Nonformal deformations of the algebras of holomorphic functions on the polydisk and on the ball in $\mathbb C^n$
Alexei Yu. Pirkovskii
TL;DR
The work introduces holomorphic, nonformal deformations of algebras of holomorphic functions on the polydisk and ball by constructing Fréchet algebras \\mathscr O_\\mathrm{def}(D) over \\mathscr O(\\mathbb C^\times). Each fiber at q reproduces the quantum holomorphic algebras \\mathscr O_q(D), and the family forms a continuous Fréchet algebra bundle that implements a strict deformation quantization in the sense of Rieffel. A noncommutative power-series perspective shows that these deformations are not topologically free, while a detailed comparison with free polydisk and free ball algebras yields an isomorphism between the two holomorphic deformation models. The paper also establishes a bridge to formal deformation theory via extension of scalars, by constructing a formal deformation with a star-product and showing it is isomorphic to the holomorphic deformation after base change. Overall, the results provide a robust, analytic framework for holomorphic nonformal deformation quantization of classical domains and connect nonformal holomorphic deformations with their formal counterparts.
Abstract
We construct Fréchet $\mathcal O(\mathbb C^\times)$-algebras $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and $\mathcal O_{\mathrm{def}}(\mathbb B^n)$ which may be interpreted as nonformal (or, more exactly, holomorphic) deformations of the algebras $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$ of holomorphic functions on the polydisk $\mathbb D^n\subset\mathbb C^n$ and on the ball $\mathbb B^n\subset\mathbb C^n$, respectively. The fibers of our algebras over $q\in\mathbb C^\times$ are isomorphic to the previously introduced ``quantum polydisk'' and ``quantum ball'' algebras, $\mathcal O_q(\mathbb D^n)$ and $\mathcal O_q(\mathbb B^n)$. We show that the algebras $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and $\mathcal O_{\mathrm{def}}(\mathbb B^n)$ yield continuous Fréchet algebra bundles over $\mathbb C^\times$ which are strict deformation quantizations (in Rieffel's sense) of $\mathbb D^n$ and $\mathbb B^n$. We also give a noncommutative power series interpretation of $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ and apply it to showing that $\mathcal O_{\mathrm{def}}(\mathbb D^n)$ is not topologically projective (and a fortiori is not topologically free) over $\mathcal O(\mathbb C^\times)$. Finally, we consider respective formal deformations of $\mathcal O(\mathbb D^n)$ and $\mathcal O(\mathbb B^n)$, and we show that they can be obtained from the holomorphic deformations by extension of scalars.