In this paper, we determine the second Hochschild cohomology group for a class of self-injective algebras of tame representation type namely, which are standard one-parametric but not weakly symmetric. These were classified up to derived equivalence by Bocian, Holm and Skowroński in [1]. We connect this to the deformation of these algebras.
[1] R. Bocian, T. Holm and A. Skowroński, “Derived Equivalence Classification of One-Parametric Self-Injective Algebras,” Journal of Pure and Applied Algebra, Vol. 207, No. 3, 2006, pp. 491-536. doi:10.1016/j.jpaa.2005.10.015
[2] E. L. Green and N. Snashall, “Projective Bimodule Resolutions of an Algebra and Vanishing of the Second Hochschild Cohomology Group,” Forum Mathematicum, Vol. 16, No. 1, 2004, pp. 17-36. doi:10.1515/form.2004.003
[3] D. Al-Kadi, “Self-Injective Algebras and the Second Hochschild Cohomology Group,” Journal of Algebra, Vol. 321, No. 4, 2009, pp. 1049-1078. doi:10.1016/j.jalgebra.2008.11.019
[4] D. Happel, “Hochschild Cohomology of Finite-Dimensional Algebras,” Lecture Notes in Mathematics, Spring-Verlag, Berlin, 1989. doi:10.1090/S0002-9947-01-02687-3
[5] E. L. Green, Ø. Solberg and D. Zacharia, “Minimal Projective Resolutions,” Transactions of the American Mathematical Society, Vol. 353, No. 7, 2001, pp. 2915-2939. doi:10.1007/BFb0084073