Visual exposition of Representation Theory (1/n)
Definition, equivalence, constructing and deconstructing
Introduction
If someone where to ask me what a representation of algebra is, then the simplest answer I can give is, if you can find a homomorphism between a given vector space with a product structure (mapping two vectors to another vector) a.k.a an Algebra to the square matrices of dimension of that algebra over another vector space, then this other vector space is said to be a representation of the algebra. Note that the vector space is the representation, not the matrices.
Before we go further, we gotta understand that the map is not the territory, it maybe that we have many different maps of the territory, so firstly we need to make sure that different maps relate well when they talk about the same piece of territory. This leads to the next section.
Equivalence of two representations
The cloud standing on top of the vector spaces are the square matrix sets. Note that on the right side of the equation underneath the blue arrow, I meant phi( rho (a) ). Unfortunately I didn’t have enough space so I just marked it with the color.
Now a homomorphism between vector spaces is a homomorphism between the vector spaces seen as representation of an algebra, when the homomorphism commutes with the action on vectors in the vector space by the algebra.
Now, we can talk about something quite interesting. Building new representation of an algebra out of old ones.
Construction and Deconstruction
For this section, tensor products are crucial. If you need a referesher on what a tensor product is, I’d recommend Math3ma’s article. It’s a pretty significant fact that if we have two representations of an algebra, then the tensor product of the representation as vector spaces is again a representation. We want to think of the reverse now, when can a representation be broken down into simpler ones? before we can go, I will make a few remarks, first on tensor products and then on dimensions.
Note on Tensor products
The tensor product is actually defined upto isomorphism!
The tensor product forms a monoid under the isomorphism classes of vector spaces over a field
The identity of this monoid is the field considered as a vector space over itself
Note on dimensions:
The dimension of a vector space is actually restrictive property. Infact, if we have the following facts as true (refer):
For a given field, all vector spaces of a given dimension are isomorphic
Isomorphic vector fields have same dimension (even for infinity dimen)
A vector space can only be factored into a tensor product of two other different vector spaces iff you can factor the dimension of that vector space
Now, a question is, when can a representation be decomposed into smaller representations? This would be the same asking, when can a vector space representing an Algebra be split into a tensor product of two other vector products? This is an important topic of representation theory. We give a special name to the vector spaces which can’t be split anymore into tensor product of other vector spaces: indecomposable spaces.
References and thanks:
Intro to representation theory MIT notes
Affinoid Voidoid (discord server)
Mathematics Stackexchange.
Wikipedia