I am really sorry if this is trivial but I come from Physics and it is really confusing for me to understand what is going on and looking at the answers on both Physics and Math SE sent me through a rabbit hole and I am more confused than ever. From what I understand from this this MathSE answer is that for finite sequence of vector spaces the direct product is essentially the same as the direct sum with the added property of linearity.
A tensor product, however, is entirely different. Most Physics texts tend to use both $\otimes$ and $\times$ to denote the tensor(?) product. So if when I say something like $SU(2)\times SU(2)$, is it actually $SU(2)\otimes SU(2)$? So, roughly, the statement of addition of angular momenta is that a tensor product state can be decomposed as a direct sum.
Next, what made me even more confused is that, I saw it in Matthew D. Schwartz's QFT and the Standard Model that the Lie Algebra of the Lorentz group (which is $SU(2)\otimes(?)SU(2)$) breaks down into $\mathfrak{so}(1,3) = \mathfrak{su}(2)\oplus\mathfrak{su}(2)$ where $\mathfrak{su}(2)$ is the Lie Algebra of $SU(2)$. But that would mean that if I have matrix respresentations of Lie Algebras $\mathfrak{h},\mathfrak{g}$ with dimensions $N$ and $M$ respectively, their direct sum has dimension $N+M$ and so the representations of the group obtained from this representation of the algebra also has dimension $N+M$ so are we talking about the direct product here? So the $SO^{+}(1,3) \cong SU(2)\times SU(2)$, actually? I am so confused.
I got downvoted in PhysicsSE, can someone please explain to me what is the distinction and if the above is true for Lie Algebras? I am absolutely clueless about this, thanks.
