Codifferential Calculi on Quantum Homogeneous Spaces

Abstract

We develop the theory of first- and higher-order codifferential calculi over coalgebras $C$ over fields $k$ with characteristic $\mathrm{char}(k)\neq 2$. For a given first-order codifferential calculus, we introduce its maximal prolongation by means of an explicit construction that associates to it a differential graded coalgebra, satisfying a universal property. For module coalgebras over a Hopf algebra $U$, we introduce the notion of an equivariant codifferential calculus. If $C$ is of the form $U\otimes_H k$ for a Hopf algebra $U$ and a right coideal subalgebra $H$ such that $U$ is faithfully flat as a left- and right $H$-module, we show that equivariant first-order codifferential calculi correspond to certain right coideals $T\subseteq \ker(\varepsilon\colon C\rightarrow k)$ called quantum tangent spaces. If $H$ is a sub bialgebra and the right $C$-coaction on $T$ is trivial, then the maximal prolongation is described in terms of a quadratic coalgebra. We further relate codifferential calculi to differential calculi and Cartan pairs over the dual algebra $C^\ast$, or more generally subalgebras thereof. We explicitly compute codifferential calculi on the coalgebra pre duals of the Podle\'s sphere and the quantized projective spaces. As an application, we give a new proof that the antiholomorphic Heckenberger--Kolb calculi on quantized projective spaces have classical dimension.