Noncommutative Discriminants of Quantum Cluster Algebras

Transcription

1 Noncommutative Discriminants of Quantum Cluster Algebras Kurt Trampel Joint work with Bach Nguyen and Milen Yakimov Louisiana State University Maurice Auslander Distinguished Lectures and International Conference 2018 Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

2 Overview 1 Background Discriminants Quantum cluster algebras 2 Quantum cluster algebras at a root of unity Definition Certain subalgebras Main results Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

3 Uses of the discriminant of an algebra The discriminant has been used in: determining automorphism groups and solving isomorphism problems for certain PI algebras. [Ceken, Palmieri, Wang, and Zhang] solving the Zariski cancellation problem (A[T ] B[T ] A B) in certain cases. [Bell and Zhang] classifing the Azumaya locus of certain algebras. [Brown and Yakimov] Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

4 Uses of the discriminant of an algebra The discriminant has been used in: determining automorphism groups and solving isomorphism problems for certain PI algebras. [Ceken, Palmieri, Wang, and Zhang] solving the Zariski cancellation problem (A[T ] B[T ] A B) in certain cases. [Bell and Zhang] classifing the Azumaya locus of certain algebras. [Brown and Yakimov] Issue The discriminant can be very difficult to compute directly. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

5 Discriminant d n (R/C) Say R is a free, finite rank n module over a subalgebra C Z(R). Then the embedding R M n (C) gives a trace map tr : R M n (C) tr Mn(C) C. The discriminant of R over C is defined by ( ) d n (R/C) := C det tr(y i y j ) where {y 1,,y n } is a C-basis of R Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

6 Quantum cluster algebras [Berenstein and Zelevinsky] A( B) A q (M, B) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

7 Quantum cluster algebras [Berenstein and Zelevinsky] A( B) A q (M, B) Ambient field Frac(Z[x 1,..., x N ]) F = Frac(T q (Λ)) T q (Λ) = Z[q ± 1 2 ]-algebra with basis X f, f Z N and relations X f X g = q Λ(f,g) 2 X f +g Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

8 Quantum cluster algebras [Berenstein and Zelevinsky] A( B) A q (M, B) Ambient field Frac(Z[x 1,..., x N ]) F = Frac(T q (Λ)) T q (Λ) = Z[q ± 1 2 ]-algebra with basis X f, f Z N and relations X f X g = q Λ(f,g) 2 X f +g Seeds ( x, B) (M, B) with compatibility between M and B M : Z N F such that T q (Λ M ) φ F (1) M(f ) = φ(f ) (2) F Frac(T q (Λ m )) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

9 Quantum cluster algebras [Berenstein and Zelevinsky] A( B) A q (M, B) Mutation k ex [1, N] µ k ( x, B) = ( x, B ) µ k ( x, B) = (ρ M b k,s ME s, E s BFs ) x k replaced by µ k M(e i ) = M(e i ), i k b ik >0 xb ik i + b ik <0 x b ik i x k µ k M(e k ) = M( e k + [b k ] + ) + M( e k [b k ] ) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

10 Quantum cluster algebras [Berenstein and Zelevinsky] A( B) A q (M, B) Mutation k ex [1, N] µ k ( x, B) = ( x, B ) µ k ( x, B) = (ρ M b k,s ME s, E s BFs ) x k replaced by µ k M(e i ) = M(e i ), i k b ik >0 xb ik i + b ik <0 x b ik i x k µ k M(e k ) = M( e k + [b k ] + ) + M( e k [b k ] ) The algebras inv [1, N]\ex A( B, inv) = Z-subalgebra A q (M, B, inv) = Z[q ± 1 2 ]-subalgebra of F generators: x 1 j for j inv generators: M(e j ) 1 for j inv x j x for (x, B ) (x, B) M (e j ) for (M, B ) (M, B) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

11 Quantum cluster algebras at a root of unity Replace q 1 2 by a primitive l th root of unity ɛ 1 2 Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

12 Quantum cluster algebras at a root of unity Replace q 1 2 by a primitive l th root of unity ɛ 1 2 Construct F = Frac(T ɛ (Λ)) and root of unity toric frames M : Z N F Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

13 Quantum cluster algebras at a root of unity Replace q 1 2 by a primitive l th root of unity ɛ 1 2 Construct F = Frac(T ɛ (Λ)) and root of unity toric frames M : Z N F Have seeds (M, B, Λ) with compatibility conditions. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

14 Quantum cluster algebras at a root of unity Replace q 1 2 by a primitive l th root of unity ɛ 1 2 Construct F = Frac(T ɛ (Λ)) and root of unity toric frames M : Z N F Have seeds (M, B, Λ) with compatibility conditions. Mutation similar to before µ k (M, B, Λ) = (ρ M b k,s ME s, E s BFs, E T s ΛE s ) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

15 Quantum cluster algebras at a root of unity Replace q 1 2 by a primitive l th root of unity ɛ 1 2 Construct F = Frac(T ɛ (Λ)) and root of unity toric frames M : Z N F Have seeds (M, B, Λ) with compatibility conditions. Mutation similar to before µ k (M, B, Λ) = (ρ M b k,s ME s, E s BFs, E T s ΛE s ) A ɛ (M, B, Λ, inv) is the Z[ɛ ± 1 2 ]-subalgebra of F generated by M(e j ) 1, j inv and by all M (e j ) of (M, B, Λ ) (M, B, Λ) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

16 Some central elements Lemma If (M, B, Λ ) (M, B, Λ), then M (e j ) l A ɛ (M, B, Λ) is central. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

17 Some central elements Lemma If (M, B, Λ ) (M, B, Λ), then M (e j ) l A ɛ (M, B, Λ) is central. Proposition For a quantum seed (M, B, Λ) and l coprime to a finite set of integers dependent on B, (µ k M(e k )) l = b ik >0 (M(e i) l ) b ik + b ik <0 (M(e i) l ) b ik M(e k ) l Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

18 Some central elements Lemma If (M, B, Λ ) (M, B, Λ), then M (e j ) l A ɛ (M, B, Λ) is central. Proposition For a quantum seed (M, B, Λ) and l coprime to a finite set of integers dependent on B, (µ k M(e k )) l = b ik >0 (M(e i) l ) b ik + b ik <0 (M(e i) l ) b ik M(e k ) l Recall x k = b ik >0 x b ik i + b ik <0 x b ik i x k Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

19 A familiar central subalgebra Theorem (Nguyen Yakimov T.) The (classical) cluster algebra A( B, inv) embeds into the center of A ɛ (M, B, Λ, inv). Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

20 A familiar central subalgebra Theorem (Nguyen Yakimov T.) The (classical) cluster algebra A( B, inv) embeds into the center of A ɛ (M, B, Λ, inv). Moreover the exchange graphs of A( B, inv), A q (M q, B, inv), and A ɛ (M, B, Λ, inv) all coincide. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

21 Certain subalgebras A ɛ (Θ) Θ is a set of seeds such that every two seeds are connected by sequence of mutations and every nonfrozen direction is mutated at least one time. Θ : µ 2 (M, B) µ 1 (M, B) µ 3 µ 2 (M, B) (M, B) µ 3 (M, B) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

22 Certain subalgebras A ɛ (Θ) Θ is a set of seeds such that every two seeds are connected by sequence of mutations and every nonfrozen direction is mutated at least one time. A ɛ (Θ) A ɛ (M, B, Λ, inv) is the subalgebra that is generated by cluster variables from seeds in Θ and the inverted frozen variables. Θ : µ 2 (M, B) µ 1 (M, B) µ 3 µ 2 (M, B) (M, B) µ 3 (M, B) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

23 Certain subalgebras A ɛ (Θ) Θ is a set of seeds such that every two seeds are connected by sequence of mutations and every nonfrozen direction is mutated at least one time. A ɛ (Θ) A ɛ (M, B, Λ, inv) is the subalgebra that is generated by cluster variables from seeds in Θ and the inverted frozen variables. C ɛ (Θ) A ɛ (Θ) is the central subalgebra generated by l th powers of cluster variables from seeds in Θ and the inverted frozen variables. Θ : µ 2 (M, B) µ 1 (M, B) µ 3 µ 2 (M, B) (M, B) µ 3 (M, B) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

24 Discriminant d n (A ɛ (Θ)/C ɛ (Θ)) The discriminant d n (R/C) defined here if R is a free, finite rank n module over a central subalgebra C. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

25 Discriminant d n (A ɛ (Θ)/C ɛ (Θ)) The discriminant d n (R/C) defined here if R is a free, finite rank n module over a central subalgebra C. A ɛ (Θ) is finitely generated over C ɛ (Θ). Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

26 Discriminant d n (A ɛ (Θ)/C ɛ (Θ)) The discriminant d n (R/C) defined here if R is a free, finite rank n module over a central subalgebra C. A ɛ (Θ) is finitely generated over C ɛ (Θ). Theorem (Nguyen Yakimov T.) When A ɛ (Θ) is free over C ɛ (Θ), then d n (A ɛ (Θ)/C ɛ (Θ)) = ) powers (noninverted frozen variables of A ɛ (Θ) Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

27 Quantum Schubert cells (or quantum unipotent groups) Definition (De Concini Kac Procesi and Lusztig) Let g be a symmetrizable Kac-Moody algebra. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

28 Quantum Schubert cells (or quantum unipotent groups) Definition (De Concini Kac Procesi and Lusztig) Let g be a symmetrizable Kac-Moody algebra. Let w W with reduced expression w = s i1 s i2... s in. Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

29 Quantum Schubert cells (or quantum unipotent groups) Definition (De Concini Kac Procesi and Lusztig) Let g be a symmetrizable Kac-Moody algebra. Let w W with reduced expression w = s i1 s i2... s in. U ɛ [w] is the subalgebra of U ɛ [g] generated by Lusztig root vectors F βj = T i1... T ij 1 (F ij ). Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

30 Quantum Schubert cells (or quantum unipotent groups) Definition (De Concini Kac Procesi and Lusztig) Let g be a symmetrizable Kac-Moody algebra. Let w W with reduced expression w = s i1 s i2... s in. U ɛ [w] is the subalgebra of U ɛ [g] generated by Lusztig root vectors F βj = T i1... T ij 1 (F ij ). Theorem (Geiß Leclerc Schröer, Goodearl Yakimov) Uɛ [w] has a canonical cluster algebra structure. The frozen variables are given by generalized minors ωi,wω i for i S(w). Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

31 Discriminants of quantum Schubert cells Theorem (Nguyen Yakimov T.) For symmetrizable Kac-Moody algebra g, w W, and ɛ a primitive l th root of unity, d n (Uɛ [w]/c ɛ ) = i S(w) ln 1 (l 1) ω i,wω i Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14

32 Thank you! Kurt Trampel (LSU) Disc. of Quantum Cluster Algebras Auslander Conference / 14