Convergent Twist Deformations
Chiara Esposito, Michael Heins, Stefan Waldmann
TL;DR
This work develops a functorial approach to promote formal Drinfeld deformations into convergent, strictly defined deformations on spaces of analytic vectors for finite-dimensional Lie algebras. By introducing malleable $\mathfrak{g}$-triples and their analytic/inductive variants, and by imposing an equicontinuity condition on the twist, the authors prove convergence of the universal deformation formula and holomorphic dependence on $\hbar$ for entire and analytic vectors, respectively. The theory is then instantiated with explicit Giaquinto–Zhang twists applied to Abelian, $ax+b$, and Heisenberg-extended Lie algebras, where equicontinuity is verified and analytic vector spaces are identified, yielding strict deformations in concrete representations. The results broaden the scope of strict deformation quantization by providing a robust, category-theoretic framework and concrete analytic tools (Cauchy estimates, projective tensor product control) to handle convergence and continuity questions in non-formal settings.
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.