Week 10: Bimodules

Welcome back to my blog! In part 1 of my Week 8 post, I mentioned an alternative formulation of the group theoretical property: that for some algebra A, the category of A-bimodules is pointed. To further explore the research question, we must better understand how to work with bimodules.

Modules and bimodules

Recall that a ring is a structure with addition and multiplication. Then, as I discussed in previous weeks, a left module is the analog of a vector space for rings. To be precise, it’s an additive group along with multiplication r•m that returns another module element. This multiplication needs to distribute with addition: r•(m+n)=r•m+r•n and (r+s)•m=r•m+s•m. (Here, r and s are ring elements, while m and n are module elements). It must also be compatible with multiplication: (rs)•m=r•(s•m).

This is the standard definition of modules, but it isn’t sufficient for our purposes. In category theory, we usually don’t have any elements to work with, so we must formulate this in terms of morphisms. Multiplication is a morphism from R x M to M with the compatibility and distributivity conditions which I mentioned earlier. These conditions make the morphism bilinear, so it uniquely factors through the tensor product R⊗M. The phrase “uniquely factors” is a fundamental idea in category theory, which I can explain if anyone is interested. For now, I’ll just say that it lets us instead work with morphisms from R⊗M to M, i.e. morphisms in Hom(R⊗M,M)

A right module is analogous, but multiplication is done on the other side, so now it uses morphisms in Hom(M⊗R,M). Then, bimodules unite these concepts. They are an additive group M along with the two multiplications such that r•(m•s)=(r•m)•s. To phrase this in categorical language, we can take two paths from R⊗M⊗R to M: we either do left multiplication or right multiplication first, and we do the other second. Then, these paths should result in the same morphisms. I’m essentially describing a commutative diagram, if you’re interested in the formulation with paths.

Semisimple algebras

By default, we can’t have modules over objects from a fusion category. The R from prior discussion must be a ring, so it needs to have multiplication. However, multiplication does not come along with the fusion category structure. Therefore, we need to construct a new multiplicative structure.

There are two basic requirements for the multiplication: it must be associative, and it must have a multiplicative identity. Both of these conditions are usually phrased in terms of elements, but can also be written categorically. For example, associativity says that the two paths from A⊗A⊗A to A (which multiply on one side and then multiply the results) are the same. However, the details of this aren’t too important for our purposes, as most multiplications we construct will have this property.

By adding multiplication, we have turned our object into an algebra, and we can now consider modules over it.  There is one more key property: if the algebra is semisimple. We say that an element a annihilates a module if a•m=0 for all m in the module. Then, the algebra is semisimple if no element other than 0 annihilates all left modules.

Alternatively, an algebra is semisimple if (and only if) it is the direct product of simple algebras, i.e. algebras which have no proper ideals. This definition provides less clarity about why it’s an important property, but because we already know how the objects decompose under the direct sum, this view sheds more light on our specific case.

Conclusion

Bimodules were the last missing piece needed to attack the research question. Now, I should have all the necessary theory to consider any group, although some might be extremely computation-expensive while others might just not be possible. Stay tuned to see what will work!