Let K $ to be an algebraically closed field. The theory of the finite-dimensional representation of polynomial algebra K $[x]$ is tamed and perfectly understood, which I will first summarize. Its indecomposable modules are determined by Jordan blocks

$$ J _ { lambda, m} = begin {pmatrix}

lambda & 1 & 0 & cdots & 0 \

0 & lambda & 1 & cdots & 0 \

vdots & vdots & ddots & ddots & vdots \

0 & 0 & cdots & lambda & 1 \

0 & 0 & cdots & 0 & lambda

end {pmatrix}. $$

Simple modules are precisely 1-dimensional modules. For each simple module $ M_1 $ defined so that the action of $ x $ sure $ M_1 $ is the scalar multiplication by $ lambda in K $, we have an infinite chain of submodules

$$ M_1 subset M_2 subset M_3 subset ldots, $$

where the action of $ x $ sure $ M_i $ is a linear transformation by $ J _ { lambda, i} $. In particular, such a chain of indecomposables induces a homogeneous tube in the quiver of Auslander-Reiten $ Gamma $ finite dimensional modules of K $[x]$and all the components of $ Gamma $ are homogeneous tubes with a simple module to the mouth. As a result, there is an irreducible monomorphism $ M_i rightarrow M_ {i + 1} $ and irreducible epimorphism $ M_ {i + 1} rightarrow M_i $ for everyone $ i $ in the chain above. Yes

$$ M-1 subset M-2 subset M -3 subset ldots, $$

is a different string (so $ M_1 not Cong M $ _1) and then $ mathrm {Hom} _ {K[x]} (M_i, M & # 39; _j) = $ 0 and $ mathrm {Hom} _ {K[x]} (M-i, M_j) = $ 0 for all $ i $ and $ j.

**My questions consist of two parts:**

(1) What do we know about finite representation representation theory (but not necessarily finite)? K $[x]$? Is the category of finely generated representations tamed (ie Is it possible to classify indecomposable representations and / or $ mathrm {Hom} $-the spaces? An explanation / summary accompanied by references would be greatly appreciated (if this is feasible, that is to say).

(2) Do we know anything about the theory of infinite dimension representation (but not necessarily generated)? K $[x]$? My instinct says that this category is probably wild (but a confirmation / rebuttal is welcome). Even in this case, are there examples or known families of infinitely generated representations that might be considered interesting or could have potential applications in other areas of mathematics (such as physics? mathematical)?