Thank you for this question. Since you are a mathematician you will understand the basic idea here. I have said that I think the universe must have "complete symmetry", meaning that there should be one degree of symmetry corresponding to every physical degree of freedom. I believe this may be the only way to explain the holographic principle to resolve the information loss paradox for black holes. If symmetry was equivalent to group invariance then another way of saying this would be that the state space of the universe must be represented by a group, or a Lie algebra. In fact I think the concept of symmetry here has to be generalised, e.g. to supersymmetry and probably beyond, but for the sake of argument let's consider the hypothesis that the algebraic structure for the universe is a group.
If you then started with no other information about the laws of physics you might say that the universe is described by a ensemble of all possible groups, a strange kind of algebraic multiverse. However, I want to go beyond that view which I think is too simplistic because it does not take into account relationships between universes. If you make some observation within the universe you get some information about it. In group theory information would take the form of some algebraic equivalence relations between group elements. Given those relations what can you then say about the group? I am sure you know the answer. To construct the largest group G satisfying any specific set of algebraic relations you start with the free group F generated by all possible group elements and then construct the normal subgroup N of elements that must be equivalent to the identity given the provided relations the group you have is then G = F/N.
If you are then given further information you can construct a normal subgroup M of G and reduce further to a group H = G/M. Each time there is a group homomorphism mapping F onto G or G onto H taking the normal subgroup onto the identity.
When we think of an ensemble of possible universes we would imagine that gaining information would simply select some subset from the ensemble. If we believe the universe is algebraic it is more natural to start with a free algebraic structure and form homomorphic images of it instead of selecting sub-algebras or sets of sub-algebras. That is basically my idea.
I will give another example of how this can work. In a theory like string theory, the world is described on top of a geometric structure called spacetime. It has too many dimensions but dimensions can be reduced by compactification. Suppose that a more general underlying theory is found which is algebraic. Spacetime would then be a feature of the algebraic structure. When we compactify the spacetime we are identifying points and reducing the algebraic structure modulo those identity relationships. I think when string theory is seen as algebraic compactification will be seen as part of a more general process of factoring out some normal subgroup or whatever the equivalent of that is in the right type of algebraic structure. Being given information about what vacuum state the universe has selected will be equivalent to factoring out the corresponding algebraic structure.
Now I must confess to how this idea fails. In quantum mechanics gaining information is equivalent to making a measurement. If you measure position and then momentum you do not get the same result as when you measure momentum and then position. The operations do not commute. The simple algebraic picture of factorisations cannot for this noncommutivity. However, I don't think all is lost. It may be possible to take a level of abstraction and replace homomorphisms with morphisms between objects having a more non-commutative structure. I've no idea how something like that can be made to work, or even whether it corresponds to something that mathematicians have already considered.
