The cohomology structure of string algebras

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8 J. C. BUSTAMANTE Remark. At this point, it is important to note that even if Lemmas 2.1, 2.2 and 2.3 were stated assuming that A = kQ/I is a string triangular algebra, the condition (S2) in the definition of string algebras has never been used. Thus, the preceding theorem holds for every monomial quadratic, triangular algebra A = kQ/I such that (Q, I) satisfies (S3). This includes, for instance, the gentle algebras. We now turn our attention to the Lie algebra structure HH ∗ (A). Given φ ∈ Cmin n (A, A), ψ ∈ Cm min (A, A), and i ∈ {1, . . . , n} we define φ ◦ i ψ ∈ (A, A) as φ ◦ i ψ = µ n+m−1 ((ω n φ) ◦ i (ω m ψ)) C n+m−1 min and from this, the bracket [−, −] is defined as in 1.5. In particular, and this is the crucial point for what follows, given α 1 · · · α n+m−1 ∈ Γ n+m−1 we have ⎧ ⎨ φ(α 1 · · · α i−1 ψ(α i · · · α i+m−1 )α i+m · · · α n+m−1 ), if φ◦ i ψ(α 1 · · · α n+m−1 ) = α 1 · · · α i−1 ψ(α i · · · α i+m−1 )α i+m · · · α n+m−1 ∈ Γ n , ⎩ 0 otherwise. Again, one can verify that δ n+m−1 [f, g] = (−1) m−1 [δ n f, g] + [f, δ m g], so that [−, −] induces a bracket, which we still denote by [−, −] at the cohomology level (note that, by construction, this bracket coincides with that of 1.5). This leads us to the following result 3.2. Theorem. Let A = kQ/I be a triangular gentle algebra. Then, for n > 1 and m > 1, we have [HH n (A), HH m (A)] = 0. Proof : We show that, in fact, under the hypothesis, the products ◦ i are equal to zero at the cochain level. This follows immediately from the discussion above. Indeed, with that notations, if α 1 · · · α n+m−1 ∈ Γ n+m−1 we have, in particular that α i−1 α i ∈ I. But A being gentle, there is no other arrow β in Q with s(β) = t(α i−1 ) such that α i−1 β ∈ I. □ The following example shows that the previous result can not be extended to string algebras which are not gentle. 3.3. Example. Let A = kQ/I where Q is the quiver 3 α 2 2 α 1 β 1 γ α 3 4 α 4 5 and I is the ideal generated by paths of length 2. This is a string algebra which is not gentle. From theorem 3.1 in [7] we get ⎧ ⎨ 1 if i ∈ {0, 2, 3, 4}, dim k HH i (A) = 2 if i = 1, ⎩ 0 otherwise. Keeping in mind the identifications of section 2, the generators of HH i (A) are the elements corresponding to (β, ¯β), and (γ, ¯γ) for i = 1; (α 2 α 3 , ¯β) for i = 2; (α 1 βα 4 , ¯γ), for i = 3; and (α 1 α 2 α 3 α 4 , ¯γ) for i = 4. A straightforward computation shows that [HH n (A), HH m (A)] = HH n+m−1 (A).
THE COHOMOLOGY STRUCTURE OF STRING ALGEBRAS 9 The first Hochschild cohomology group of an algebra A is by its own right a Lie algebra. In case the algebra A is monomial, this structure has been studied in [13]. The following result gives information about the role played by HH 1 (A) in the whole Lie algebra HH ∗ (A) in our context. 3.4. Proposition. Let A = kQ/I be a triangular monomial and quadratic algebra. Then [HH n (A), HH 1 (A)] = HH n (A), whenever n > 1. Proof : In fact, for every f ∈ Cmin n (A, A) there exists g ∈ C1 min (A, A) such that [f, g] = f, and g ≠ 0 in HH 1 (A). Clearly it is enough to consider an arbitrary basis element f ∈ Cmin n (A, A). Let f be such an element, corresponding to (α 1 · · · α n , p). Define g ∈ Cmin 1 (A, A) by { α1 if γ = α g(γ) = 1 , 0 otherwise It is straightforward to verify that f ◦ g = f ◦ 1 g, and, since we assume A triangular and n > 1, that g ◦ f = 0 so that [f, g] = f. Moreover, direct computations show that ḡ ≠ 0 in HH 1 (A). □ Acknowledgements. The author gratefully thanks professors E.N. Marcos, for several interesting discussions and comments, as well as professors E.N. Marcos and I. Assem for a carefully reading of preliminar versions of this work. This work was done while the author had post-doctoral fellowship at the University of São Paulo, Brazil. He gratefully acknowledges the University for hospitality during his stay there, as well as financial support from F.A.P.E.S.P. References [1] I. Assem and J.A. de la Peña. The fundamental groups of a triangular algebra. Comm. Algebra, 24(1):187–208, 1996. [2] I. Assem and A. Skowroński. Iterated tilted algebras of type Ã. Math. Zeitschrift, 195(2):269– 290, 1987. [3] M. J. Bardzell. The alternating syzygy beheavior of monomial algebras. J. Algebra, 188:69–89, 1997. [4] K. Bongartz and P. Gabriel. Covering spaces in representation theory. Invent. Math, 65(3):331–378, 1981-1982. [5] M.C.R. Butler and C.M. Ringel. Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra, 15(1-2):145–179, 1987. [6] C. Cibils. Cohomology of incidence algebras and simplicial complexes. J. Pure Appl. Algebra, 56:221–232, 1989. [7] C. Cibils. Hochschild coohomology of radical square zero algebras. In I. Reiten, S. Smalø, and Ø. Solberg, editors, Algebras and Modules II, number 24 in CMS Conf. proceedings, pages 93–101, Geiranger, Norway, 1998. [8] M. Gerstenhaber. On the deformations of rings and algebras. Ann. Math. Stud., 79(2):59–103, 1964. [9] M. Gerstenhaber. The Cohomology structure on an associative ring. Ann. of Math., 78(2):267–288, 1978. [10] E.L. Green, N. Snashall, and Ø. Solberg. The hochschild cohomology ring modulo nilpotence of a monomial algebra. www.arXiv:math/RT/0401446v1, 2004. [11] E.L. Green and Ø. Solberg. Hochschild cohomology rings and triangular rings. In Beijing Norm. Univ. Press, editor, Representations of algebra., volume I, pages 192–200, 2002. [12] D. Happel. Hochschild cohomology of finite-dimensional algebras, pages 108–126. Number 1404 in Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg New York Tokyo, 1989. [13] C. Strametz. The Lie algebra structure of the first Hochschild cohomology group for monomial algebras. C. R., Math., Acad. Sci. Paris, 334(9):733–738, 2002.
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