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.
Formal deformation quantization has been introduced in [2 2, 3 3] and its guiding principle is to deform the commutative algebra of smooth functions C ∞ (M ) on a Poisson manifold M into a noncommutative algebra C ∞ (M ) ℏ of formal power series with respect to a formal parameter ℏ by means of a star product ⋆. This is an associative product, C ℏ -bilinear, whose zeroth-order term reproduces the usual pointwise multiplication, while the firstorder commutator recovers the Poisson bracket. Existence of such formal star products was first established for symplectic manifolds [13 13, 21 21], and later for arbitrary Poisson manifolds by Kontsevich [29 29], see also [48 48] for an introduction.
Despite the impressive impact of these results and their many subsequent applications, genuine physical interpretations demand going beyond purely formal power series, since the deformation parameter ℏ should be regarded as Planck’s constant and not as a formal parameter. This motivates the interest for strict versions of deformation quantization. A common route toward strictness replaces formal deformations with C * -algebraic deformations. This approach, initiated by Rieffel (see in particular [38 38,39 39]) and developed further by many others, see e.g. [5 5, 6 6, 8 8, 34 34], typically relies on oscillatory integral expressions for the star product, which allow estimates strong enough to construct C * -norms. While powerful, a key obstacle is that there is, in general, no universal way to produce star products via oscillatory integrals. This leads to an alternative strategy [49 49] based on functional analytic techniques: start from formal star products and study their convergence properties directly. In several families of examples this can work as follows. First, one identifies a “small” subalgebra of functions on which the star product converges for relatively straightforward reasons. Since general theorems are lacking, one proceeds case by case. Next, one equips this subalgebra with a suitable locally convex topology that makes the star product continuous. Again, this step is mostly example-driven. If successful, completing the subalgebra then yields a larger locally convex algebra-typically a Fréchet algebra-suitable for further analysis.
In finite dimensions this program may not look more decisive than earlier methods, as it still offers no general existence results. Nevertheless, it covers different kinds of examples, including analogues of algebras of unbounded operators. In addition, it is well suited to infinite-dimensional situations, where oscillatory integral techniques are generally unavailable. In this sense, the approach complements the established strict deformation quantizations by providing new and structurally different examples. A detailed overview can be found in [ 50 50] and more recent works include [1 1, 4 4, 18 18, 20 20, 27 27, 28 28, 32 32, 33 33, 37 37, 41 41-43 43].
This paper focuses on some special star products, obtained from the idea of Drinfeld of using symmetries to construct formal deformations. More explicitly, given an action ▷ by derivations of a Lie algebra g on an associative algebra (A, µ A ), we consider a Drinfeld twist [15 15, 16 16] F ∈ U(g) ⊗ U(g) ℏ (1.1) defined by the following conditions:
i.)
Here we denoted by ∆ and ε the standard coproduct and counit that make the universal enveloping algebra U(g) of g into a bialgebra. Formal twists allow us to obtain an associative formal deformation of A by means of a universal deformation formula (UDF)
for all a, b ∈ A ℏ .
(1.2)
Here ▷ is the action of g extended to the universal enveloping algebra U(g) and then to U(g) ⊗ U(g) acting on A ⊗ A in a diagonal fashion. Drinfeld twists and the associated universal deformation formula continue to attract considerable attention, with several recent contributions exploring both structural aspects and concrete constructions; see, for instance, [7 7, 11 11, 17 17, 19 19].
In this work, we address the question originally posed by Giaquinto and Zhang [23 23] regarding the existence of UDF analogs that yield strict deformation quantizations. Our main abstract results, presented in Section 3 3, are structured as follows. First, in Theorem 3.7 3.7, we prove that assuming the continuity of the deformation allows for the construction of continuous universal deformations on spaces of entire vectors-those analytic vectors with an infinite radius of convergence. We then extend our results by introducing the notion of malleable g-triples. In Theorem 3.7 3.7 and Theorem 3.19 3.19, we establish that for the analytic vectors of such triples, the resulting series not only converge but also define continuous multilinear mappings. Both constructions follow a rigorous methodology that ensures the preservation of the g-triple structure when passing to sub-spaces of well-behaved vectors, imposing an equicontinuity condition on the twist.
Finally, in Section 4 4, we put the developed theory into prac
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