I already saw examples, where we ''converted'' a group G into a category with one object and the morphisms are exactly the group elements and the composition of the morphisms equals the composition of the group. Analogously we can ''convert'' a K-algebra in such a so called K-linear category with one object, but the morphismset is now a K-vectorspace, with... You know what I mean. Is there a similarly construction with Lie-algebras?
My main goal is to motivate representation theory of these objects (already done for groups and K-algebras - they are just the functors into Vect_K).
Maybe one of you could help me? Maybe as simple as possible, since I'm a newbie regarding Category theory. Thank you for your time :)
Edit: My K-algebras are associative and have a unital element.
As mentioned in the comments, Lie algebras do not lend themselves to a direct translation into some sort of category with one object, because associativity is built into the definition of categories.
However, they do act in very interesting ways on categories, and this has led to some of the most interesting connections between various areas of representation theory. Here is the fundamental example: consider representations of the symmetric group $S_n$ over a field $F$ of positive characteristic $p$. Write $C_n$ for the category of such representations. There is a bi-adjoint pair of functors of restriction and induction $E:C_n \to C_{n-1}$ and $F:C_{n-1} \to E_n$ that relate the various categories as $n$ varies.
Evidently any element of the centralizer of the group algebra $F S_{n-1}$ in $F S_n$ acts by endomorphisms on the functors $E$ and $F$. It turns out that the $S_{n-1}$ orbit sum $$\phi_n=(1n)+(2n)+\cdots+(n-1,n)$$ together with the center of $FS_{n-1}$ generates this centralizer (this is sometimes referred to as the $n$th Jucys-Murphy-Young element).
The possible eigenvalues of $\phi_n$ are the elements $0,1,\dots,p-1$ of the prime field $F_p=\mathbf{Z}/p \mathbf{Z}$ of $F$. Decomposing $E$ and $F$ into eigen-functors gives bi-adjoint pairs $(E_0,F_0),\dots,(E_{p-1},F_{p-1})$. On the level of Grothendieck groups, the operators induced by these produce a highest weight representation of the affine Lie algebra $\widehat{\mathfrak{sl}}_p$. There is an additional piece of structure that is also important for this story: the endomorphism of the functor $E^2$ given by the simple transposition $(n,n+1)$; together with the action of the Jucys-Murphy-Young elements this produces an action of an (degenerate) affine Hecke algebra.
All these ideas are present in Grojnowski's beautiful paper Affine $\mathfrak{sl}_p$ controls the modular representation theory of the symmetric group and related Hecke algebras. These ideas were further developed and used to prove Broué's conjecture on derived equivalences between blocks of different symmetric groups in Chuang-Rouquier's landmark Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification. Since these foundational works, the area has exploded. The key words to look for are $2$-categories and higher representation theory.