Noncommutative differential calculus on a quadratic algebra

Abstract

We consider the algebra k[X2,XY,Y2] where characteristic of the field k is zero. We compute a differential calculus, introduced earlier by the authors, by associating an algebraic spectral triple with this algebra. This algebra can also be viewed as the coordinate ring of the singular variety UV–W2 and hence, is a quadratic algebra. We associate two canonical algebraic spectral triples with this algebra and its quadratic dual, and compute the associated Connes’ calculus. We observe that the resulting Connes’ calculi are also quadratic algebras, and they turn out to be quadratic dual to each other.