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Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies

Institution:	1. School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China;2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, Guangdong, China;1. Department of Mathematics, University of Rijeka, Radmile Matej?i? 2, 51000 Rijeka, Croatia;2. Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia;1. Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, Campus Morelia, Morelia, Michoacán, 58089, Mexico;2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, Circuito Exterior S/N, Cd. Universitaria, Colonia Copilco el Bajo, Alcaldía Coyoacán, 04510, México D.F., Mexico;1. Instituto de Matemática e Estatística, Universidade Federal Fluminense, Campus Gragoatá, Rua Alexandre Moura 8 - São Domingos, 24210-200 Niterói, Rio de Janeiro, Brazil;2. Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 30, 44121 Ferrara, Italy

Abstract:	We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra $T_{p}$ for any integer $p > 2$ which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of $T_{p}$ , as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. Also, we find a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber's original formula for Hochschild cohomology.

Keywords:	Hochschild cohomology Hopf algebra cohomology Gerstenhaber bracket Taft algebra
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