The first Hochschild cohomology group of quantum matrices and the   quantum special linear group
  We calculate the first Hochschild cohomology group of quantum matrices, the quantum general linear group and the quantum special linear group in the generic case when the deformation parameter is not a root of unity. As a corollary, we obtain information about twisted Hochschild homology of these algebras. 
S Launois and T H Lenagan