Algebras and Representation Theory

, Volume 21, Issue 1, pp 61–86

# Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

• María Julia Redondo
• Lucrecia Román
Article

## Abstract

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.

## Keywords

Hochschild cohomology Cup product Lie bracket String algebras

16E40 16W99

## Notes

### Acknowledgments

The first author is a researcher and the second author has a fellowship from CONICET, Argentina. This work has been supported by the project PICT-2011-1510.

## References

1. 1.
Assem, I., Happel, D.: Generalized tilted algebras of type A n. Comm. Algebra 9(20), 2101–2125 (1981)
2. 2.
Assem, I., Simson, D., Skowroński, A.: Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006). x+458pp
3. 3.
Assem, I., Skowroński, A.: Iterated tilted algebras of type $$\tilde {\textbf {A}}_{n}$$. Math. Z. 195(2), 269–290 (1987)
4. 4.
Avella-Alaminos, D., Geiss, C.: Combinatorial derived invariants for gentle algebras. J. Pure Appl. Algebra 212(1), 228–243 (2008)
5. 5.
Bardzell, M.J.: The alternating syzygy behavior of monomial algebras. J. Algebra 188(1), 69–89 (1997)
6. 6.
Bardzell, M.J., Locateli, A.C., Marcos, E.N.: On the Hochschild cohomology of truncated cycle algebras. Comm. Algebra 28(3), 1615–1639 (2000)
7. 7.
Bustamante, J.C.: The cohomology structure of string algebras. J. Pure Appl. Algebra 204(3), 616–626 (2006)
8. 8.
Butler, M.C.R., Ringel, C.M.: Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15(1-2), 145–179 (1987)
9. 9.
Cibils, C.: On the Hochschild cohomology of finite-dimensional algebras. Comm. Algebra 16(3), 645–649 (1988)
10. 10.
Cibils, C.: Hochschild cohomology algebra of radical square zero algebras, Algebras and modules, II (Geiranger 1996), CMS Conf. Proc., vol. 24, pp. 93–101. Amer. Math. Soc., Providence, RI (1998)Google Scholar
11. 11.
Cibils, C., Redondo, M.J., Saorín, M.: The first cohomology group of the trivial extension of a monomial algebra. J. Algebra Appl. 3(2), 143–159 (2004)
12. 12.
Crawley-Boevey, W.: Tameness of biserial algebras. Arch. Math. (Basel) 65(5), 399–407 (1995)
13. 13.
Erdmann, K.: Algebras and dihedral defect groups. Proc. London Math. Soc. (3) 54(1), 88–114 (1987)
14. 14.
Fuller, K.R.: Biserial rings, Ring theory (Proc. Conf., Univ. Waterloo, Waterloo, 1978), Lecture Notes in Math., vol. 734, pp. 64–90. Springer, Berlin (1979)Google Scholar
15. 15.
Gel’fand, I.M., Ponomarev, V.A.: Indecomposable representations of the Lorentz group. Uspehi Mat. Nauk 23(2 (140)), 3–60 (1968). (Russian)
16. 16.
Gerstenhaber, M.: The cohomology structure of an associative ring. Ann. of Math. (2) 78, 267–288 (1963)
17. 17.
Happel, D.: Hochschild cohomology of finite-dimensional algebras, Séminaire d’Algèbre Paul Dubreil et Marie-Paul Malliavin, 39ème Année (Paris 1987/1988), Lecture Notes in Math., vol. 1404, pp. 108–126. Springer, Berlin (1989)Google Scholar
18. 18.
Janusz, G.J.: Indecomposable modules for finite groups. Ann. of Math. (2) 89, 209–241 (1969)
19. 19.
Kawada, Y.: On Köthe’s problem concerning algebras for which every indecomposable module is cyclic. I. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 7(1962), 154–230 (1962)
20. 20.
Ladkani, S.: Hochschild cohomology of gentle algebras, available at arXiv:1208.2230v1[math.RT]
21. 21.
Le, J., Zhou, G.: On the Hochschild cohomology ring of tensor products of algebras. J. Pure Appl. Algebra 218(8), 1463–1477 (2014)
22. 22.
Liu, Y., Zhou, G.: The Batalin-Vilkovisky structure over the Hochschild cohomology ring of a group algebra. J. Noncommut. Geom. 10(3), 811–858 (2016)
23. 23.
Nakayama, T.: On Frobeniusean algebras. I. Ann. of Math. (2) 40, 611–633 (1939)
24. 24.
Redondo, M.J.: Hochschild cohomology via incidence algebras. J. Lond. Math. Soc. (2) 77(2), 465–480 (2008)
25. 25.
Redondo, M.J., Román, L.: Hochschild cohomology of triangular string algebras and its ring structure. J. Pure Appl. Algebra 218(5), 925–936 (2014)
26. 26.
Ringel, C.M.: The indecomposable representations of the dihedral 2-groups. Math. Ann. 214, 19–34 (1975)
27. 27.
Ringel, C.M.: Kawada’s theorem, Abelian group theory (Oberwolfach 1981), Lecture Notes in Math., vol. 874, pp. 431–447. Springer, Berlin-New York (1981)Google Scholar
28. 28.
Ringel, C.M.: The minimal representation-infinite algebras which are special biserial, Representations of algebras and related topics, EMS Ser. Congr. Rep., pp. 501–560. Eur. Math. Soc., Zürich (2011)Google Scholar
29. 29.
Sánchez-Flores, S.: La structure de Lie de la cohomologie de Hochschild d’algèbres monomiales, Université Montpellier II - Sciences et Techniques du Languedoc. HAL ID: tel-00464064 (2009)Google Scholar
30. 30.
Sánchez-Flores, S.: The Lie module structure on the Hochschild cohomology groups of monomial algebras with radical square zero. J. Algebra 320(12), 4249–4269 (2008)
31. 31.
Schroll, S.: Trivial extensions of gentle algebras and Brauer graph algebras. J. Algebra 444, 183–200 (2015)
32. 32.
Shepler, A.V., Witherspoon, S.: Group actions on algebras and the graded Lie structure of Hochschild cohomology. J. Algebra 351, 350–381 (2012)
33. 33.
Sköldberg, E.: A contracting homotopy for Bardzell’s resolution. Math. Proc. R. Ir. Acad. 108(2), 111–117 (2008)
34. 34.
Skowroński, A., Waschbüsch, J.: Representation-finite biserial algebras. J. Reine Angew. Math. 345, 172–181 (1983)
35. 35.
Snashall, N., Taillefer, R.: The Hochschild cohomology ring of a class of special biserial algebras. J. Algebra Appl. 9(1), 73–122 (2010)
36. 36.
Strametz, C.: The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra. J. Algebra Appl. 5(3), 245–270 (2006)
37. 37.
Suárez-Álvarez, M.: Applications of the change-of-rings spectral sequence to the computation of Hochschild cohomology, available at arXiv:0707.3210 [math.KT]
38. 38.
Wald, B., Waschbüsch, J.: Tame biserial algebras. J. Algebra 95(2), 480–500 (1985)
39. 39.
Witherspoon, S., Zhou, G.: Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions. Pacific J. Math. 283(1), 223–255 (2016)