Quantum exterior algebra

Question

In Generalisation of the quantum exterior algebra the quantum exterior algebra is discussed: $$ K\langle x_1,\dotsc x_n\rangle/(x_i^2,x_i x_j + q_{i,j}x_j x_i), $$ with nonzero field elements $q_{i,j}$ for $i<j$.

Where was this algebra first introduced and is there a standard text about such things? Is the standard name really the quantum exterior algebra? I have searched for it but not had any luck finding a good reference.

Moreover, where is the dimension of such algebras calculated? It must surely be the same as the normal exterior algebra . . .

Answer 1

Quantum affine space has been studied since the nineties. The coordinate ring of quantum affine space is a quantum polynomial algebra, whose definition I think you can imagine, and the Koszul dual of a quantum polynomial algebra is a quantum exterior algebra. There are conditions on the $q_{i,j}$ in order for the algebras constructed this way to be associative (or, if you like, to have the dimension you imagine them to have), and I'm sure you can work these out yourself. Quantum complete intersections started appearing soon afterwards, and have been studied by many people including Lucho Avramov. They started becoming relevant to the study of block theory of finite groups due to an example of Jon Alperin, that I further studied with Ed Green, and later with Radha Kessar. Hochschild cohomology and support varieties have also been studied, see the work of Petter Bergh, of Karin Erdmann, and of Steffen Oppermann.