Monday, 17 December 2018

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)×SU(2)×U(1) and SU(3)SU(2)U(1) is. The × is the Cartesian product while the 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 VW 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 GH is nonsense, mathematically speaking.


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


II) If V and W are two vector spaces, then the tensor product VW is again a vector space. Also the direct or Cartesian product V×W of vector spaces is isomorphic to the direct sum VW 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 VW and the direct sum VW are representation spaces for the Cartesian product group G×H.


(The direct sum representation space VW(VF)(FW) for the Cartesian product group G×H can be viewed as a direct sum of two G×H representation spaces, and is hence a composite concept. Recall that any group has a trivial representation.)


This interplay between the tensor product VW and the Cartesian product G×H may persuade some authors into using the misleading notation GH for the Cartesian product G×H. Unfortunately, this often happens in physics and in category theory.



III) In contrast to groups, note that Lie algebras (say g, h) are always vector spaces, so tensor products gh of Lie algebras do make sense. However due to exponentiation, it is typically the direct sum gh of Lie algebras that is relevant. If exp:gG and exp:hH denote exponential maps, then exp:ghG×H.


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