Categorification of a linear algebra identity and factorization of Serre functors

We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.


Introduction
The general philosophy behind categorification, as explained for example in [2], is that numbers should be interpreted as sets, sets as categories, equalities as isomorphisms and so on. When one considers linear operators, the following suggested interpretation makes sense, see also [15] for a similar definition.
Given the data of a free Z-module V of finite rank and linear maps f 1 , f 2 , . . . , f n , g : V → V satisfying g = f 1 · f 2 · . . . · f n , a (weak) categorification of this data consists of an abelian or triangulated category B whose Grothendieck group K 0 (B) is isomorphic to V , together with exact functors F i : B → B and G : B → B, such that: • F 1 , F 2 , . . . , F n , G induce linear maps on K 0 (B) which, under the isomorphism with V , coincide with f 1 , f 2 , . . . , f n , g; • There is an isomorphism of functors between G and the composition F 1 · F 2 · . . . · F n . When V carries additional structure, such as a bilinear form, it is preferable that this structure lifts to B as well.
1.1. A linear algebra identity. The following well-known statement concerns products of reflection-like matrices defined by a square matrix. Proposition 1.1. Let B be any square n × n matrix over a commutative ring. Then . . · r B n , where the matrices B + and B − are the upper and lower triangular parts of B, defined by otherwise, otherwise, (so that B = B + + B T − ), and for each 1 ≤ i ≤ n, the square matrix r B i is obtained from the identity matrix by subtracting the i-th row of B, that is, This statement originally appeared as an exercise in the book of Bourbaki [5,Ch. 5,§6,no. 3], following an argument presented in Coxeter's paper [9]. Various specific cases have since then appeared in the literature, including A'Campo [1] in the bipartite case and Howlett [14] in the symmetric case. The general form is stated and proved in an article by Coleman [8], and an alternative proof can be found in [16].
As important special case is when B = C + C T is the symmetrization of an upper triangular square matrix C with ones on its main diagonal. In this case the matrices r B i are reflections, and the proposition implies that (1.3) − C −1 C T = r B 1 · r B 2 · . . . · r B n . This equality provides us with two points of view on the so-called Coxeter transformation. First, as known in Lie theory, it is the product of the simple reflections, as given by the right hand side of (1.3). Second, as follows from the left hand side, it can also be described as the automorphism Φ satisfying x, y C = − y, Φx C where ·, · C is the bilinear form defined by the matrix C and x, y are any two vectors, as known in the representation theory of algebras, see [17].

Categorical interpretation.
Our categorical interpretation of equations (1.1) and (1.3) is achieved by using functors on triangulated categories arising from finite dimensional algebras. In order to state our result in precise terms, we need to recall a few notions from the representation theory of finite dimensional algebras.
For a finite dimensional algebra A over a field k, denote by D b (A) the bounded derived category of finite dimensional right A-modules, and by per A its full triangulated subcategory consisting of all complexes quasiisomorphic to perfect complexes, that is, bounded complexes whose terms are finitely generated projective A-modules.
The Grothendieck group K 0 (per A) is free abelian of finite rank, with a basis consisting of the classes of the indecomposable projective A-modules. It is equipped with a bilinear form induced by the Euler form The algebra A is called triangular if there exist primitive orthogonal idempotents e 1 , . . . , e n of A such that e i Ae j = 0 for any j < i and e i Ae i ≃ k for 1 ≤ i ≤ n. The modules P i = e i A then form a complete collection of indecomposable projectives. Taking their classes as a basis for K 0 (per A), it will be convenient for us to order them [P n ], . . . , [P 1 ] and to define the Cartan matrix C A as the matrix of ·, · A with respect to that basis, namely so that C A is upper triangular with ones on its main diagonal.
Similarly, for a (finite dimensional) A-A-bimodule M we can define a matrix C M by (C M ) ij = dim k e n+1−j M e n+1−i , and call M triangular if C M is upper triangular, or equivalently, e i M e j = 0 for any j < i. We have C T M = C DM , where DM is the dual of M , defined as DM = Hom k (M, k).
The trivial extension Λ = A ⋉ DM is the k-algebra which has A ⊕ DM as its underlying vector space, with the multiplication defined by (a, µ)(a ′ , µ ′ ) = (aa ′ , aµ ′ + µa ′ ). Its indecomposable projectives are in bijective correspondence with those of A, and its Cartan matrix is given by Theorem 1.2. Let A be a finite dimensional triangular algebra over a field and let A M A be a triangular A-A-bimodule. Set Λ = A ⋉ DM to be the trivial extension of A with the dual of M .
Then there exist, for 1 ≤ i ≤ n = rank K 0 (per Λ), triangulated functors (a) Each functor R i induces a linear map on K 0 (per Λ) whose matrix with respect to the basis of indecomposable projective Λ-modules is and commute up to a natural isomorphism of functors.
The vertical arrows of (1.4) induce an isomorphism K 0 (per Λ) → K 0 (A) sending projectives to projectives. Thus, by considering the diagram (1.4) at the level of the Grothendieck groups, we get the following commutative diagram (where I n is the n × n identity matrix), which explains why the theorem can be seen as a categorical interpretation of (1.1) for B = C Λ .
To complement this result, we note that any integral square matrix B with non-negative entries and positive entries on its main diagonal can be realized as a Cartan matrix of a suitable Λ as in the theorem. More precisely, there exist a finite dimensional triangular algebra A and a triangular bimodule M over A with B + = C A and B − = C M , see Section 2.7 for the details.
1.3. Application to Serre functors. A triangular finite dimensional algebra A is of finite global dimension, hence its bounded derived category D b (A) admits a Serre functor ν A in the sense of Bondal and Kapranov [4]. By a result of Happel [12], it is given by the left derived functor of the Nakayama functor, ν A = − commute up to a natural isomorphism of functors.
Thus, one can lift (a shift of) the Serre functor on D b (A) to a product of the "reflections" R i in per T (A). As before, the diagram (1.5) can be regarded as a categorical interpretation of equation (1.3) for C = C A , the Cartan matrix of A. This can be done for any upper triangular integral matrix C with non-negative entries and 1 on its main diagonal, see Section 2.7.
1.4. On the proof. Section 2 is devoted to the proof of the theorem and its corollaries. A key ingredient in the proof is the proper definition and analysis of the functors R i . They are defined, for each 1 ≤ i ≤ n, as the left derived functors of tensoring with a two-term complex of bimodules, where m denotes the multiplication map and e 1 , . . . , e n are the primitive orthogonal idempotents.
The functors R Λ i have already been considered in the works of Rouquier-Zimmermann [19] on braid group actions on derived categories of Brauer tree algebras without exceptional vertex, and by Hoshino and Kato [13] in relation with constructions of two-sided tilting complexes for self-injective algebras. When the algebra Λ is symmetric and dim e i Λe i = 2, the functor R Λ i can be viewed as a twist functor in the sense of Seidel and Thomas [20] with respect to the 0-spherical object e i Λ. Our result shows the importance of the functors R Λ i for a wider class of algebras Λ, which are not necessarily restricted to be self-injective or symmetric.
In the course of the proof we establish the special case of Theorem 1.2 where the bimodule M is zero, namely that for any finite-dimensional tri- Plugging in this statement the definition of R A i , we obtain a (finite) projective resolution of the triangular algebra A as a bimodule over itself. A similar construction, with relation to Hochschild cohomology computations, was presented by Cibils in [7].
1.5. Previous work. Another categorical interpretation of (1.3), in the realm of representation theory of quivers, is given by a result of Gabriel [10], correcting previous paper by Brenner and Butler [6].
For a quiver Q without oriented cycles, one can consider two exact autoequivalences on the bounded derived category of its path algebra. The first is the Auslander-Reiten translation, corresponding to the left hand side of (1.3), and the second is the so-called Coxeter functor, which was defined by Bernstein, Gelfand and Ponomarev [3] as a product of their reflection functors, corresponding to the right hand side of (1.3).
In [10], it is shown that for any quiver whose underlying graph is a tree, or more generally, does not contain a cycle of odd length, the Auslander-Reiten translation is isomorphic to the Coxeter functor, thus interpreting the equality in (1.3) as an isomorphism of functors.
In Section 3 we explain this result in more detail and compare it with our approach.
Another approach to the factorization of Serre functors for certain finite dimensional algebras, including ones arising from category O associated to semi-simple complex Lie algebras, is presented by Mazorchuk and Stroppel in [18].
Acknowledgements. The results in this paper were first presented at the workshop on Spectral methods in representation theory of algebras and applications to the study of rings of singularities that was held at Banff in September 2008. I would like to thank J. A. de la Peña, C. Ringel and C. Stroppel for their helpful comments and suggestions.
2. Proof of the theorem 2.1. The building blocks -the functors R Λ i . Let Λ be a basic finite dimensional algebra over a field k and let P 1 , . . . , P n be a complete collection of the non-isomorphic indecomposable projectives in mod Λ, the category of finite dimensional right Λ-modules. Let e 1 , . . . , e n be primitive orthogonal idempotents in Λ such that P i = e i Λ for 1 ≤ i ≤ n.
Fix 1 ≤ i ≤ n and consider the following complex of Λ-Λ-bimodules where Λ is in degree 0 and m is the multiplication map. Taking the tensor product − ⊗ Λ C i yields an endofunctor on the category C b (Λ) of bounded complexes of finite dimensional right Λ-modules, which induces an endofunctor on its homotopy category K b (Λ).
Since its terms are projective as left Λ-modules, the complex C i defines a triangulated functor on the derived category D b (Λ) of mod Λ. Moreover, as the terms are also projective as right Λ-modules, this functor restricts to a functor on the triangulated subcategory per Λ of complexes quasi-isomorphic to perfect ones (that is, bounded complexes of finitely generated projectives).
In the sequel, when no confusion arises, we shall denote all the above functors by R Λ i . These functors were considered by Rouquier and Zimmermann [19] in relation with braid group actions on the derived categories of Brauer tree algebras with no exceptional vertex, and by Hoshino and Kato [13] in relation with constructions of two-sided tilting complexes for self-injective algebras.
The Grothendieck group K 0 (per Λ) is a free abelian group on the generators [P 1 ], . . . , [P n ] equipped with a bilinear form induced by the Euler form i is triangulated, it is enough to verify this equality on the basis elements [P j ]. This follows directly from Lemma 2.1.
The next lemma provides an explicit description of compositions of functors R Λ i , which will be useful in the sequel.
with the differentials d r ϕ defined on each summand by Proof. By definition, (where we replaced L ⊗ by ⊗ since the terms of C i are projective as left (as well as right) modules), so it is enough to show that where the right hand side is an iterated tensor product of complexes.
We prove this by induction on s, the case s = 1 being merely the definition of R Λ ϕ (1) . Now assume the claim for s, consider a function ϕ : {1, . . . , s+1} → {1, . . . , n} and denote by ϕ ′ its restriction to {1, . . . , s}. By the induction hypothesis, we need to show that . Recall that the tensor product of two complexes X Λ and Λ Y is defined by where the left summand vanishes for r = s + 1 and the right vanishes for r = 0. Expanding these summands according to (2.1), we get a sum over all the r-tuples 1 ≤ i 1 < · · · < i r ≤ s + 1, where the left summand corresponds to the tuples with i r ≤ s while the right to the tuples with i r = s + 1. Hence the term equals T Λ,r ϕ . Concerning the differentials, we have the following picture (−1) r−1 ⊗m u u l l l l l l l l l l l l l l l T Λ,r−1 which shows that they coincide with the d r ϕ as defined in (2.2). As a side application, we show the following commutativity result which is analogous to the fact that in a Weyl group corresponding to a generalized Cartan matrix B, the two simple reflections r B i and r B j commute when B ij = 0 = B ji , compare with Proposition 2.12 of [20].
A special role is played by the composition R Λ n · . . . · R Λ 2 · R Λ 1 corresponding to the identity function on {1, . . . , n}. We thus denote by T Λ = T Λ id the corresponding complex of bimodules of Lemma 2.3, so that

Triangular algebras.
In this section we study the complexes T A for triangular algebras A. Lemma 2.6. Let A be triangular and let 1 ≤ i ≤ j ≤ n. Then Proof. If i < j, then Hom A (P i , P j ) ≃ e j Ae i = 0, hence by Lemma 2.1, Proposition 2.7. Let A be triangular. Then: (c) T A is contractible as a complex of right A-modules as well as a complex of left A-modules.
Proof. A repeated application of Lemma 2.6 shows that for 1 ≤ j, s ≤ n, is homotopic to zero. Since A generates D b (A), the first assertion follows. Now the second assertion follows from (2.3). For the third, observe that all the terms of T A are projective both as right and as left A-modules (in fact, the above argument shows directly the contractibility of T A as a complex of right A-modules).
Remark 2.8. Since all its terms at negative degrees are also projective as A-A-bimodules, the complex T A yields a projective resolution of A as an A-A-bimodule, which can be useful when computing Hochschild cohomology. Indeed, a similar resolution is given by Cibils [7], where an explicit contraction (of k-modules) is also given. Remark 2.9. Since T A is contractible as a complex of left A-modules, the tensor product X ⊗ A T A yields a projective resolution of a right module X A . Similarly, T A ⊗ A Y gives a projective resolution of a left module A Y .
The statement of Proposition 2.7 is no longer true when the assumption that A is triangular is removed, even under the condition that A has finite global dimension. This is demonstrated by the following example.
Example 2.10. Let Λ be the path algebra of the quiver modulo the ideal generated by the path βα. The algebra Λ is 5-dimensional, its primitive orthogonal idempotents e 1 , e 2 correspond to the vertices of the quiver and its global dimension is 2. However, Λ is not triangular as its Cartan matrix equals 2 1 1 1 .

Moreover, the complex
is not acyclic since its Euler characteristic as a complex of vector spaces (that is, the alternating sum of dimensions) is 3 · 1 · 2 − (3 · 3 + 2 · 2) + 5 = 0. Note that when k = C, the category mod Λ is equivalent to the principal block of category O of the complex Lie algebra sl 2 , see [21, §5.1.1].
For a triangular algebra A, the compositions of R A i in the reverse order have a very simple form. This is recorded in the next proposition, which will not be used in the sequel.
Proof. Apply Lemma 2.3 for the function ϕ defined by ϕ(t) = i t for 1 ≤ t ≤ s and observe that all the terms T A,r ϕ vanish when r > 1 as e it Ae i t+1 = 0 for all 1 ≤ t < s.

Triangular bimodules and their trivial extensions. Let
A be a basic finite-dimensional algebra with primitive orthogonal idempotents e 1 , . . . , e n . Let M be a triangular bimodule and let DM = Hom k (M, k) be its dual. Consider the trivial extension Λ = A ⋉ DM , that is, the k-algebra which has A ⊕ DM as an underlying vector space, with the multiplication defined by (a, µ)(a ′ , µ ′ ) = (aa ′ , aµ ′ + µa ′ ).
The ring homomorphisms A ι − → Λ π − → A given by ι(a) = (a, 0) π(a, µ) = a give rise to the bimodules A Λ A and Λ A Λ (where a ∈ A acts via multiplication by ι(a) and λ ∈ Λ acts via multiplication by π(λ)). In particular we have the exact functors The left derived functors of their adjoints give rise to The elements ι(e 1 ), . . . , ι(e n ) are primitive orthogonal idempotents of Λ. We shall denote them by e 1 , . . . , e n when there is no risk of confusion. Proposition 2.13. Let A be a finite-dimensional basic algebra and M be a triangular bimodule. Then there exist short exact sequences of complexes of bi-modules Proof. We prove only the exactness of the first sequence, as the proof for the other is similar.
Let 1 ≤ r ≤ n and consider the terms at degree −r of Λ ⊗ A T A and T Λ ⊗ Λ A as direct sums running over the tuples 1 ≤ i 1 < · · · < i r ≤ n. By our hypothesis that M is a triangular bimodule, e j M e i = 0 hence e i DM e j = 0 for all i < j. Therefore we can identify e i Ae j with e i Λe j (via either ι or π) so that the terms (Λ⊗ A T A ) −r and (T Λ ⊗ Λ A) −r are isomorphic via the map Moreover, by considering the explicit forms of the right A-action on Λ and the left Λ-action on A, λ 0 · a 1 = λ 0 ι(a 1 ), a r−1 a r = ι(a r−1 ) · a r , ι(a j a j+1 ) = ι(a j )ι(a j+1 ) for 1 ≤ j < r − 1, we see that these isomorphisms commute with the differentials as long as r > 1.
Finally, note that ( with the top and bottom differentials given by respectively. Summarizing, we get the following commutative diagram of complexes of A-Λ-bimodules which shows the desired exact sequence. Proof. Since T A is contractible as a complex of left A-modules, the complex Λ⊗ A T A is contractible as a complex of left Λ-modules, hence it is isomorphic to zero in D b (Λ op ⊗ A). Now the assertion follows from the first short exact sequence in (2.4). The proof of the second assertion is similar.
Part (b) of Theorem 1.2 now follows from Corollary 2.14 by setting R i = R Λ i for 1 ≤ i ≤ n and using (2.3). Remark 2.15. When M is zero, Λ = A and we recover Proposition 2.7.
2.5. K-theoretic interpretation. We now prove part (a) of Theorem 1.2 and explain how that theorem can be regarded as a categorification of equation (1.1). In fact, we will recover this equation through a process known as decategorification, by looking at the effect of the functors appearing in the theorem on the corresponding Grothendieck groups.
Indeed, as the functors and − L ⊗ Λ A are triangulated, they induce linear maps on the corresponding Grothendieck groups, which we describe explicitly in terms of the so-called Cartan matrices of A and Λ.
For an arbitrary (basic) finite dimensional algebra Λ with indecomposable projectives P 1 , . . . , P n , it will be convenient to reorder them in reverse order and to consider the basis of the Grothendieck group K 0 (per Λ). We denote by C Λ the matrix of the Euler form ·, · Λ with respect to that basis, known as the Cartan matrix of Λ. In explicit terms, Lemma 2.16. Let 1 ≤ i ≤ n. Then the matrix of the linear map on K 0 (per Λ) induced by R Λ i is given by r C Λ n+1−i . Proof. The j-th column of that matrix is equal to the class of R Λ i (P n+1−j ) in K 0 (per Λ) which, according to Corollary 2.2, equals For an algebra A with primitive orthogonal idempotents e 1 , . . . , e n , the condition that A is triangular implies that the matrix C A is upper triangular with ones on its main diagonal. Similarly to the definition of C A , one can define for any A-A-bimodule X, a Cartan matrix C X by so that X is triangular if and only if C X is upper triangular. Lemma 2.17. Let A be a triangular algebra and X an A-A-bimodule. Then the matrix of the linear map on K 0 (per A) induced by the functor − L ⊗ A X is given by C −1 A C X . Proof. Denote that matrix (with respect to the basis ε 1 , . . . , ε n ) by x. Since the functor − L ⊗ A X sends each P j to P j ⊗ A X ≃ e j X, we have for all 1 ≤ i ≤ n. Now, for any 1 ≤ l ≤ n, When A is triangular and M is a triangular bimodule, the Cartan matrix of the trivial extension Λ = A ⋉ DM equals C Λ = C A + C DM = C A + C T M . Hence (C Λ ) + = C A is the upper triangular part of C Λ and (C Λ ) − = C M is its lower triangular part, as defined in Proposition 1.1.
Combining everything together, observing that the functor − L ⊗ Λ A sends the projective ι(e i )Λ to e i A and thus induces the identity matrix between the isomorphic groups K 0 (per Λ) and K 0 (per A), we conclude the following.

2.7.
Realization of matrices as Cartan matrices. We now show that Theorem 1.2 categorifies (1.1) for any integral square matrix B with nonnegative entries and positive entries on its main diagonal. We start with an observation on the realization of such matrices as Cartan matrices of finite dimensional algebras.
Lemma 2.20. Let C be an integral n × n matrix with C ij ≥ 0 and C ii > 0 for 1 ≤ i, j ≤ n, and let k be a field. Then there exists a finite dimensional algebra A over k whose Cartan matrix equals C.
Proof. We construct A from a quiver with relations. Let Q be the quiver whose vertices are {1, 2, . . . , n}, with the number of arrows from i to j set to |{arrows i → j}| = C n+1−j,n+1−i − δ ij . Let I ⊆ kQ be the ideal in the path algebra of Q generated by all the paths of length 2. Then the Cartan matrix of A = kQ/I equals C, since dim k Hom A (P n+1−i , P n+1−j ) is the number of paths in Q from n + 1 − j to n + 1 − i of length at most one, which equals C ij by construction.
Observe that the algebra A constructed in the lemma is triangular if and only if C is upper triangular with ones on its main diagonal. We now consider the realization of bimodules with prescribed Cartan matrix over such algebras.
Lemma 2.21. Let C ′ be an integral n × n matrix with non-negative entries and A a finite dimensional algebra as constructed in the previous lemma, which is furthermore triangular. Then there exists a bimodule M over A such that C M = C ′ .
Proof. Let A = kQ/I be as in the previous lemma, and consider the quiver Q op × Q whose vertices are the pairs (i, j) for 1 ≤ i, j ≤ n, with "horizontal" arrows (i, j) → (i, j ′ ) for every arrow j → j ′ in Q and "vertical" arrows (i ′ , j) → (i, j) for every arrow i → i ′ in Q. By our assumption on A, there are no loops in this quiver.
We construct a representation M of Q op × Q as follows. For any vertex (i, j), let M (i,j) be a k-vector space of dimension C ′ n+1−j,n+1−i . For any arrow α : (i, j) → (i ′ , j ′ ), take the linear map M α : M (i,j) → M (i ′ ,j ′ ) to be zero if i = j and arbitrary otherwise.
Observe that for any path in Q op × Q of length 2, the corresponding map in M vanishes, hence in particular M can be viewed as a bimodule over A.
Note that in the proof we could have taken all the maps M α to be zero. However, the construction presented in the lemma is slightly more general and in particular one can realize A as a bimodule over itself in this way.
Combining the above two observations, we deduce the following. In particular we see that Corollary 1.3 categorifies (1.3) for any square upper triangular integer matrix C with non-negative entries and 1 on its main diagonal (or equivalently, for any square symmetric integer matrix B with non-negative entries and 2 on its main diagonal).

Discussion and Comparison
In this section we recall previous work on path algebras of quivers that can be considered as a categorical interpretation of equation (1.3), and compare it with our approach.
3.1. A result of Gabriel. Fix an algebraically closed field k. For a quiver Q without oriented cycles, denote by kQ its path algebra and by D b (Q) the bounded derived category of finite-dimensional right kQ-modules. Recall that a vertex s ∈ Q is called a sink if there are no arrows in Q starting at s. The reflection of Q at s, denoted σ s Q, is the quiver obtained from Q by inverting all the arrows ending at s while leaving all the others intact, so that s becomes a source in σ s Q.
In [3], Bernstein, Gelfand and Ponomarev defined the reflection functor from the category of representations of Q to those of σ s Q (where s is a sink in Q). In the language of derived categories (see for example [11,(IV.4, Exercise 6)]), this functor induces a derived equivalence Order now the vertices of Q in an admissible ordering, that is, enumerate them in a sequence 1, 2, . . . , n such that there are no arrows j → i in Q for i < j. In this case, the vertex 1 ≤ i ≤ n is a sink in the quiver σ i+1 σ i+2 . . . σ n Q. Moreover, the quiver σ 1 . . . σ n Q is equal to Q. Thus, the composition of the (derived) reflection functors defines an autoequivalence R 1 · R 2 · . . . · R n of D b (Q), known as the Coxeter functor.
Another autoequivalence of D b (Q) is given by the Auslander-Reiten translation τ , which is related to the Serre functor ν = − L ⊗ kQ D(kQ) on D b (Q) by τ = ν[−1], see [12]. The following result of Gabriel [10] relates it with the Coxeter functor.
Theorem 3.1 ( [10]). If the underlying graph of Q is a tree, or more generally, does not contain a cycle of odd length, then Similarly to Corollary 2.18, the relation with equation (1.3) is obtained through decategorification, by considering the Grothendieck group K 0 (Q) of the triangulated category D b (Q) together with its Euler form ·, · kQ , but this time using bases of simple modules rather than the indecomposable projective ones.
Let S i be the simple module corresponding to the vertex 1 ≤ i ≤ n. The classes [S 1 ], . . . , [S n ] form a basis of K 0 (Q), and the matrix C Q of ·, · kQ with respect to that basis has an explicit combinatorial description, namely When the vertices are ordered in an admissible order, the matrix C Q is upper triangular with ones on its main diagonal.
It is well known that the matrix of the linear map on K 0 (Q) induced by τ is given by −C −1 Q C T Q . On the other hand, for a sink s, the reflection functor R s induces a linear map K 0 (Q) → K 0 (σ s Q) whose matrix with respect to the bases of simples is given by the reflection r B Q s , where B Q = C Q + C T Q is the symmetrization of C Q , see [3]. Moreover, B σsQ = B Q since is independent on the orientation of the arrows.
Therefore, by looking at the Grothendieck groups, Theorem 3.1 implies the following commutative diagram recovering equation (1.3) for C = C Q as a K-theoretical consequence of the isomorphism of the functors τ and R 1 · R 2 · . . . · R n on D b (Q). The use of the basis of simples is a rather special feature of hereditary algebras. Indeed, recall that for a quiver Q and a sink s, one has C σsQ = r T s C Q r s where r s is the corresponding reflection built from the symmetrization of C Q . However, for a general triangular algebra A whose Euler form is given by an upper triangular matrix C with respect to the basis of simples, there may be no algebra whose Euler form is given by the matrix r T s Cr s , where s is a sink or a source in the quiver of A and r s is the corresponding reflection built from the symmetrization of C.