Qi Wang Osaka University @Qingdao, July 15, 2019 Outline Introduction and motivation r-tilting theory Mutation Poset structure Simply connected algebras Staircase algebras References Auslander-Reiten translation Throughout, let Λ be a finite dimensional basic algebra over an algebraically closed field K. For a A-module M with a minimal projective presentation
we have TrM := coker Homハ(d1, Λ). The Auslander-Reiten translation is defined by
where D = Homk(-,K). In 2014, Adachi-Iyama-Reiten introduced support τ-tilting modules for any A and constructed the (left or right) mutation of them, which has the following nice properties: ▶ Mutation (left or right) is always possible. ▶ There is a partial order on the set of (isomorphism classes of) basic support τ-tilting modules such that its Hasse quiver realizes the left mutation. This is considered as a generalization of the classical tilting theory via mutations. We call A r-tilting finite if there are only finitely many (iso. classes of) basic support τ-tilting Λ-modules. Motivation Note that representation-finite algebras are τ-tilting finite and the converse is not true in general. For example, let
then Λ n is τ-tilting finite. But Λ n is ▶ representation-finite if n = 2,3,4,5. ▶ tame if n = 6. ▶ wild if n ≥ 7. Therefore, we want to know which kind of algebras meets the following conditions:ｧ-tilting finite ⇔ representation-finite. Besides, many scholars have studied the support τ-tilting modules of various algebras.  For example, ▷ Gentle algebras (Plamondon, 2018). ▶ Tilted and cluster-tilted algebras (Zito, 2019). ▶ Algebras with radical square zero (Adachi, 2016). ▶ Brauer graph algebras (Adachi-Aihara-Chan, 2018). ▶ Preprojective algebras of Dynkin type (Aihara-Mizuno, 2016, Mizuno, 2014).ｧ-tilting theory We denote by |M| the number of (iso. classes of) indecomposable direct summands of M. Definition 1.1 (Adachi-Iyama-Reiten, 2014) Let M be a right Λ-module and P ∈ proj Λ. (1) M is called τ-tilting if Hom Λ ( M, τ M ) = 0 and | M | = | Λ | . (2) M is called support τ-tilting if M is a τ-tilting (Λ/ 〈e〉)-module, where e is an idempotent of Λ. We denote by s -tilt Λ the set of (iso. classes of) basic supportァ-tilting Λ-modules. Example
Then,
Thus, for example, ▶ P1 ⊕ P2, P1 ⊕ S1 and S2 ⊕ P2 are τ-tilting modules. ▶ S1 and S2 are support r-tilting modules. Mutation We denote by add( M ) (respectively, Fac( M )) the full subcategory whose objects are direct summands (respectively, factor modules) of finite direct sums of copies of M. Definition 1.2 (Adachi-Iyama-Reiten, 2014) Let T = M ⊕ N be a basic τ-tilting module, where M̸ξ Fac( N ) is an indecomposable summand. We take an exact sequence with a minimal left add( N )-approximation π:
we call μ M (T) := U ⊕ N the left mutation of T with respect to M . Remark π is called a minimal left add( N )-approximation if N ′ ∈ add( N ) and it satisfies the following conditions: (i) every h : N ′ → N ′ that satisfies h ○ π = π is an automorphism.
Example
Poset structure Definition 1.3 (Adachi-Iyama-Reiten, 2014) For M, N ∈ s τ -tilt Λ, we say M ≥ N if Fac( N ) ⊆ Fac( M ). Example Let Λ be the algebra given before. The Hasse quiver of sτ-tilt Λ is
Proposition 1.4 (Adachi-lyama-Reiten, 2014) The mutation quiver Q(sr-tilt Λ) and the Hasse quiver H(sr-tilt Λ) coincide. Proposition 1.5 (Adachi-lyama-Reiten, 2014) If the mutation quiver Q(sr-tilt A) contains a finite connected component, then it exhausts all support τ-tilting modules. Reduction theorems It is well-known that any idempotent truncation of a τ-tilting finite algebra is also τ-tilting finite.  Furthermore, we have Proposition 1.6 (Adachi-lyama-Reiten, 2014) There exists a poset isomorphism between sτ-tilt Λ and sτ-tilt Λ°Ρ. Proposition 1.7 (Eisele-Janssens-Raedschelders, 2018) Let / be a two-sided ideal generated by elements which are contained in the center and the radical, then there exists a poset isomorphism between sτ-tilt Λ and sτ-tilt (Λ/Ι). Minimal representation-infinite algebras An algebra Λ is called minimal rep.-infinite if Λ is rep.-infinite, but Λ/ΛeΛ is rep.-finite for any non-zero idempotent e of Λ. We denote by ΓΛ the Auslander-Reiten quiver of Λ.  A connected component C of ΓΛ is called preprojective if ▶ there is no oriented cycle in C, and ▶ any module in C is of form τ−k(P) for some k ∈ N and some indecomposable projective module P. Proposition 1.8 (Happel-Vossieck, 1983) A m.r.i. algebra with preprojective component is either a n-Kronecker algebra (n ≥ 2) or a tame concealed algebra, which is of type An, Dn(n > 4), E6, E7 or E8. Tilted algebras A tilting A-module T provided: ► | T | = | Λ | ; ► gl. dim. T ≤ 1; ► Ext λ ( T , T ) = 0. A (concealed) tilted algebra of type Q is the endomorphism algebra of a (preprojective) tilting module over a hereditary algebra KQ. Lemma 1.9 (Zito, 2019) Let Λ be a tilted or cluster-tilted algebra, then Λ is τ-tilting finite if and only if Λ is representation-finite. Simply connected algebras Let Λ = KQ/I be an algebra with a quiver Q = ( Q 0 , Q 1 , s, t ) and an admissible ideal I. For each arrow α ∈ Q 1 , let α − be its formal inverse with s(α−) = t(α) and t(α−) = s(α). Then, we set
A walk is a formal composition w = w1w2 . . . wn with w; E Q1 U Q1 for all 1 ≤ i ≤ n. Then, we set
and denote by 1 x the trivial path at vertex x. For walks w and u with s(u) = t(w), the composition wu is defined in the obvious way. Let ～ be the smallest equivalence relation on the set of all walks in Q satisfying the following conditions: ▶ For each α : x → y in Q 1 , we have αα − ～ 1 x and α − α ～ 1 y . ► For each minimal relation ∑ j =1 λ jω j in I, we have ω j ～ ω j for all 1 ≤ i,j ≤ n. ▶ If u, v, w and w' are walks and u ~ v, then wuw' ~ ww' whenever these compositions are defined. We denote by [w] the equivalence class of a walk w. Let x ∈ Q 0 . The set Π 1 (Q, I, x) of equivalence classes of all walks w with s(w) = t(w) = x is a group via [u] · [v] = [uv], and one can show that it does not depend on the choice of x. Thus, we define the fundamental group of ( Q, I ) as follows. Π 1 (Q, I) := Π 1 (Q, I, x). Recall that Λ = KQ/I is called triangular if Q is acyclic. Definition 2.1 (Assem-Skowroński, 1988) A connected triangular algebra Λ is simply connected if, for every presentation (Q, I) of A, the fundamental group fll(Q, I) is trivial. We have the following examples. (1) All tree algebras are simply connected. (2) A hereditary algebra is simply connected if and only if its quiver is a tree. Theorem 2.2 (W, 2019) Let Λ be a simply connected algebra, then it is τ-tilting finite if and only if Λ is representation-finite. A full subquiver Q' of Q is a convex subquiver if any path in Q with source and target in Q' lies entirely in Q'. Let I' := KQ' ∩ I, then KQ'/I' is called a convex subalgebra of KQ/I. An algebra is called critical if it is rep.-infinite, but any proper convex subalgebra is rep.-finite. Note that the path algebra KQ with the following quiver Q, is critical but not m.r.i.. A grading of a tree T is a function g : Tvertex → N satisfying ▶ g-1(0) ≠ ∅. ▶ g(x) – g(y) ∈ 1 + 2 Z, whenever x and y are neighbours in T. A graded tree is a pair ( T , g ) formed by a tree T and a grading g . Sketch of the proof: By [Bongartz, 1984], Λ is rep.-finite if and only if it does not contain a critical convex subalgebra, which arises from a graded tree. On the other hand, such a critical algebra is a m.r.i. algebra with preprojective component. Staircase algebras Let λ = (λ 1 , λ 2 , . . . , λ ε ) be a (non-increasing) partition of a positive integer n. It is well-known that we can visualize λ by the corresponding Young diagram Y( λ ). For example,
For any partition λ ├ n, let ▷ Q x is a quiver such that the vertices are given by the boxes in Y( λ ), the arrows are given by drawing arrows from right to left and from bottom to top. ▷ /\ is a two-sided ideal generated by all commutativity relations for all squares appearing in Q λ . Then the algebra A(λ) := KQ λ / λ is called a staircase algebra. If λ = (3,2,1), then Q λ is given by Then, the corresponding staircase algebra A( λ ) is defined by
Proposition 3.1 (Boos, 2017) For any λﾋn, Α(λ) is triangular and simply connected. Proposition 3.2 (Boos, 2017) A staircase algebra A(λ) with λﾋn is (1) representation-finite if and only if one of the following holds: ▶ λ ∈ {(n),(n − k,1k),(n − 2,2),(22,1n−4)} for k ≤ n. ► n < 8 and λ # {(4,3,1),(3 2 ,2),(3,2 2 ,1),(4,2,1 2 )}. (2) tame concealed if and only if λ comes up in the following list: (6, 3), (6, 2, 1), (5, 22), (4, 3, 1), (4, 2, 12), (3, 2², 1), (3², 1³), (2³, 1³), (3, 2, 1⁴). (3) tame, but not tame concealed if and only if λ comes up in the following list: (5²), (5, 4), (4², 1), (3³), (3², 2), (3, 2³), (25), (24, 1). Otherwise, A(λ) is wild. Corollary 3.3 (W, 2019) A staircase algebra A(A) with λﾋn is τ-tilting finite if and only if one of the following holds:
Question 3.4 We have known that if λ = (n) or (n − k, 1k), the number of support τ-tilting Α(λ)-modules is How about others?