Okay, this is going to sound pretty silly and trivial, but I’ve been grading today. There is one representation we always have for any group, called the zero representation .
Pretty obviously this is built on the unique zero-dimensional vector space . It shouldn’t be hard to convince yourself that is the trivial group, and so any group has a unique homomorphism to this group. Thus there is a unique representation of on the vector space .
We should immediately ask: is this representation a zero object? Suppose we have a representation . Then there is a unique arrow sending every vector to . Similarly, there is a unique arrow sending the only vector to the zero vector . It’s straightforward to show that these linear maps are intertwinors, and thus that the zero representation is indeed a zero object for the category of representations of .
This is all well and good for groups, but what about representing an algebra ? This can only make sense if we allow rings without unit, which I only really mentioned back when I first defined a ring. This is because there’s only one endomorphism of the zero-dimensional vector space at all! The endomorphism algebra will consist of just the element , and a representation of has to be an algebra homomorphism to this non-unital algebra. Given this allowance, we do have the zero representation, and it’s a zero object just as for groups. It’s sort of convenient, so we’ll tacitly allow this one non-unital algebra to float around just so we can have our zero representation, even if we allow no other algebras without units.