Finite-dimensional faithful representation of a matrix algebra is completely reducible

Question

I am reading "Noncommutative Geometry and Particle Physics" by van Suijlekom. I have somequestions about the finite representations of matrix algebras.

It was said that any finite-dimensional faithful representation $\mathcal{H}$ of a matrix algebra $\mathcal{A}$ is completely reducible, i.e. a direct sum of irreducible representations. I know that $M_n(\mathbb{C})$ has the up to isomorphism unique irreducible representation $\mathbb{C}^n$. How can I prove the above statement ?

Also later it was mentioned that every finite-dimensional Hilbert-space representation can be written as $\mathcal{H}\cong\bigoplus_{i=1}^N \mathbb{C^{n_i}}\otimes V_i$, where the $V_i$ are vector spaces. I was expecting only the direct sum of the $\mathbb{C}^{n_i}$. Why do the tensor products with vector spaces occur?

Thanks for your help.

Answer 1

In fact the more general fact is true: for any algebra $A$ the algebras $A, Mat_{n}(A)$ has isomorphic category of representations. See wiki/Morita_equivalence

Here is the proof in our case. Let $W$ be $Mat_n(F)$ module. Denote by $V$ the vector space of columns of length $n$ as right $Mat_n(F)$ module and by $V'$ the vector space of rows of length $n$ as left $Mat_n(F)$ module.

We have the following isomorphism of $(Mat_n(F), Mat_n(F))$ bimodules:

$$V\otimes_F V'\simeq Mat_n(F).$$

So:

$$W\simeq Mat_n(F)\otimes_{Mat_n(F)} W\simeq (V\otimes_F V')\otimes_{Mat_n(F)} W\simeq V\otimes_F (V'\otimes_{Mat_n(F)} W)\simeq V^{\oplus\dim (V'\otimes_{Mat_n(F)}W)}.$$

Answer 2

In this book a matrix algebra was defined as an algebra of the form $\mathfrak{A}=\bigoplus_{i=1}^N M_{n_i}(\mathbb{C})$

It was said that any finite-dimensional faithful representation $\mathcal H$ of a matrix algebra $\mathcal A$ is completely reducible

More is true: what you have defined as a matrix algebra is a subclass of rings called semisimple rings (also see maybe the Wedderburn-Artin theorem), and for semisimple rings, all modules are direct sums of simple modules, and those modules are what you'd call completely reducible representations for $\mathcal A$.

So as you've defined things, it does not matter if $\mathcal H$ is finite dimensional or faithful. However, I thought I'd heard this theorem stated with $\mathcal A$ something more relaxed than semisimple algebras, and in that case it did depend on those hypotheses.