The number of subdirect powers of an finite algebraic structure is a question that has appeared at various points in recent history. The situation is known in full for groups, and to some extent in semigroups. We answer the question for unary algebras, and look at how the situation is more complicated for infinite algebras. We then discuss what these results can tell us about the general question, and how it ties into other topics such as boolean separation.
The number of countable subdirect powers of finite unary algebras
Feb. 15, 2022 3pm (MATH 350)
Topology
Howie Jordan (CU Boulder)
X
Fusion categories are complex linear categories with tensor products, duals, and satisfying simplicity conditions. These categories have a finite class of simple objects out of which all other objects are built, and these along with their duals form the representations of the pure states of particles and anti-particles in TQFTs. By adding a braiding onto the monoidal structure and twist maps compatible with the braiding we arrive at a ribbon category. A final nondegeneracy condition gives us the notion of a Modular Tensor Category (MTC). In particular, MTCs have a nondegenerate inner product structure allowing for probabilities to be assigned to states inline with traditional quantum theory.
In this talk, we unpack the definitions of fusion and ribbon categories and build up to the notion of Modular Tensor Category as a nondegenerate ribbon fusion category. We will consider these axiomatically as well as skeletal versions in terms of 6j symbols and diagrammatic calculi.
The number of countable subdirect powers of finite unary algebras