ALGEBRA OF DIFFERENTIAL FORMS WITH
EXTERIOR DIFFERENTIAL $d^3=0$
IN DIMENSIONS ONE AND TWO

N. BAZUNOVA

Abstract:

In this paper, we construct the algebra of differential forms with exterior differential satisfying $d^3=0$ over an associative algebra with one and $n$ generators satisfying quadratic relations. Supposing $d^2\ne0$, we introduce the second order differentials $d^2x^i$. We also assume that the homomorphism defining a first order differential calculus is linear in variables, and that there are no relations between the terms $(dx^i)^2$ and $d^2x^j$. A graded $q$-differential algebra with $d^3=0$ is constructed by means of the Wess-Zumino method. The commutation relations between generators $x^i$, $dx^j$, $d^2x^k$ of the algebra of differential forms in pairs and themselves are found. In the case of the algebra with $n$ generators, the commutation relations between noncommutataive derivatives $\partial_i$ and generators $d^2x^j$ also are found, and the consistency conditions are described.