  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.
