Chapter 11 - Modules over monoid algebras and bimonoids in species

Summary

In this chapter, we forge a connection between bimonoids in species and representation theory of monoid algebras. More precisely, the category of cocommutative bimonoids is equivalent to the category of left modules over the Tits algebra. Similarly, commutative bimonoids relate to right modules over the Tits algebra, bicommutative bimonoids to modules over the Birkhoff algebra, arbitrary bimonoids to modules over the Janus algebra, and more generally, q-bimonoids to modules over the q-Janus algebra. Moreover, these equivalences are compatible with duality and base change and also have signed analogues. Some illustrative examples are as follows. The (bicommutative) exponential bimonoid corresponds to the trivial module over the Birkhoff algebra. Both are self-dual in an appropriate sense. The (bicommutative) bimonoid of flats corresponds to the Birkhoff algebra viewed as a module over itself. Similarly, the (cocommutative) bimonoid of chambers corresponds to the left module of chambers over the Tits algebra, while the (cocommutative) bimonoid of faces corresponds to the Tits algebra viewed as a left module over itself. The bimonoid of bifaces corresponds to the Janus algebra viewed as a left module over itself.We approach these results through characteristic operations on bimonoids. They can also be derived by computing the Karoubi envelopes of the Birkhoff monoid, Tits monoid, Janus monoid, and using the interpretation of bimonoids as functor categories.