Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Let K be a field of characteristic 0 and L be a G-graded Lie PIalgebra where the support of L is a finite subset of G. We define the G-graded Gelfand-Kirillov dimension (GK) of L in k-variables as the GK dimension of its G-graded relatively free algebra having k homogeneous variables for each element of the support of L. We compute the G-graded GK dimension of sl(2)(K), where G is any abelian group. Then, we compute the exact value for the Zn-graded GK dimension of sl(n)(K) endowed with the Zn-grading of Vasilovsky.654752767FAPESP of Brazil [2013/06752-4, 2013/04590-7]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)