The restricted quantum double of the Yangian

This article is a continuation of [WDYhg], which studied the Yangian double $\mathrm{D}Y_{\hbar}\mathfrak{g}$ associated to an arbitrary symmetrizable Kac–Moody algebra $\mathfrak{g}$ through the lens of a $\mathbb{Z}$-graded algebra homomorphism

Here $\mathrm{L}\widehat{Y_{\hbar}\mathfrak{g}}_{z}$ is a naturally defined $\mathbb{Z}$-graded $\mathbb{C}[\![\hbar]\!]$-algebra, described explicitly in Lemma 4.5, and $Y_{\hbar}\mathfrak{g}$ is the Yangian of $\mathfrak{g}$, defined over $\mathbb{C}[\![\hbar]\!]$. This homomorphism, called the formal shift operator, naturally extends the so-called shift homomorphism $\tau_{z}$ on the Yangian, and has a number of remarkable properties. For instance, it induces a family of isomorphisms between completions of $\mathrm{D}Y_{\hbar}\mathfrak{g}$ and $Y_{\hbar}\mathfrak{g}$, realizes $Y_{\hbar}\mathfrak{g}$ as a degeneration of $\mathrm{D}Y_{\hbar}\mathfrak{g}$, and is injective provided $\mathfrak{g}$ is of finite type or of simply laced affine type. In addition, it was applied in [GWPoles] to characterize the category of finite-dimensional representations of $\mathrm{D}Y_{\hbar}\mathfrak{g}$, for $\hbar\in\mathbb{C}^{\times}$ and $\mathfrak{g}$ of finite type, as the tensor-closed Serre subcategory of that of the Yangian consisting of those representations which have no poles at zero.

In this article, we narrow our focus to the case where $\mathfrak{g}$ is a finite-dimensional simple Lie algebra, and apply these results in conjunction with those of the recent paper [GTLW19] to prove one of the main conjectures from the work [KT96] of Khoroshkin and Tolstoy. Namely, we establish that $\mathrm{D}Y_{\hbar}\mathfrak{g}$, which is defined by doubling the generators of $Y_{\hbar}\mathfrak{g}$ (see Definition 4.1), is isomorphic to the restricted quantum double of the Yangian $Y_{\hbar}\mathfrak{g}$, where the prefix “restricted” indicates that all duality operations are taken so as to respect the underlying gradings. As a consequence of this result and its proof, we find that $\Phi_{z}$ identifies the universal $R$-matrix of $\mathrm{D}Y_{\hbar}\mathfrak{g}$, which arises from the quantum double construction, with Drinfeld’s universal $R$-matrix $\mathcal{R}(z)\in(Y_{\hbar}\mathfrak{g}\otimes Y_{\hbar}\mathfrak{g})[\![z^{-1}]\!]$. Our argument makes essential use of the constructive proof of the existence of $\mathcal{R}(z)$ given in [GTLW19], which is independent from Drinfeld’s cohomological construction of $\mathcal{R}(z)$ from the foundational paper [Dr].

Let us now sketch our main results in detail. The two results alluded to above form Parts (1) and (2) of the following theorem.

This is a combination of the three main results of this article: Theorems 7.5, 8.5 and 9.8. Part (1) is the statement of our second main result — Theorem 8.5 — and is precisely the variant of the conjecture from [KT96]*§2 which we establish in the present paper. Our approach to proving it is, in a certain sense, dual to the strategy outlined in [KT96] which was brought to fruition for $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$. In more detail, our argument hinges on the fact, proven in Proposition 7.1, that the universal $R$-matrix $\mathcal{R}(z)$ of the Yangian gives rise to a $\mathbb{C}[\![\hbar]\!]$-algebra homomorphism

which is compatible with the Hopf algebra structure on $Y_{\hbar}\mathfrak{g}$ and the co-opposite Hopf structure on the dual $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}$ of the Yangian $Y_{\hbar}\mathfrak{g}$ taken in the category of $\mathbb{Z}$-graded quantized enveloping algebras. That is, $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}$ is the restricted (or graded) dual of the Drinfeld–Gavarini [DrQG, Gav02] subalgebra $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}\subset Y_{\hbar}\mathfrak{g}$, defined in Section 5, and provides a homogeneous quantization of the restricted dual $t^{-1}\mathfrak{g}[t^{-1}]$ to the $\mathbb{N}$-graded Lie bialgebra $\mathfrak{g}[t]$, as we prove in detail in Section 6; see Theorem 6.7.

Using the construction of $\mathcal{R}(z)$ given in [GTLW19] and properties of $\Phi_{z}$ established in [WDYhg], we deduce that the image of $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}$ under $\Phi_{z}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ is contained in the image of $\Phi_{z}$. We may thus compose $\Phi_{z}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}$ with $\Phi_{z}^{-1}$ to obtain a $\mathbb{C}[\![\hbar]\!]$-algebra homomorphism

where $Y_{\hbar}\mathfrak{g}^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}:=(\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}){\vphantom{)}}^{\scriptscriptstyle\mathrm{cop}}$. In our first main result — Theorem 7.5 — we show that there is a unique $\mathbb{Z}$-graded Hopf algebra structure on $\mathrm{D}Y_{\hbar}\mathfrak{g}$ for which both $\check{\imath}$ and the natural inclusion $\imath:Y_{\hbar}\mathfrak{g}\to\mathrm{D}Y_{\hbar}\mathfrak{g}$ are injective homomorphisms of graded Hopf algebras. This is exactly the Hopf structure alluded to in the statement of Theorem I, and is such that $\mathrm{D}Y_{\hbar}\mathfrak{g}$ provides a homogeneous quantization of the restricted Drinfeld double $\mathfrak{g}[t^{\pm 1}]$ of $\mathfrak{g}[t]$. Using Theorem 7.5, it is then not difficult to establish Part (1) above (i.e., Theorem 8.5) using the double cross product realization of the restricted quantum double (see Section 8.2).

Our third and final main result, Theorem 9.8, is a strengthening of Part (2) above. Indeed, it outputs the Gauss decomposition for ${\boldsymbol{\EuScript{R}}}$ while identifying each factor appearing in this decomposition with the factors $\mathcal{R}^{\pm}(z)$ and $\mathcal{R}^{0}(z)$ of $\mathcal{R}(z)$, which were studied in detail in [GTLW19, GTL3].

Part (2) of Theorem I implies that $\mathcal{R}(z)$ can be recovered from the canonical element defined by the pairing between $Y_{\hbar}\mathfrak{g}$ and its restricted dual $Y_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}\subset\mathrm{D}Y_{\hbar}\mathfrak{g}$ by applying the injection $\Phi_{\shortminus{z}}$ to its second tensor factor:

Here we refer the reader to Theorem 9.8 for further details, which takes into account the topological subtleties surrounding this statement. Obtaining this interpretation of $\mathcal{R}(z)$ is in fact our original motivation for addressing the conjecture of [KT96]*§2, and brings the theory surrounding the universal $R$-matrix of the Yangian to a more equal footing with that of the (extended, untwisted) quantum affine algebra $U_{q}(\widehat{\mathfrak{g}})$ and the quantum loop algebra $U_{q}(L\mathfrak{g})$. The differences are, however, still quite pronounced. Indeed, $U_{q}(\widehat{\mathfrak{g}})$ is itself nearly the quantum double of its (quantum Kac–Moody) Borel subalgebra $U_{q}(\mathfrak{b}^{+})$, and its universal $R$-matrix $\mathscr{R}$ lies in a completion of $U_{q}(\mathfrak{b}^{+})\otimes U_{q}(\mathfrak{b}^{-})$. One then recovers the universal $R$-matrix $\mathscr{R}_{L\mathfrak{g}}$ of $U_{q}(L\mathfrak{g})$ as a truncation of $\mathscr{R}$, and its $z$-dependent analogue is

where $D_{z}$ is given by the $\mathbb{Z}$-grading on $U_{q}(L\mathfrak{g})$; see [EFK-Book]*§9.4 or [FR92]*§4, for instance. Crucially, $\mathscr{R}$ can be constructed by computing dual bases with respect to the pairing between $U_{q}(\mathfrak{b}^{+})$ and $U_{q}(\mathfrak{b}^{-})$, and was done explicitly by Damiani in [Damiani98]. In contrast, the Yangian $Y_{\hbar}\mathfrak{g}$ is not of Kac–Moody type and does not arise as a Hopf algebra from the quantum double construction applied to any analogue of $U_{q}(\mathfrak{b}^{+})$¹¹1We refer the reader to [YaGu3]*§4 for a related construction of $Y_{\hbar}\mathfrak{g}$ with respect to its deformed Drinfeld coproduct, which does not endow $Y_{\hbar}\mathfrak{g}$ with the structure of a Hopf algebra.. In addition, $\mathcal{R}(z)$ and $\mathscr{R}_{L\mathfrak{g}}(z)$ exhibit significantly different analytic behaviour when evaluated on finite-dimensional representations [GTLW19]. Nonetheless, the results of this article further cement that there are very strong parallels to be drawn between the two pictures. Indeed, one obtains the $Y_{\hbar}\mathfrak{g}$-analogue of the above story by replacing $U_{q}(\mathfrak{b}^{+})$ by $Y_{\hbar}\mathfrak{g}$, $U_{q}(L\mathfrak{g})$ by the Yangian double $\mathrm{D}Y_{\hbar}\mathfrak{g}$, and $D_{z}$ by the formal shift operator $\Phi_{z}$.

It should be noted that it appears that this realization of $\mathcal{R}(z)$ has been anticipated for some time in the mathematical physics community; see for instance [Stukopin07]*§5, which considers its super-analogue. The direction taken therein is, however, based on both the conjecture from [KT96]*§2 at the heart of the present article, and on the infinite product formulas for the factors ${\boldsymbol{\EuScript{R}}}^{\pm}$ of ${\boldsymbol{\EuScript{R}}}$ given [KT96]*§5, which remain conjectural. Some more discussion on this point is given in Section 9.4.

Let us now give a few brief remarks. Firstly, it is essential that the Yangian double $\mathrm{D}Y_{\hbar}\mathfrak{g}$ is defined as a topological algebra over $\mathbb{C}[\![\hbar]\!]$ for the above results to hold true. To expand on this, $\mathrm{D}Y_{\hbar}\mathfrak{g}$ can be realized as the $\hbar$-adic completion of a $\mathbb{Z}$-graded $\mathbb{C}[\hbar]$-algebra $\mathds{D}Y_{\hbar}\mathfrak{g}^{\jmath}$ defined by generators and relations; see Remark 4.2. One can further specialize $\hbar$ to any nonzero complex number $\zeta$ to obtain a $\mathbb{C}$-algebra $\mathds{D}Y_{\zeta}\mathfrak{g}=\mathds{D}Y_{\hbar}\mathfrak{g}^{\jmath}/(\hbar-\zeta)\mathds{D}Y_{\hbar}\mathfrak{g}^{\jmath}$, whose category of finite-dimensional representations was characterized in terms of that of the corresponding Yangian $Y_{\zeta}(\mathfrak{g})$ in [GWPoles]. Though this category has a tensor structure which corresponds to the Hopf structure on $\mathrm{D}Y_{\hbar}\mathfrak{g}$, it is important to note that $\mathds{D}Y_{\zeta}\mathfrak{g}$ is not a Hopf algebra over $\mathbb{C}$, and in particular it does not coincide with the (restricted) quantum double of $Y_{\zeta}(\mathfrak{g})$ defined in any reasonable sense.

That being said, $\mathds{D}Y_{\zeta}\mathfrak{g}$ admits a natural $\mathbb{Z}$-filtration corresponding to the $\mathbb{Z}$-grading on $\mathrm{D}Y_{\hbar}\mathfrak{g}$, and the expectation is that the formal completion of $\mathds{D}Y_{\zeta}\mathfrak{g}$ with respect to this filtration coincides with the (restricted) quantum double of $Y_{\zeta}(\mathfrak{g})$ taken in the appropriate category of $\mathbb{Z}$-filtered, complete topological Hopf algebras. This is in fact the version of Part (2) of Theorem I conjectured in [KT96], and is consistent with the situation that transpires in type A for the $R$-matrix realization of the Yangian, which has been developed in great detail in the recent paper [Naz20]. For our purposes, it is more natural to work over $\mathbb{C}[\![\hbar]\!]$ within the framework of quantized enveloping algebras first developed by Drinfeld [DrQG], where we may study $\mathrm{D}Y_{\hbar}\mathfrak{g}$ from the point of view of quantization of Lie bialgebras. At the same time, many of our results are “global” (in the sense of [Gav07]) and admit an interpretation over both $\mathbb{C}[\hbar]$ and $\mathbb{C}$, including the realization of $\mathcal{R}(z)$ provided by Theorem 9.8; see Appendix A.

Theorem LABEL:T:R-can and Proposition LABEL:P:Rchi-can imply that the universal $R$-matrix $\mathcal{R}(z)$ and its Gauss components can be computed, at least in principal, combinatorially as the canonical tensors associated to the pairing $\langle\langle\,,\rangle\rangle$ and its appropriate restrictions. Since $\mathcal{R}^{+}(z)=\mathcal{R}_{21}^{-}(-z)^{-1}$ and $\mathcal{R}^{0}(z)$ is known explicitly, obtaining a closed expression for $\mathcal{R}(z)$ in this form reduces to computing the canonical tensor $\mathds{R}^{-}\sim\mathcal{R}^{-}(z)$ with respect to the pairing $\dot{\mathsf{Y}}_{\hbar}^{-}(\mathfrak{g})\times\imath^{\smash{\scalebox{0.7}[1.4]{\rotatebox{90.0}{{\T1\guilsinglleft}}}}}(\dot{\mathsf{Y}}_{\hbar}^{-}\mathfrak{g}{\vphantom{)}}^{\star})\to\mathbb{C}[\![\hbar]\!]$.

This problem was considered in the foundational work [KT96] of Khoroshkin and Tolstoy, where an infinite product formula for $\mathds{R}^{-}$ (equivalently, for $\mathcal{R}^{-}(z)$) was obtained for $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2}$; see Theorem 5.1 therein. A distinct formula for $\mathcal{R}^{-}_{\mathfrak{s}\mathfrak{l}_{2}}(z)$ was obtained in Theorem 5.5 of [GTLW19] by solving the recurrence relation (3.23).

The paper is written so as to provide a complete picture, accessible to non-experts, where possible. For this reason, we take great care to lay the foundation needed to state and prove the results outlined in Section 1.2. The first three sections — Sections 2, 3 and 4 – are intended to serve a preliminary role: Section 2 surveys the theory of $\mathbb{Z}$-graded topological $\mathbb{C}[\![\hbar]\!]$-modules, algebras and Hopf algebras, including homogeneous quantizations of graded Lie bialgebras. This theory plays a prominent role throughout the article. In Section 3, we review the definition and main properties of the Yangian $Y_{\hbar}\mathfrak{g}$, defined both over $\mathbb{C}[\hbar]$ and $\mathbb{C}[\![\hbar]\!]$. Notably, this includes a review of the construction of the universal $R$-matrix $\mathcal{R}(z)$ carried out in [GTLW19]. Section 4 is focused on the Yangian double $\mathrm{D}Y_{\hbar}\mathfrak{g}$ and, in particular, on reviewing the main results of [WDYhg]; see Theorems 4.6 and 4.8.

In Sections 5 and 6, we study the Drinfeld–Gavarini subalgebra $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}$ of the Yangian, its $\mathbb{C}[\hbar]$-form, and its restricted dual $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}$ in detail. This includes a detailed proof that $\dot{\mathsf{Y}}_{\hbar}\mathfrak{g}{\vphantom{)}}^{\star}$ provides a homogeneous quantization of $t^{-1}\mathfrak{g}[t^{-1}]$, equipped with the Yangian Lie bialgebra structure; see Definition 6.1 and Theorem 6.7.

The last three sections of the article contain its three main results: Theorems 7.5, 8.5 and 9.8. We refer the reader to Section 1.2 above, where these are outlined in detail. Finally, in Appendix A we explain how to translate the construction of the universal $R$-matrix given in [GTLW19] for $\hbar\in\mathbb{C}^{\times}$ to the setting of the present paper, in which $\hbar$ is a formal variable; see Proposition A.1, which appears in Section 3.6 as Theorem 3.9.

I would like to thank Alex Weekes, Andrea Appel, Sachin Gautam, and Valerio Toledano Laredo for the many insightful discussions and helpful comments they have been the source of over the last few years. These have played a significant role in shaping this article.

Recall that a $\mathbb{C}[\![\hbar]\!]$-module $\mathsf{M}$ is separated if the intersection of the family of submodules $\hbar^{n}\mathsf{M}$ is trivial, and it is complete if the natural $\mathbb{C}[\![\hbar]\!]$-linear map

is surjective, where the inverse limit is taken over the set $\mathbb{N}$ of non-negative integers. In particular, $\mathsf{M}$ is both separated and complete if and only if the above map is an isomorphism. If $\mathsf{M}$ is separated, complete and torsion free as a $\mathbb{C}[\![\hbar]\!]$-module, then it is said to be topologically free. This is equivalent to the existence of a $\mathbb{C}[\![\hbar]\!]$-module isomorphism $\mathsf{M}\cong\mathsf{V}[\![\hbar]\!]$ for a complex vector space $\mathsf{V}$. Such an isomorphism is specified by a choice of complement $\mathsf{V}\subset\mathsf{M}$ to $\hbar\mathsf{M}$:

More generally, if $\mathsf{M}$ is any $\mathbb{C}[\![\hbar]\!]$-module, then the space $\mathsf{V}=\mathsf{M}/\hbar\mathsf{M}$ is called the semiclassical limit of $\mathsf{M}$. Similarly, the semiclassical limit of a $\mathbb{C}[\![\hbar]\!]$-linear map $\uptau:\mathsf{M}\to\mathsf{N}$ is the $\mathbb{C}$-linear map $\bar{\uptau}:\mathsf{M}/\hbar\mathsf{M}\to\mathsf{N}/\hbar\mathsf{N}$ uniquely determined by the commutativity of the diagram

As the following elementary result illustrates, the semiclassical limit of a $\mathbb{C}[\![\hbar]\!]$-module homomorphism encodes important information about the original map.

The topological tensor product $\mathsf{M}\,{\widehat{\otimes}}\,\mathsf{N}$ of two $\mathbb{C}[\![\hbar]\!]$-modules $\mathsf{M}$ and $\mathsf{N}$ is the $\hbar$-adic completion of the algebraic tensor product $\mathsf{M}\otimes_{\mathbb{C}[\![\hbar]\!]}\mathsf{N}$:

If $\mathsf{M}$ and $\mathsf{N}$ are topologically free with $\mathsf{M}\cong\mathsf{V}[\![\hbar]\!]$ and $\mathsf{N}\cong\mathsf{W}[\![\hbar]\!]$, then $\mathsf{M}\,{\widehat{\otimes}}\,\mathsf{N}$ is topologically free and isomorphic to $(\mathsf{V}\otimes_{\mathbb{C}}\mathsf{W})[\![\hbar]\!]$.

In this article, we shall say that $\mathsf{M}$ is a topological module over $\mathbb{C}[\![\hbar]\!]$ if $\mathsf{M}$ is a $\mathbb{C}[\![\hbar]\!]$-module which is both separated and complete. For any such module, we have

Similarly, by a topological algebra $\mathsf{A}$ over $\mathbb{C}[\![\hbar]\!]$ we shall always mean that $\mathsf{A}$ is a $\mathbb{C}[\![\hbar]\!]$-algebra which is both separated and complete as a module over $\mathbb{C}[\![\hbar]\!]$. In particular, the multiplication $m$ can be viewed as a $\mathbb{C}[\![\hbar]\!]$-linear map

A topological Hopf algebra $\mathsf{H}$ over $\mathbb{C}[\![\hbar]\!]$ is a topological $\mathbb{C}[\![\hbar]\!]$-algebra equipped with a counit $\varepsilon:\mathsf{H}\to\mathbb{C}[\![\hbar]\!]$, a coproduct $\Delta:\mathsf{H}\to\mathsf{H}\,\widehat{\otimes}\,\mathsf{H}$ and an antipode $S:\mathsf{H}\to\mathsf{H}$ which collectively satisfy the axioms of a Hopf algebra with all tensor products given by the topological tensor product $\widehat{\otimes}$. By modifying these definitions in the expected way, one obtains the notion of a topological coalgebra and bialgebra over $\mathbb{C}[\![\hbar]\!]$.

If $\mathsf{M}$ and $\mathsf{N}$ are topological $\mathbb{C}[\![\hbar]\!]$-modules and $\mathsf{N}\cong\mathsf{W}[\![\hbar]\!]$ is topologically free, then the space of $\mathbb{C}[\![\hbar]\!]$-module homomorphisms $\mathrm{Hom}_{\mathbb{C}[\![\hbar]\!]}(\mathsf{M},\mathsf{N})$ is separated, complete and torsion free. If in addition $\mathsf{M}$ is topologically free with $\mathsf{M}\cong\mathsf{V}[\![\hbar]\!]$, then one has

In particular, the $\mathbb{C}[\![\hbar]\!]$-linear dual $\mathsf{M}^{\ast}:=\mathrm{Hom}_{\mathbb{C}[\![\hbar]\!]}(\mathsf{M},\mathbb{C}[\![\hbar]\!])$ of a topologically free $\mathbb{C}[\![\hbar]\!]$-module $\mathsf{M}\cong\mathsf{V}[\![\hbar]\!]$ satisfies $\mathsf{M}^{\ast}\cong\mathsf{V}^{\ast}[\![\hbar]\!]$.

Let us now turn towards the $\mathbb{Z}$-graded analogues of the above definitions. Henceforth, we view $\mathbb{C}[\hbar]=\bigoplus_{k\in\mathbb{N}}\mathbb{C}\hbar^{k}$ as an $\mathbb{N}$-graded ring. For brevity, we shall denote its $\mathbb{N}$-graded quotient $\mathbb{C}[\hbar]/\hbar^{n}\mathbb{C}[\hbar]$ by $\mathsf{K}_{n}$, for each $n\in\mathbb{N}$.

A $\mathbb{C}[\![\hbar]\!]$-module homomorphism $\mathsf{M}\to\mathsf{N}$ between $\mathbb{Z}$-graded topological $\mathbb{C}[\![\hbar]\!]$-modules $\mathsf{M}$ and $\mathsf{N}$ is said to be $\mathbb{Z}$-graded if the induced morphisms

are all $\mathbb{Z}$-graded. More generally, it is $\mathbb{Z}$-graded of degree $a\in\mathbb{Z}$ if each of these induced morphisms is homogeneous of degree $a$.

The category of $\mathbb{Z}$-graded topological modules is closed under the tensor product $\widehat{\otimes}$. Indeed, this follows from the elementary observation that, given two $\mathbb{C}[\![\hbar]\!]$-modules $\mathsf{M}$ and $\mathsf{N}$, one has

which can be equipped with the standard tensor product grading, provided $\mathsf{M}/\hbar^{n}\mathsf{M}$ and $\mathsf{N}/\hbar^{n}\mathsf{N}$ are both $\mathbb{Z}$-graded.

Of course, one also has the notion of a $\mathbb{Z}$-graded topological coalgebra and bialgebra, which are defined by making the obvious modifications to the above definition.

The prototypical example of a $\mathbb{Z}$-graded topological module over $\mathbb{C}[\![\hbar]\!]$ is $\mathsf{M}=\mathsf{V}[\![\hbar]\!]$ where $\mathsf{V}=\bigoplus_{k\in\mathbb{Z}}\mathsf{V}_{k}$ is a $\mathbb{Z}$-graded complex vector space. In this case one has

which is naturally graded, as $\mathsf{V}[\hbar]$ is graded with $\mathsf{V}[\hbar]_{k}=\bigoplus_{n\geq 0}\hbar^{n}\mathsf{V}_{k-n}$, and $\hbar^{n}\mathsf{V}[\hbar]$ is a graded submodule. The assertion that $\mathsf{M}$ is $\mathbb{Z}$-graded may be recaptured as follows. For each $k\in\mathbb{Z}$, set

Then each $\mathsf{M}_{k}$ is a closed subspace of $\mathsf{M}$ satisfying $\mathsf{M}_{k}/\hbar^{n}\mathsf{M}_{k-n}\cong(\mathsf{M}/\hbar^{n}\mathsf{M})_{k}$, and $\mathsf{M}$ contains the $\mathbb{Z}$-graded $\mathbb{C}[\hbar]$-module $\bigoplus_{k\in\mathbb{Z}}\mathsf{M}_{k}$ as a dense $\mathbb{C}[\hbar]$-submodule. Moreover, the $\hbar$-adic topology on this submodule coincides with the subspace topology, so $\mathsf{M}$ is the $\hbar$-adic completion of $\bigoplus_{k\in\mathbb{Z}}\mathsf{M}_{k}$. If in addition $\mathsf{V}$ is $\mathbb{N}$-graded, then $\bigoplus_{k\in\mathbb{Z}}\mathsf{M}_{k}$ coincides with the polynomial space $\mathsf{V}[\hbar]\subset\mathsf{V}[\![\hbar]\!]$.

The below lemma provides an equivalent characterization of the definition of a $\mathbb{Z}$-graded topological module and algebra which generalizes this picture.

If $\mathsf{M}$ is a $\mathbb{Z}$-graded topological module then the $k$-th component $\mathsf{M}_{k}$ of $\mathsf{M}_{\mathbb{Z}}$ from Lemma 2.4 is uniquely determined and recovered as the inverse limit

Moreover, one has $(\mathsf{M}/\hbar^{n}\mathsf{M})_{k}\cong\mathsf{M}_{k}/\hbar^{n}\mathsf{M}_{k-n}$ for all $n\in\mathbb{N}$ and $k\in\mathbb{Z}$.

We further observe that, for each $k\in\mathbb{Z}$, the system of linear projections $\pi_{n,k}:\mathsf{M}/\hbar^{n}\mathsf{M}\to(\mathsf{M}/\hbar^{n}\mathsf{M})_{k}$ gives rise to a projection

which restricts to the projection of $\mathsf{M}_{\mathbb{Z}}$ onto its $k$-th homogeneous component. In particular, a $\mathbb{C}[\![\hbar]\!]$-linear map $\uptau:\mathsf{M}\to\mathsf{N}$ between $\mathbb{Z}$-graded topological modules is graded if and only if $\pi_{k}^{\mathsf{N}}\circ\uptau=\uptau\circ\pi_{k}^{\mathsf{M}}$ for each $k\in\mathbb{Z}$.

We conclude this preliminary subsection with two corollaries of the above discussion. The first shows that any topologically free $\mathbb{Z}$-graded $\mathbb{C}[\![\hbar]\!]$-module is of the form described above Lemma 2.4.

Let us now shift our attention to the case where $\dot{\mathsf{M}}=\bigoplus_{k\in\mathbb{N}}\dot{\mathsf{M}}_{k}$ is an $\mathbb{N}$-graded $\mathbb{C}[\hbar]$-module. Any such module is automatically separated, and so embeds into its $\hbar$-adic completion

which is an $\mathbb{N}$-graded topological $\mathbb{C}[\![\hbar]\!]$-module. Moreover, if $\dot{\mathsf{M}}$ is a torsion free $\mathbb{C}[\hbar]$-module, then $\mathsf{M}$ is topologically free as a $\mathbb{C}[\![\hbar]\!]$-module. Since $\dot{\mathsf{M}}_{k-n}$ is trivial for $n>k$, the submodule $\mathsf{M}_{k}$ of $\mathsf{M}$ (see (2.1)) coincides with $\dot{\mathsf{M}}_{k}$ and so, in the notation of Lemma 2.4, one has $\dot{\mathsf{M}}=\mathsf{M}_{\mathbb{N}}$. These observations, coupled with Corollary 2.5 and that $\mathsf{V}[\![\hbar]\!]_{k}=\mathsf{V}[\hbar]_{k}$ when $\mathsf{V}$ is $\mathbb{N}$-graded, yield the following.

Note that if $\dot{\mathsf{M}}$ is a $\mathbb{N}$-graded $\mathbb{C}[\hbar]$-algebra or Hopf algebra, then $\mathsf{M}$ is automatically an $\mathbb{N}$-graded topological algebra or Hopf algebra, respectively.

For a given $\mathbb{Z}$-graded complex vector space $\mathsf{V}=\bigoplus_{n}\mathsf{V}_{n}$, we let $\mathsf{V}^{\star}=\bigoplus_{n}(\mathsf{V}^{\star})_{n}\subset\mathsf{V}^{\ast}$ denote the restricted, or graded, dual of $\mathsf{V}$, where

and $\mathbb{C}$ is given the trivial grading with $\mathbb{C}_{0}=\mathbb{C}$ and $\mathbb{C}_{m}=\{0\}$ for $m\neq 0$. One can similarly define the restricted dual $\mathsf{M}^{\star}\subset\mathsf{M}^{\ast}$ in the category of $\mathbb{Z}$-graded topological $\mathbb{C}[\![\hbar]\!]$-modules. In this subsection we will recall some properties of this duality operation in the $\mathbb{N}$-graded setting which will be applied to construct the dual Yangian in Section 6.

Suppose that $\mathsf{M}$ is an $\mathbb{N}$-graded, topological $\mathbb{C}[\![\hbar]\!]$-module with $\mathbb{N}$-graded $\mathbb{C}[\hbar]$-submodule $\mathsf{M}_{\mathbb{N}}=\bigoplus_{k\in\mathbb{N}}\mathsf{M}_{k}$ as in Lemma 2.4. For each $n\in\mathbb{N}$, let $\mathsf{J}_{k}$ denote the $\hbar$-adic completion of the ideal $\bigoplus_{k\geq n}\mathsf{M}_{k}$ of $\mathsf{M}_{\mathbb{N}}$. The gradation topology on $\mathsf{M}$ is the topology associated to the descending filtration