g -polytopes of Brauer graph algebras Toshitaka Aoki Graduate - PowerPoint PPT Presentation

g -polytopes of Brauer graph algebras Toshitaka Aoki Graduate School of Mathematics, Nagoya University August 27, 2019 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 1 / 21

Aim of this talk To introduce the g -polytopes of f. d. algebras, firstly studied by [Asashiba-Mizuno-Nakashima (2019)] ▶ cones of g -vectors ▶ simplicial complexes of two-term silting complexes ▶ lattice polytopes Convexity and symmetry of (the closure of) g -polytopes of Brauer graph algebras. Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 2 / 21

Motivation Motivation: idea of L. Hille Q : an acyclic quiver with vertices 1,..., n . k = k : a field. In [Hille (2006, 2015)], he studied a simplicial complex of tilting modules over kQ as ∪ C ( M ) ⊆ R n , M where M = ⊕ n i =1 M i runs over all f. g. tilting kQ -modules, C ( M ) := { ∑ n i =1 a i dim M i | a i ∈ R ≥ 0 } Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 3 / 21

Motivation Motivation: idea of L. Hille Q : an acyclic quiver with vertices 1,..., n . k = k : a field. In [Hille (2006, 2015)], he studied a simplicial complex of tilting modules over kQ as ∪ C ≤ 1 ( M ) ⊆ R n , M where M = ⊕ n i =1 M i runs over all f. g. tilting kQ -modules, C ≤ 1 ( M ) := { ∑ n i =1 a i dim M i | a i ∈ R ≥ 0 , ∑ n i =1 a i ≤ 1 } = conv { 0 , dim M i | 1 ≤ i ≤ n } . Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 3 / 21

� Motivation ∪ M C ≤ 1 ( M ) Q 1 : 1 • = indec . rigid module 2 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 4 / 21

�� � Motivation ∪ M C ≤ 1 ( M ) Q 1 : 1 Q 2 : 1 2 2 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 4 / 21

�� � ��� Motivation ∪ M C ≤ 1 ( M ) Q 1 : 1 Q 2 : 1 Q 3 : 1 2 2 2 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 4 / 21

Motivation Motivation: L. Hille’s idea Theorem [Hille (2015)] If Q is of Dynkin type A , then ∪ M C ≤ 1 ( M ) is convex. In this case, we have ∪ C ≤ 1 ( M ) = conv { 0 , dim X | X: indec. kQ-module } M = conv ( { e i } n i =1 ∪ { e i + · · · + e j | 1 ≤ i < j ≤ n } ∪ { 0 } ) and it does not depend on the orientation of Q. For type D and E , ∪ M C ≤ 1 ( M ) is non-convex. Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 5 / 21

Motivation Motivation: L. Hille’s idea Theorem [Hille (2015)] If Q is of Dynkin type A , then ∪ M C ≤ 1 ( M ) is convex. In this case, we have ∪ C ≤ 1 ( M ) = conv { 0 , dim X | X: indec. kQ-module } M = conv ( { e i } n i =1 ∪ { e i + · · · + e j | 1 ≤ i < j ≤ n } ∪ { 0 } ) and it does not depend on the orientation of Q. For type D and E , ∪ M C ≤ 1 ( M ) is non-convex. Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 5 / 21

Motivation Mutation Let M = ⊕ n i =1 M i be a tilting kQ -module and f ⊕ → M ′ 0 − → M i − → X λ − i − → 0 ( X λ : indec . ) λ ∈ Λ where f is a left minimal add ( M / M i )-apx. of M . N := M / M i ⊕ M ′ i is called mutation of M if it is tilting. Lemma In the above, the following hold: C ≤ 1 ( M ) , C ≤ 1 ( N ) intersect only at their boundary. 1 If #Λ ≤ 2 , then C ≤ 1 ( M ) ∪ C ≤ 1 ( N ) is convex. 2 If Q is of type A , then #Λ ≤ 2 is always satisfied. 3 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 6 / 21

g -polytopes This talk [H] this talk Object tilting module 2-term silting cpx. Numerical data dim. vector g -vector Cones C ( M ) C ( T ) ∪ ∪ Polytope C ≤ 1 ( M ) C ≤ 1 ( T ) M T Intersections mutation silting mutation the middle term Locally convexity defined similarly of mutation seq. Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 7 / 21

g -polytopes A : a finite dimensional k -algebra P (1) , . . . , P ( n ): indecomposable projective A -modules Due to [Asashiba-Mizuno-Nakashima (2019)], we define the following subset ∆( A ) of R n , which we call g-polytope of A : ∪ C ≤ 1 ( T ) ⊆ R n , where ∆( A ) := T T = ⊕ n i =1 T i runs over all two-term silting complexes C ≤ 1 ( T ) := { ∑ n i =1 a i g T i | a i ∈ R ≥ 0 , ∑ n i =1 a i ≤ 1 } For a two-term complex T = ( T − 1 → T 0 ) ∈ K b (proj A ), g T := ( m 1 − m ′ n ) ∈ Z n : the g -vector of T , 1 , . . . , m n − m ′ where T 0 ∼ i =1 P ( i ) m i and T − 1 ∼ = ⊕ n = ⊕ n i =1 P ( i ) m ′ i . Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 8 / 21

g -polytopes Silting mutation Let T = ⊕ n i =1 T i be a two-term silting complex for A and f ⊕ → T ′ T i − → X λ − i → T i [1] , ( X λ : indec ) ( ∗ ) λ ∈ Λ where f is a minimal left add ( T / T i )-apx. of T i . Then U := T / T i ⊕ T ′ i is again a silting complex, and is called a two-term silting mutation of T if it is two-term. Lemma (analogues of tilting modules) In the above, the following hold: C ≤ 1 ( T ) , C ≤ 1 ( U ) intersect only at their boundary. 1 If #Λ ≤ 2 , then C ≤ 1 ( T ) ∪ C ≤ 1 ( U ) is convex. 2 Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 9 / 21

g -polytopes Definition We say that A is locally convex if #Λ ≤ 2 in ( ∗ ) always satisfied for any two-term silting complex T and any two-term silting mutation of T . Theorem [Asashiba-Mizuno-Nakashima (2019)] Assume that # { basic two-term silting complexes for A } / isom < ∞ . Then the following conditions are equivalent: (1) A is locally convex. (2) ∆( A ) is convex. In this case, ∆( A ) = conv { g X | X: indec. two-term presilt. } Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 10 / 21

g -polytopes Definition We say that A is locally convex if #Λ ≤ 2 in ( ∗ ) always satisfied for any two-term silting complex T and any two-term silting mutation of T . Theorem [Asashiba-Mizuno-Nakashima (2019)] Assume that # { basic two-term silting complexes for A } / isom < ∞ . Then the following conditions are equivalent: (1) A is locally convex. (2) ∆( A ) is convex. In this case, ∆( A ) = conv { g X | X: indec. two-term presilt. } Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 10 / 21

g -polytopes Brauer tree algebras Brauer tree algebras are f. d. symmetric algebras defined by Brauer trees(= trees embedded in a disk). containing the trivial extension of path algebras of type A closed under derived equivalent # { basic two-term silting complexes for A } / isom < ∞ Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 11 / 21

g -polytopes Theorem [Asashiba-Mizuno-Nakashima (2019)] Let A G be a Brauer tree algebra associated to a Brauer tree G. Then the following hold: ∆( A G ) is convex. 1 ∆( A G ) is symmetric with respect to origin 2 (i.e. ∆( A G ) = − ∆( A G ) ). Corollary For Brauer tree algebras, the g-polytope provides a derived invariant in the sense that ⇒ ∆( A G ) ∼ der A G ′ = A G ∼ = SL ∆( A G ′ ) M A n := conv ( {± e i } n i =1 ∪{± ( e i + · · · + e j ) | 1 ≤ i < j ≤ n } ) Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 12 / 21

g -polytopes Theorem [Asashiba-Mizuno-Nakashima (2019)] Let A G be a Brauer tree algebra associated to a Brauer tree G. Then the following hold: ∆( A G ) is convex. 1 ∆( A G ) is symmetric with respect to origin 2 (i.e. ∆( A G ) = − ∆( A G ) ). Corollary For Brauer tree algebras, the g-polytope provides a derived invariant in the sense that ⇒ ∆( A G ) ∼ der A G ′ = A G ∼ = SL ∆( A G ′ ) M A n := conv ( {± e i } n i =1 ∪{± ( e i + · · · + e j ) | 1 ≤ i < j ≤ n } ) Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 12 / 21

g -polytopes Theorem [Asashiba-Mizuno-Nakashima (2019)] Let A G be a Brauer tree algebra associated to a Brauer tree G. Then the following hold: ∆( A G ) is convex. 1 ∆( A G ) is symmetric with respect to origin 2 (i.e. ∆( A G ) = − ∆( A G ) ). Corollary For Brauer tree algebras, the g-polytope provides a derived invariant in the sense that ⇒ ∆( A G ) ∼ SL ∆( A G ′ ) ∼ der A G ′ = A G ∼ = SL M A n = M A n := conv ( {± e i } n i =1 ∪{± ( e i + · · · + e j ) | 1 ≤ i < j ≤ n } ) Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 12 / 21

g -polytopes n=2 G 1 : • • • A G 1 ∼ = the trivial extension of k (1 → 2) Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 13 / 21

g -polytopes n=3 • G 2 : • • • • G 3 : • • • A G 2 ∼ = Triv ( k (1 → 2 ← 3)) A G 3 ∼ = Triv ( k (1 → 2 → 3)) Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 13 / 21

Main result Main Result Toshitaka Aoki (Nagoya University) g -polytopes of Brauer graph algebras August 27, 2019 14 / 21