One might generalise the classical exterior algebra as follows to the quantum exterior algebra: $K<x_1,...x_n>/(x_i^2,x_i x_j + q_{i,j}x_j x_i)$ with nonzero field elements $q_{i,j}$ for $i<j$. Now this algebra still has charateristic properties of the exterior algebra: It is a local (quiver) algebra and it is selfinjective, meaning in this case it has a simple socle spanned by $x_1 ... x_n$. Such algebras also generalise some algebras appearing as counterexample of some conjectures, such as the Liu-Schulz algebra. Thus it might be a good idea to find an ultimative generalisation of such algebras. This generalisation should still have some of the old properties like:
-being a local quiver algebra and finite dimensional (this means the ideal $I$ should satisfy $J^r \subseteq I \subseteq J^2$ for some $r$ when $J=<x_1,...x_n>$) (1)
-being selfinjective (meaning it has a simple socle). (2)
Now algebras like $K<x_1,x_2>/(x_1x_2-x_2x_1 , x_1^2, x_2^3)$ still have those properties. So a general try would be as follows: A=$K <x_1,...x_n> / (x_i^{n_i},x_i x_j + q_{i,j}x_j x_i $ for $(i,j) \in S_2, x_i x_j x_k + q_{i,j,k} x_{o(i)} x_{o(j)} x_{o(k)}$ for $(i,j,k,o) \in S_3, ...)$ Here $S_r$ for $r \geq 2$ is a set of $r$-tupels together with a permutation of $r$, thus elements look like $(i_1,i_2,...,i_r,o)$ where $i_j$ are different indices and $o$ is a r-permutation. And q with some indices are nonzero field elements. Here $n_i \geq 2$ are some numbers. For example one obtains the quantum exterior algebra again when choosing $S_r$ to be empty for $r \geq 3$ and $S_r$ the set of all 2 tupels $(i,j)$ with $i<j$ and $n_i=2$ for all i. For which choices of such sets $S_r$ and field elements one obtains an algebra having still properties (1) and (2)? And is there a better generalisation (there could of couse also be relations like $x_1 x_2=x_3 x_4 x_5$, which can not happen in my generalisation)?
