Alladi Ramakrishnan Hall

An Invitation to Product Systems

S. Sundar

IMSc

Product systems are `fibred objects' that come with a multiplication.
They are fibred over a semigroup/monoid/group. The fibres could be
sets/Hilbert spaces/ bimodules etc.....The most important axiom that
we impose is that the multiplication is a bijection at the level
of fibres.

The proper definition of product systems of sets is as follows.
Let S be a semigroup and X be another semigroup together with
a surjection $p:X \to S$. For $s \in S$, let $X_s$ be the fibre.
We say X is a product system if the map

$$X_s \times X_t
i (x,y) \to xy \in X_{st}$$

is a bijection.

Can we classify such objects? We show that the classification
problem is trivial if S is a group or if S is the semigroup
of natural numbers. I will give a countably many non-equivalent
examples of product systems over $[0,\infty)$.

Towards the end of the talk, I will try to give a bit of history
about how or from where product systems come from...

The talk will be accessible to first-year students, who are encouraged
to attend.