Quantum cluster algebra at roots of unity and discriminant formula

Transcription

1 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

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 ].

3 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)).

4 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.

5 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].

6 Quantum Cluster Algebra Torus frame M : Z N F Exchange matrix B Quantum seed (M, B) Quantum cluster variables M(e j ) s

7 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)).

8 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.

9 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.

10 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 (M, B) µ 1 µk (M, B ) M(e j ) M (e j )

11 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 (M, B) µ 1 µk (M, 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).

12 Quantum Cluster Algebra at Roots of Unity Let ɛ 1/2 be a primitive l th root of unity and we work over Z[ɛ ±1/2 ]. The based quantum torus is now T ɛ (Γ)

13 Quantum Cluster Algebra at Roots of Unity Let ɛ 1/2 be a primitive l th root of unity and we work over Z[ɛ ±1/2 ]. The based quantum torus is now T ɛ (Γ) Note that we are not specialize T q (Γ) at ɛ but simply define a quantum torus over Z[ɛ ±1/2 ].

14 Quantum Cluster Algebra at Roots of Unity Let ɛ 1/2 be a primitive l th root of unity and we work over Z[ɛ ±1/2 ]. The based quantum torus is now T ɛ (Γ) Note that we are not specialize T q (Γ) at ɛ but simply define a quantum torus over Z[ɛ ±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).

15 Quantum Cluster Algebra at Roots of Unity Let ɛ 1/2 be a primitive l th root of unity and we work over Z[ɛ ±1/2 ]. The based quantum torus is now T ɛ (Γ) Note that we are not specialize T q (Γ) at ɛ but simply define a quantum torus over Z[ɛ ±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 ɛ (M, B, Γ, inv) is a Z[ɛ ±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]

16 Quantum Cluster Algebra at Roots of Unity Let A( B) be the cluster algebra associated to the exchange matrix B.

17 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 ɛ (M, B, Γ) and A( B) are all isomorphic. Moreover, the root of unity quantum cluster algebra satisfies the Laurent phenomenon.

18 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 ɛ (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 ɛ (M, B, Γ). Moreover, the central subalgebra generated by them is isomorphic to the cluster algebra A( B).

19 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.

20 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.

21 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.

22 Discriminant of Quantum Cluster Algebra Suppose Θ is a finite set of seeds in A ɛ (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.

23 Discriminant of Quantum Cluster Algebra Suppose Θ is a finite set of seeds in A ɛ (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 ɛ (Θ) be the subalgebra of A ɛ (M, B, Γ) generated by the cluster variables in Θ. Let C ɛ (Θ) be the central subalgebra of A ɛ (Θ) generated by the l th power of the cluster variables. Then A ɛ (Θ) is finitely generated as a C ɛ (Θ)-module.

24 Discriminant of Quantum Cluster Algebra Theorem 4 (N. Trampel Yakimov) Suppose A ɛ (Θ) is free over C ɛ (Θ). Then d(a ɛ (Θ)/C ɛ (Θ)) = (noninverted frozen variables).

25 Discriminant of Quantum Cluster Algebra Theorem 4 (N. Trampel Yakimov) Suppose A ɛ (Θ) is free over C ɛ (Θ). Then d(a ɛ (Θ)/C ɛ (Θ)) = (noninverted frozen variables) Ċonsider 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].

26 Discriminant of Quantum Cluster Algebra Theorem 4 (N. Trampel Yakimov) Suppose A ɛ (Θ) is free over C ɛ (Θ). Then d(a ɛ (Θ)/C ɛ (Θ)) = (noninverted frozen variables) Ċonsider 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]. Theorem 5 (N. Trampel Yakimov) For any symmetrizable Kac Moody algebra g, w W, and ɛ an odd primitive root of unity, where t is the length of w. d(u ɛ (n + w(n ))/C ɛ ) = lt+1 (l 1) ω i,wω i