Convergent Twist Deformations

TL;DR

This paper develops a functorial framework for the convergence of Drinfeld's Universal Deformation Formula on analytic vector spaces, ensuring both convergence and continuity of deformed structures, with applications to explicit twists.

Contribution

It introduces a new functorial approach linking convergence of UDF with equicontinuity, enabling explicit analysis of analytic vectors in concrete representations.

Findings

Established convergence and continuity of deformed bilinear mappings.

Applied theory to explicit Drinfeld twists by Giaquinto and Zhang.

Confirmed the possibility of strict formal twists in specific cases.

Abstract

This paper establishes a functorial framework for convergence of Drinfeld's Universal Deformation Formula (UDF) on spaces of analytic vectors. This is accomplished by matching the order of the latter with an equicontinuity condition on the Drinfeld twist underlying the deformation. Throughout, we work with representations of finite-dimensional Lie algebras by continuous linear mappings on locally convex spaces. This allows us to establish not only convergence of the formal power series, but the continuity of the deformed bilinear mappings as well as the entire holomorphic dependence on the deformation parameter $\hbar$. Finally, we demonstrate the effectiveness of our theory by applying it to the explicit Drinfeld twists constructed by Giaquinto and Zhang, where we establish both the equicontinuity condition and determine the corresponding spaces of analytic vectors for concrete representations. Thereby we answer a question posed by Giaquinto and Zhang whether a strict version of their formal twists is possible in the positive.