Graded $q$-differential matrix algebra

Presenter: Md. Raknuzzaman, University of Tartu, Estonia

Joint work with Viktor Abramov

We consider the algebra of square matrices of order $N$ as graded $q$-differential algebra, where $q$ is a primitive $N$th root of unity. We study the differential calculus of this algebra determined by a differential $d$, which satisfies the graded $d$-Leibniz rule and $d^{N}=0$. A graded $q$-differential algebra can be viewed as a generalization of graded differential algebra, and we consider the higher degree elements of graded $q$-differential matrix algebra as analogs of differential forms. We use two matrices $x,\xi$, which generate the algebra of square matrices of order $N$ in order to express these analogs of differential forms in terms of "coordinate" $x$ and its differentials $dx,d^{2}x,\cdots,d^{N-1}x$. We study an analog of connection and calculate its curvature.