Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$ is. The $\times$ is the Cartesian product while the $\otimes$ is the tensor product. I gave the example of the Standard Model gauge group but it can be any product of groups. My question is when talking about global or gauge groups, do we mean Cartesian products or tensor products? And what is the real difference between them anyway?

Answer

I) The main point is that we usually only consider tensor products $V \otimes W$ of vector spaces $V$, $W$ (as opposed to general sets $V$, $W$). But groups (say $G$, $H$) are often not vector spaces. If we only consider tensor products of vector spaces, then the object $G \otimes H$ is nonsense, mathematically speaking.

With further assumptions on the groups $G$ and $H$, it is sometimes possible to define a tensor product $G \otimes H$ of groups, cf. my Phys.SE answer here and links therein.

II) If $V$ and $W$ are two vector spaces, then the tensor product $V \otimes W$ is again a vector space. Also the direct or Cartesian product $V\times W$ of vector spaces is isomorphic to the direct sum $V \oplus W$ of vector spaces, which is again a vector space.

In fact, if $V$ is a representation space for the group $G$, and $W$ is a representation space for the group $H$, then both the tensor product $V\otimes W$ and the direct sum $V\oplus W$ are representation spaces for the Cartesian product group $G\times H$.

(The direct sum representation space $V\oplus W\cong (V\otimes \mathbb{F}) \oplus(\mathbb{F}\otimes W)$ for the Cartesian product group $G\times H$ can be viewed as a direct sum of two $G\times H$ representation spaces, and is hence a composite concept. Recall that any group has a trivial representation.)

This interplay between the tensor product $V\otimes W$ and the Cartesian product $G\times H$ may persuade some authors into using the misleading notation $G\otimes H$ for the Cartesian product $G\times H$. Unfortunately, this often happens in physics and in category theory.

III) In contrast to groups, note that Lie algebras (say $\mathfrak{g}$, $\mathfrak{h}$) are always vector spaces, so tensor products $\mathfrak{g}\otimes\mathfrak{h}$ of Lie algebras do make sense. However due to exponentiation, it is typically the direct sum $\mathfrak{g}\oplus\mathfrak{h}$ of Lie algebras that is relevant. If $\exp:\mathfrak{g}\to G$ and $\exp:\mathfrak{h}\to H$ denote exponential maps, then $\exp:\mathfrak{g}\oplus\mathfrak{h}\to G\times H$.