  The category of finite dimensional module over the quantum superalgebra
U_q(sl(2|1)) is not semi-simple and the quantum dimension of a generic
U_q(sl(2|1))-module vanishes. This vanishing happens for any value of q (even
when q is not a root of unity). These properties make it difficult to create a
fusion or modular category. Loosely speaking, the standard way to obtain such a
category from a quantum group is: 1) specialize q to a root of unity; this
forces some modules to have zero quantum dimension, 2) quotient by morphisms of
modules with zero quantum dimension, 3) show the resulting category is finite
and semi-simple. In this paper we show an analogous construction works in the
context of U_q(sl(2|1)) by replacing the vanishing quantum dimension with a
modified quantum dimension. In particular, we specialize q to a root of unity,
quotient by morphisms of modules with zero modified quantum dimension and show
the resulting category is generically finite semi-simple. Moreover, we show the
categories of this paper are relative G-spherical categories. As a consequence
we obtain invariants of 3-manifold with additional structures.
