Hochschild cohomology rings of Temperley-Lieb algebras

Abstract

The authors first construct an explicit minimal projective bimodule resolution (ℙ, δ) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Δ: ℙ → ℙ ⊗ _A ℙ. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.

Keywords

Hochschild cohomology / Cup product / Temperley-Lieb algebra / q-Schur algebra

References

[1] Ardizzoni A., Menini C., Stefan D.. Hochschild cohomology and smoothness in monoidal categories. J. Pure Appl. Algebra, 2007, 208: 297-330

[2] Buchweitz R. O., Green E. L., Snashall N., Solberg Ø.. Multiplicative structures for Koszul algebras. Quart. J. Math., 2008, 59(4): 441-454

[3] Bulter M. C. R., King A. D.. Minimal resolution of algebras. J. Algebra, 1999, 212: 323-362

[4] Bustamante J. C.. The cohomology structure of string algebras. J. Pure Appl. Algebra, 2006, 204: 616-626

[5] Birman J., Wenzl H.. Braids, link polynomials and a new algebra. Trans. Amer. Math. Soc., 1989, 313: 249-273

[6] Cibils C.. Rigidity of truncated quiver algebras. Adv. Math., 1990, 79: 18-42

[7] Erdmann K., Holm T.. Twisted bimodules and Hochschild cohomology for self-injective algebras of class An. Forum Math., 1999, 11: 177-201

[8] Erdmann K., Schroll S.. On the Hochschild cohomology of tame Hecke algebras. Arch. Math., 2010, 94: 117-127

[9] Erdmann K., Snashall N.. On Hochschild cohomology of preprojective algebras, II. J. Algebra, 1998, 205: 413-434

[10] Fan J. M., Xu Y. G.. On Hochschild cohomology ring of Fibonacci algebras. Frontiers of Mathematics in China, 2006, 1(4): 526-537

[11] Gerstenhaber M.. On the deformation of rings and algebras. Ann. Math., 1964, 79: 59-103

[12] Green E. L., Hartman G., Marcos E. N., Solberg Ø.. Resolutions over Koszul algebras. Arch. Math., 2005, 85: 118-127

[13] Green E. L., Solberg Ø.. Hochschild cohomology rings and triangular rings. Beijing Normal University Press, Beijing, 2002, 2: 192-200

[14] Green E. L., Solberg Ø., Zacharia D.. Minimal projective resolutions. Trans. Amer. Math. Soc., 2001, 353: 2915-2939

[15] Happel D.. Hochschild cohomology of finite-dimensional algebras, Lecture Notes in Mathematics, 1404. Springer-Verlag, New York, 1989 108-126

[16] Hochschild G.. On the cohomology groups of an associative algebra. Ann. Math., 1945, 46(1): 58-67

[17] De la Pena J. A., Xi C. C.. Hochschild cohomology of algebras with homological ideals. Tsukuba J. Math., 2006, 30(1): 61-80

[18] Jones V. F. R.. Index for subfactors. Invent. Math., 1983, 72: 1-25

[19] Jones V. F. R.. A polynomial invariant for links via von Neumann algebras. Bulletin of the Amer. Math. Soc., 1985, 129: 103-112

[20] Kauffman, L. H., Knots in Physics, World Scientic Press, River Edge, NJ, 1994.

[21] Strametz C.. The Lie algebra structure on the first Hochschild cohomology group of a monomial algebra. Comptes Rendus Mathematique, 2002, 334: 733-738

[22] Skowronski A.. Simply connected algebras and Hochschild cohomology. Can. Math. Soc. Proc., 1993, 14: 431-447

[23] Siegel S. F., Witherspoon S. J.. The Hochschild cohomology ring of a group algebra. Proc. London Math. Soc., 1999, 79(3): 131-157

[24] Temperley H. N. V., Lieb E. H.. Relations between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: Some exact results for the percolation problem. Proc. R. Soc. Lon. (Ser. A), 1971, 322: 251-280

[25] Westbury B. W.. The representation theory of the Temperley-Lieb algebras. Math. Z., 1995, 219(4): 539-565

[26] Xi C. C.. On representation types of q-Schur algebras. J. Pure Appl. Algebra, 1993, 84: 73-84

[27] Xu Y. G., Xiang H. L.. Hochschild cohomology rings of d-Koszul algebras. J. Pure Appl. Algebra, 2011, 215: 1-12