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 ∈ M N ( Z ) we define a quantum torus T q (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 ∈ M N ( Z ) we define a quantum torus T q (Γ) over Z [ q ± 1 / 2 ] . • A map M : Z N − → F is a toric frame if there exist Γ such that it defines an embedding T q (Γ) ֒ → F where F ∼ = Fract ( T q (Γ)) .
Quantum Cluster Algebra We will be working over Z [ q ± 1 / 2 ] for an formal variable q . • For a skew symmetric matrix Γ ∈ M N ( Z ) we define a quantum torus T q (Γ) over Z [ q ± 1 / 2 ] . • A map M : Z N − → F is a toric frame if there exist Γ such that it defines an embedding T q (Γ) ֒ → F where F ∼ = Fract ( T q (Γ)) . • Fix n ≤ N , and let ex ⊆ [1 , N ] such that | ex | = n . An integral matrix ˜ B N × ex is called exchange matrix if the submatrix B ex 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 ∈ M N ( Z ) we define a quantum torus T q (Γ) over Z [ q ± 1 / 2 ] . • A map M : Z N − → F is a toric frame if there exist Γ such that it defines an embedding T q (Γ) ֒ → F where F ∼ = Fract ( T q (Γ)) . • Fix n ≤ N , and let ex ⊆ [1 , N ] such that | ex | = n . An integral matrix ˜ B N × ex is called exchange matrix if the submatrix B ex is skew symmetrizable. • A compatible pair ( M , ˜ B ) is called a quantum seed and its corresponding quantum cluster variables are M ( e j ) for j ∈ [1 , N ] .
Quantum Cluster Algebra Torus frame Exchange M : Z N − matrix ˜ → F B Quantum seed ( M , ˜ B ) Quantum cluster variables M ( e j )’s
Quantum Cluster Algebra • For each k ∈ ex , one has mutation µ k which takes quantum seed to quantum seed. µ k (( M , ˜ B )) = ( µ k ( M ) , µ k ( ˜ B )) .
Quantum Cluster Algebra • For each k ∈ ex , one has mutation µ k which takes quantum seed to quantum seed. µ k (( M , ˜ B )) = ( µ k ( M ) , µ k ( ˜ B )) . • The cluster variables indexed by [1 , N ] \ ex are called frozen variables .
Quantum Cluster Algebra • For each k ∈ ex , one has mutation µ k which takes quantum seed to quantum seed. µ k (( M , ˜ B )) = ( µ k ( M ) , µ k ( ˜ 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 µ k which takes quantum seed to quantum seed. µ k (( M , ˜ B )) = ( µ k ( M ) , µ k ( ˜ B )) . • The cluster variables indexed by [1 , N ] \ ex are called frozen variables . • Let inv ⊆ [1 , N ] \ ex . • Mutation-equivalent of quantum seeds µ 1 µ k ( M , ˜ → ( M ′ , ˜ B ′ ) B ) − → · · · − M ′ ( e j ) M ( e j )
Quantum Cluster Algebra • For each k ∈ ex , one has mutation µ k which takes quantum seed to quantum seed. µ k (( M , ˜ B )) = ( µ k ( M ) , µ k ( ˜ B )) . • The cluster variables indexed by [1 , N ] \ ex are called frozen variables . • Let inv ⊆ [1 , N ] \ ex . • Mutation-equivalent of quantum seeds µ 1 µ k ( M , ˜ → ( M ′ , ˜ B ′ ) B ) − → · · · − M ′ ( e j ) M ( e j ) • The quantum cluster algebra A q ( M , ˜ B , inv ) is the algebra generated by all cluster variables M ′ ( e j ) , j ∈ [1 , N ] and M ′ ( e k ) − 1 , k ∈ inv for all quantum seeds ( M ′ , ˜ B ′ ) which are mutation-equivalent to ( M , ˜ B ) .
Quantum Cluster Algebra at Roots of Unity Let ǫ 1 / 2 be a primitive ℓ th root of unity and we work over Z [ ǫ ± 1 / 2 ]. • The based quantum torus is now T ǫ (Γ)
Quantum Cluster Algebra at Roots of Unity Let ǫ 1 / 2 be a primitive ℓ th root of unity and we work over Z [ ǫ ± 1 / 2 ]. • The based quantum torus is now T ǫ (Γ) • Note that we are not specialize T q (Γ) at ǫ but simply define a quantum torus over Z [ ǫ ± 1 / 2 ].
Quantum Cluster Algebra at Roots of Unity Let ǫ 1 / 2 be a primitive ℓ th root of unity and we work over Z [ ǫ ± 1 / 2 ]. • The based quantum torus is now T ǫ (Γ) • Note that we are not specialize T q (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 ǫ 1 / 2 be a primitive ℓ th root of unity and we work over Z [ ǫ ± 1 / 2 ]. • The based quantum torus is now T ǫ (Γ) • Note that we are not specialize T q (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 A ǫ ( M , ˜ B , Γ , inv ) is a Z [ ǫ ± 1 / 2 ]-algebra generated by all cluster variables M ′ ( e j ) , j ∈ [1 , N ] and M ′ ( e k ) − 1 , k ∈ inv for all root of 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 A q ( M , ˜ B ) , A ǫ ( 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 A q ( M , ˜ B ) , A ǫ ( 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 ′ ( e j ) ℓ , j ∈ [1 , N ] and M ′ ( e k ) − ℓ , k ∈ inv are central in A ǫ ( 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 = { y 1 , ..., y n } ⊂ A . We define discriminant of Y to be d ( Y : tr ) = det[ tr ( y i y j )] ∈ 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 = { y 1 , ..., y n } ⊂ A . We define discriminant of Y to be d ( Y : tr ) = det[ tr ( y i y j )] ∈ C . • When A is free of rank n over some central subalgebra C , we use the map tr : A ֒ → M n ( C ) − → C . Then discriminant of A over 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 A ǫ ( 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 A ǫ ( 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 A ǫ (Θ) be the subalgebra of A ǫ ( M , ˜ B , Γ) generated by the cluster variables in Θ . Let C ǫ (Θ) be the central subalgebra of A ǫ (Θ) generated by the ℓ th power of the cluster variables. Then A ǫ (Θ) is finitely generated as a C ǫ (Θ) -module.
Discriminant of Quantum Cluster Algebra Theorem 4 (N.–Trampel–Yakimov) Suppose A ǫ (Θ) is free over C ǫ (Θ) . Then � d ( A ǫ (Θ) / C ǫ (Θ)) = ( noninverted frozen variables ) .
Discriminant of Quantum Cluster Algebra Theorem 4 (N.–Trampel–Yakimov) Suppose A ǫ (Θ) is free over C ǫ (Θ) . Then � d ( A ǫ (Θ) / C ǫ (Θ)) = ( noninverted frozen variables ) . Consider the quantum group U q ( g ) for a symmetrizable Kac–Moody algebra g , then the quantum Schubert cell algebra U q ( n + ∩ w ( n − )) is a cluster algebra due to [Geiss–Leclerc–Schroer, Goodearl–Yakimov].
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