Consider a matrix $M$ which elements $M_{ij}$ are generators of some algebra $K$, impose new relations: $M^2=0$ and get a new algebra $K_{2}$.
Question 1: Is it true that $K_2$ is Koszul algebra when $K$ is commutative polynomial algebra ? $K$ - free associative algebra ? $K$ is standard Manin matrix algebra ? $K$ is Grassman algebra ? More generally when $K$ is quadratic Koszul algebra with $n^2$ generators ?
Question 2: Assume $A$ is Koszul algebra, take $M$ to be its Manin's matrix of non-commutative symmetries of $A$, again define $K_{2}$ by $M^2=0$ - is it Koszul algebra ? (See description of Manin's construction e.g. MO.)
Question 3: What can be Hilbert–Poincaré series of these algebras in terms of original algebras ?
Variations:
Another source of quadratic relations: $MN=0$ (matrix product), we can ask similar questions when $M,N$ are filled by the generators of similar algebras.
One can combine requirement $M^2=0$ with other natural quadratic relations for matrices: e.g. split $M$ into blocks and require that blocks are commuting/Manin/anticommuting/Grassman - we get additional quadratic relations and so one may ask whether $M^2=0,$ + these additional relations produce the Koszul algebra.
One may ask similar question for symmetric/antisymmetric variants of the of the matrices.
PS
It seems in general it is not always easy to check is the algebra Koszul or not (see comment to MO), so extending set of examples might be useful.
