Direct Product vs Tensor Product

Question

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product:

1. On page 67, it mentioned that

   "you can take a direct product of two $j = 1/2$ representations"

   and build representations of higher j.

2. On page 70, it mentioned

   "we can think of [the Lorentz Group] as the direct product $SU(2) \times SU(2)$."

In each of the above, does the author mean Direct Product or Tensor Product?

Answer 1

1. On p. 67 Tseng means a tensor product of representations.

2. On p. 70 Tseng means a direct product of groups.

   Note however that the actual statement about the Lorentz group is wrong/imprecise as explained in e.g. this Phys.SE post.

   Concerning direct product vs. tensor product of groups, see also my related Phys.SE answer here.

Answer 2

In both the cases the author is talking about direct product. Addition of two angular momentums $j_1$ and $j_2$ is represented using the direct product of the two angular momentum spaces. The wave functions or kets in the product space are represented using tensor product notation.