A barred module algebra is a pair ( A , ¯), where A is a U q ( g )-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u ( a ) = u ( a ) for u ∈ U q ( g ) and a ∈ A . Example ( A q [ Mat m , n ] , ¯) is a barred module over U q ( sl m ), where ¯ : A q [ Mat m , n ] → A q [ Mat m , n ] is the unique anti-linear algebra anti-involution such that x i , j = x i , j . Karl Schmidt Based Quantum Cluster Algebras

Definition A barred module algebra is a pair ( A , ¯), where A is a U q ( g )-module algebra and ¯ : A → A is an anti-linear algebra anti-involution such that u ( a ) = u ( a ) for u ∈ U q ( g ) and a ∈ A . Example ( A q [ Mat m , n ] , ¯) is a barred module over U q ( sl m ), where ¯ : A q [ Mat m , n ] → A q [ Mat m , n ] is the unique anti-linear algebra anti-involution such that x i , j = x i , 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 B m , n be the dual canonical basis for A q [ Mat m , n ]. Karl Schmidt Based Quantum Cluster Algebras

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