Everything is a representation

last edited around 2019-01-09

A pattern

$\DeclareMathOperator{End}{\operatorname{End}}\DeclareMathOperator{Aut}{\operatorname{Aut}}$ When investigating some structure, it is a common strategy to relate it to something with a similar structure that we understand very well. For instance, in the case of Lie groups, we understand matrix groups such as $GL(V)$ and subgroups of it quite well, so it is only natural to investigate possible maps from an abstract lie group $L$ into $GL(V)$. Such maps are called representations. We can crack down the idea even more:

Vague definition. A Representation of some “structure” $X$ Is a map $\rho\colon X\to W$, where $W$ belongs to a fixed class of objects such as the group of automorphisms $\operatorname{Aut}(V)$ or the monoid of endomorphisms $\operatorname{End}(V)$ of a potentially different structure $V$.

Note that there are multiple ways of considering Endomorphisms as a structure: In many contexts where we have an addition of morphisms, like when considering vector spaces and linear maps (more generally: In any abelian category), we can consider $\End(V)$ as either a Ring, monoid, or abelian group. But let's head over to some examples.

Examples

A group action is just a group homomorphism $G\to\Aut_{\underline{Set}} (X)$
If $R$ is a ring with unit, an $R$-module is just a ring homomorphism $R\to \End(V)$
If $A$ is an $F$-Algebra ($F$ a field), that is, A ring $A$ together with an injective ring homomorphism $F\to A$ lying in the center, then an $A$-Module is similarly an $F$-Algebra-Homomorphism $A\to \End(V)$, because the endomorphisms of an $F$-vector space also form an $F$-algebra
If $A$ is a Lie-Algebra, a representation is a lie-algebra homomorphism to $End(V)$ equipped with the commutator as a lie-bracket.

Definitions $\newcommand{\C}{\mathscr C}\newcommand{\D}{\mathscr D}\newcommand{\N}{\mathbb N}$

Let $\C$ be a concrete category, $A\in \operatorname{Obj}(\C)$ and $\D$ be a category whose endomorphisms of objects are objects in $\C$. A representation of $A$ is a morphism $A\to \End X$, where $X$ is an object of $Y$.

TODO define subrepresentation (invariance), irreducibility, complete reducibility.

TODO show that group representations are either completely reducible or irreducible (iff transitive).

TODO What about the regular semigroup representation (ie., regular action) of $(\N, +)$ on itself? → Subrepresentations (invariant spaces): all rays $(n,\infty)$. Note that a „complement“ (direct sum in sets is coproduct!) to a subrep $S\subseteq \N$ is just $\N\setminus S$, but for this to be invariant would require that every element $n\in \{0..n\}$ to stay inside that interval when being acted upon with an arbitrary natural number by additionfrom the left. Obviously bonkers. So the regular rep of $(\N, +)$ is not completely reducible. Q: What are irreps of $(\N, +)$?