Question: Consider quadratic algebra with four generators $a,b,c,d,d$ and three relations $ac=ca,bd = db, ad-da = cb - bc$ . Is it a Koszul algebra ? (i.e. Koszul complex is resolution of ground field $k$). (Checking algebra is Koszul, is something algorithmic or more kind of creative task ? )
Context: The relations above first appeared in Yuri Manin's papers on "Noncommuative geometry and quantum groups" (around 1986-7). And define "Manin matrices", in the simplest 2x2 case it looks like that, consider 2x2 matrix:$$\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$$The relations above ensure that despite non-commutativity - such matrices behave like commutative ones - one can define the determinant: $ad-cb$ , it will be multiplicative, Cramer's inversion formula works, Cayley-Hamilton, MacMahon, Newton, etc ... "all" linear algebra statements - holds true in such a relaxed commutativity setup.
Generalizations, applications: There are many applications to representation theory of loop algebras - explicit description of central elements, local Langlands GL-oper, Capelli's identities, etc. See e.g. A. Molev's lectures: https://www.maths.usyd.edu.au/u/alexm/talks/seoul-manin.pdf
The question above have various generalizations - we can consider not 2x2 Manin matrices, but $n\times m$ with generators $M_{ij}$, q-analogues, super-analogues, Manin matrices of types B, C, D, generalization to other qudratic algebras, modifications for algebras with involution and so on. And ask the same question - whether obtained algebras are Koszul. At least when underlying algebra is Koszul.
In particular one the questions - mentioned in A.Molev's lectures (see page 261/78) - for the super-Manin matrices we do not know the Hilbert series and basis generating the algebra. Koszulity might help on that.