Peter-Weyl bases, preferred deformations, and Schur-Weyl duality

TL;DR

This paper explores a preferred deformation of the function algebra on a reductive Lie group, utilizing Peter-Weyl bases, and examines its relation to Schur-Weyl duality, with implications for quantum group structures.

Contribution

It introduces a specific preferred deformation of the function algebra on reductive Lie groups using Peter-Weyl bases and connects it to Schur-Weyl duality.

Findings

The deformation preserves the comultiplication structure constants.

Multiplication structure constants are governed by quantum 3j symbols.

Links between preferred deformations and Schur-Weyl duality are established.

Abstract

We discuss the deformed function algebra of a simply connected reductive Lie group G over the complex numbers using a basis consisting of matrix elements of finite dimensional representations. This leads to a preferred deformation, meaning one where the structure constants of comultiplication are unchanged. The structure constants of multiplication are controlled by quantum 3j symbols. We then discuss connections earlier work on preferred deformations that involved Schur-Weyl duality.

Equations

$$\mu (a,b)$$

$$\Delta (a)$$

$$\Delta E_i=E_i\otimes e^{-h{H}_i}+1\otimes E_i$$

$$\Delta F_i=F_i\otimes 1+e^{h{H}_i}\otimes F_i$$

$$\Delta H_i=H_i\otimes 1+1\otimes H_i$$

$$EF-FE=\frac{e^{hH}-e^{-hH}}{e^{\hslash }-e^{-\hslash }}.$$

$$X=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}{e}_{11}^{\ast }& e_{12}^{\ast }\\ e_{21}^{\ast }& e_{22}^{\ast }\end{array}\right]$$

$$X_1{X}_2=\left[\begin{array}{cccc}{a}^2& ab& ba& b^2\\ ac& ad& bc& bd\\ ca& cb& da& db\\ c^2& cd& dc& d^2\end{array}\right],X_2{X}_1=\left[\begin{array}{cccc}{a}^2& ba& ab& b^2\\ ca& da& cb& db\\ ac& bc& ad& bd\\ c^2& dc& cd& d^2\end{array}\right].$$

$$R=\left[\begin{array}{cccc}q& 0& 0& 0\\ 0& 1& 0& 0\\ 0& q-q^{-1}& 1& 0\\ 0& 0& 0& q\end{array}\right]$$

$$ab=qba,ac=qca,bd=qdb,cd=qdc,$$

$$\text{mbox}bc=cb,ad-da=(q-q^{-1})bc.$$

$$\Delta \left(\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\right)=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\otimes \left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=\left[\begin{array}{cc}a\otimes a+b\otimes c& a\otimes b+b\otimes d\\ c\otimes a+d\otimes c& c\otimes b+d\otimes d\end{array}\right]$$

$$\Delta (a^2)=a^2\otimes a^2+ab\otimes ac+ba\otimes ca+b^2\otimes c^2=a^2\otimes a^2+(1+q^{-2})ab\otimes bc+b^2\otimes c^2.$$

$${\mathcal{O}}_{\hslash }(G)\simeq {\oplus }_{\lambda }\text{End}(V_{\lambda }{)}^{\ast },$$

$$\underset{\lambda }{⨆}\{Y^{\ast }\otimes X:X,Y\in B_{\lambda }\}.$$

$$⟨Y^{\ast }\otimes X,u⟩=Y^{\ast }u(X).$$

$$\Delta (Y^{\ast }\otimes X)=\sum _{Z\in B_{\lambda }}(Y^{\ast }\otimes Z)\otimes (Z^{\ast }\otimes X).$$

$$((Y_1^{\ast }\otimes X_1)(Y_2^{\ast }\otimes X_2))(u)=(Y_1^{\ast }\otimes Y_2^{\ast })\Delta u(X_1\otimes X_2).$$

$$X_{1}\otimes X_{2}=\sum_{\nu}\sum_{1\leq k\leq c_{\lambda,\mu}^{\nu}}\sum_{X_{3}\in B_{\nu}}\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ X_{1}&X_{2}&X_{3}\end{array}\right)_{k}\phi_{k}(X_{3}).$$

$$\phi_{k}^{*}(Y_{1}^{*}\otimes Y_{2}^{*})=\sum_{Y_{3}^{*}\in B_{\nu}^{*}}\overline{\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ Y_{1}^{*}&Y_{2}^{*}&Y_{3}^{*}\end{array}\right)}_{k}Y_{3}^{*}.$$

$$(Y_{1}^{*}\otimes X_{1})(Y_{2}^{*}\otimes X_{2})=\sum_{\begin{subarray}{c}\nu\in P_{+}\\ {{X_{3},Y_{3}\in B_{\nu}}}\end{subarray}}\left[\sum_{1\leq k\leq c_{\lambda,\mu}^{\nu}}\overline{\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ Y_{1}^{*}&Y_{2}^{*}&Y_{3}^{*}\end{array}\right)}_{k}\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ X_{1}&X_{2}&X_{3}\end{array}\right)_{k}\right]Y_{3}^{*}\otimes X_{3}.$$

$$\sum_{1\leq k\leq c_{\lambda,\mu}^{\nu}}\overline{\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ Y_{1}^{*}&Y_{2}^{*}&Y_{3}^{*}\end{array}\right)}_{k}\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ X_{1}&X_{2}&X_{3}\end{array}\right)_{k}$$

$$fg=f\otimes g\in \text{End}(V\otimes W{)}^{\ast }\simeq \text{End}(V{)}^{\ast }\otimes \text{End}(W{)}^{\ast }.$$

$${\oplus }_n((\text{End}{V}^{\otimes n}{)}^{S_n}{)}^{\ast },$$

$$V^{\otimes n}\simeq {\oplus }_{\lambda }{V}_{\lambda }⊠W_{\lambda },$$

$${\oplus }_n(\text{End}{V}^{\otimes n}{)}^{S_n}\simeq {\oplus }_{\lambda }\text{End}{V}_{\lambda }.$$

$$P((c^{\ast }\otimes m_w^{\ast })\otimes (b\otimes w_m))$$

$$P((b^{\ast }\otimes w_m^{\ast })\otimes (c\otimes m_w))=\frac{1}{n!}\sum _{\sigma \in S_n}(b^{\ast }\otimes g{w}_m^{\ast })\otimes (c\otimes g{m}_w).$$

$$fg=P_{n+m}(f\otimes g).$$

$$\Delta (Y^{\ast }\otimes X)=\sum _{Z\in C}(Y^{\ast }\otimes Z)\otimes (Z^{\ast }\otimes X).$$

$$\Delta (Y^{\ast }\otimes X)=(P\otimes P)\sum _{Z\in C}(Y^{\ast }\otimes Z)\otimes (Z^{\ast }\otimes X).$$

$${\oplus }_n((\text{End}(V^{\otimes n}{)}^{H_n}{)}^{\ast }\simeq {\oplus }_{\lambda }\text{End}(V_{\lambda }{)}^{\ast }=\mathcal{O}(M_n),$$

Full text

1. Introduction

Let $G$ be a connected reductive Lie group over ${\mathbb{C}}$, and let $\mathfrak{g}$ be its Lie algebra. Associated to this data are two Hopf algebras, the commutative function algebra $\mathcal{O}(G)$ and the cocommutative universal enveloping algebra $U(\mathfrak{g})$. During the 1980s, various non-commutative and non-commutative quantizations of these Hopf algebras were independently introduced. The first example, now known as $U_{\hbar}(\mathfrak{sl}_{2})$, was discovered by Kulish and Reshetikhin in [KR81] in relation to the quantum inverse scattering method. Later, Drinfeld and Jimbo independently introduced the well studied quantized universal enveloping algebra $U_{\hbar}(\mathfrak{g})$. On the dual side, several approaches to quantizations of $\mathcal{O}(G)$ have been studied. The first was the quantum matrix bialgebra $\mathcal{O}_{\hbar}(M_{2})$ introduced by Faddeev and Takhtajan in [FT86], constructed using the monodromy matrix for the quantum Lax operator of the Liouville model. This approach was fully developed in the landmark work [FRT90] of Faddeev, Reshetikhin and Takhtajan in which a quantum Yang-Baxter $R$-matrix is used to deform the defining relations of the classical series of coordinate algebras $\mathcal{O}(G)$.

We should also mention a few other early approaches. In [W87a, W87b], Woronowicz developed the theory of compact quantum groups in the $C^{*}$-algebra framework by introducing the quantization $SU_{\mu}(2)$ in which the parameter $\mu$ is a positive real number. Matrix coefficients of finite dimensional representations play a key role in this theory. Another approach due to Manin [Man87] constructs quantum coendomorphism bialgebras as universal objects coacting on a pair of quantum linear spaces.

Since $U(\mathfrak{g})$ is rigid as an algebra and $\mathcal{O}(G)$ is rigid as a coalgebra, the fact that $U_{\hbar}(\mathfrak{g})$ and $\mathcal{O}_{\hbar}(G)$ are formal deformations implies that their finite dimensional representations and corepresentations correspond exactly to those for $U(\mathfrak{g})$ and $\mathcal{O}(G)$ respectively. In particular, these categories are semi-simple/cosemisimple. With this in mind, a dual approach may be taken by first studying the monoidal categories of corepresentations of $\mathcal{O}_{\hbar}(G)$ and $\mathcal{O}(G)$. Once enough is known about these categories, one can follow the generalized Tannaka-Krein theory to reconstruct the Hopf algebras, which must necessarily be isomorphic as coalgebras, see [JS90].

Focusing more sharply on our main point, it is a natural question to find a so-called “preferred” presentations of $U_{\hbar}(\mathfrak{g})$ and $\mathcal{O}_{\hbar}(G)$, where the algebra structure is completely unchanged for $U_{\hbar}(\mathfrak{g})$ and the coalgebra structure is completely unchanged for $\mathcal{O}_{\hbar}(G)$. With the usual generators and relations descriptions this seems to be hard since, for the natural bases, all structures are varying.

The purpose of this note is to discuss a preferred presentation for $\mathcal{O}_{\hbar}(G)$. The starting point is to view $\mathcal{O}_{\hbar}(G)$ as the restricted dual Hopf algebra of $U_{\hbar}(\mathfrak{g})$. The preferred presentation is achieved from a Peter-Weyl basis of $\mathcal{O}_{\hbar}(G)$ – a basis consisting of matrix elements of finite dimensional representations. The structure constants for the preferred presentation make use of quantum $3j$-symbols from physics, which encode the decomposition of a tensor product of irreducible representations into irreducibles. These coefficients have numerous applications and in the rank one case have been extensively studied, see [KK89, KR88, V89].

We finish by describing how this relates to Schur-Weyl duality in type A, and hence to some earlier work by Gerstenhaber, Giaquinto and Schack [GGS92, Gia92] on preferred deformations. In these papers, the quantum matrix bialgebra $\mathcal{O}_{\hbar}(M_{n})$ is viewed as the invariant or “quantum symmetric” elements of the tensor algebra $T(M_{n}^{*})$ which are fixed by the action of a certain quantum symmetric group. This is a subgroup of the cactus group studied in e.g. [KT09], and as discussed there is related to using Drinfeld’s unitarized $R$-matrix from [Dr90] in place of the usual $R$-matrix. If $V$ is the vector representation, then the image of this group in $\rm{End}(V^{\otimes n})$ generates the usual action of the Hecke algebra on this space.

In [GGS92, Gia92] the decomposition of tensor space $V^{\otimes n}$ into quantum symmetric elements is obtained with the aid of the Woronowicz quantization of $U(\mathfrak{sl}_{n})$ which acts as skew derivations of $V^{\otimes n}$ associated to certain automorphisms. These automorphisms coincide with the exponentials $\exp(\hbar\,H_{i})$ of the standard Cartan generators $H_{i}$ of $\mathfrak{sl}_{n}$. Thus the images of the Woronowicz quantization and $U_{\hbar}(\mathfrak{sl}_{n})$ in $\rm{End}(V^{\otimes n})[[\hbar]]$ coincide and so the Schur-Weyl decompositions of $V^{\otimes n}$ are the same for either of these two quantizations. The use of the Woronowicz quantization was motivated by the fact that its finite dimensional representations correspond exactly to those of $U_{\hbar}(\mathfrak{sl}_{n})$ or $U(\mathfrak{sl}_{n})$, and one does not have to exclude the non-type-1 representation that appear for the rational form $U_{q}(\mathfrak{sl}_{n})$. The disadvantage is that the Woronowicz quantization does not give a bialgebra structure (see [GGS92, p26]).

We do not carefully address the preferred presentation of $U_{\hbar}(\mathfrak{g})$ in this note. The dual Peter-Weyl basis does give a preferred presentation, but of a certain completion of $U_{\hbar}(\mathfrak{g})$. Finding a preferred presentation for the $U_{\hbar}(\mathfrak{g})$ itself seems more difficult. An algebra isomorphism from $U_{\hbar}(\mathfrak{sl}_{2})$ to $U(\mathfrak{sl}_{2})$ was given in [CP94, Proposition 6.4.6], giving the preferred presentation in that case. Only recently in [AG17] was an explicit trivialization of $U_{\hbar}(\mathfrak{sl}_{n})$ given by Appel and Gautam. This isomorphism is induced by a map between the quantum loop algebra of $\mathfrak{sl}_{n}$ and a completion of the Yangian.

This note is organized as follows. In §2 we discuss some background on deformation theory and the notion of preferred deformations. In §3 we construct a preferred deformation using a Peter-Weyl basis. In §4 we restrict to type $A$ and reformulate the construction using Schur-Weyl duality, then discuss how this relates to some older work.

2. Deformations

2.1. Formal deformations

Let $B$ be a bialgebra over $\mathbb{C}$. A $\mathbb{C}[[\hbar]]$-bialgebra $B_{\hbar}$ is a formal deformation of $B$ if it is a topologically free $\mathbb{C}[[\hbar]]$-module together with an isomorphism $B_{\hbar}/hB_{\hbar}\simeq B$. If $B_{\hbar}$ is a formal deformation of $B$, we can choose an identification of $B_{\hbar}$ with $B[[\hbar]]$ as a $\mathbb{C}[[\hbar]]$-module. The multiplication $\mu$ and comultiplication $\Delta$ of $B_{\hbar}$ then necessarily have the form

[TABLE]

where $\mu_{i}:A\otimes A\to A$ and $\Delta_{i}:A\to A\otimes A$ are linear maps and $\mu_{0}$ and $\Delta_{0}$ are the undeformed multiplication and comultiplication of $B$.

2.2. Equivalence and preferred presentations

Deformations $B_{\hbar}$ and $B_{\hbar}^{\prime}$ are equivalent if there is a ${\mathbb{C}}[[\hbar]]$-bialgebra isomorphism $\phi:B_{\hbar}\to B_{\hbar}^{\prime}$ which reduces to the identity modulo $\hbar$. It is known that every deformation of the universal enveloping algebra $U(\mathfrak{g})$ is equivalent to one in which $\mu_{\hbar}=\mu$. That is, it is a trivial deformation of the algebra structure, and so the representation theory of $U_{\hbar}(\mathfrak{g})$ and $U(\mathfrak{g})$ is identical. Dually, every deformation of $\mathcal{O}(G)$ is equivalent to one in which $\Delta_{\hbar}=\Delta$, so the co-representation theory is unchanged. A preferred presentation of a deformation of $U(\mathfrak{g})$ or $\mathcal{O}(G)$ is one with unchanged multiplication or comultiplication.

A natural question is to find preferred presentations of $U_{\hbar}(\mathfrak{g})$ and $\mathcal{O}_{\hbar}(G)$. To do so requires an identification of their underlying vector spaces with $U(\mathfrak{g})[[\hbar]]$ and $\mathcal{O}(G)[[\hbar]]$ as $\mathbb{C}[[\hbar]]$-modules. This can be accomplished, for example, by finding bases of $U_{\hbar}(\mathfrak{g})$ and $\mathcal{O}_{\hbar}(G)$ which reduce to bases of $U(\mathfrak{g})$ and $\mathcal{O}(G)$ modulo $\hbar$. However, most choices of bases do not provide preferred presentations: both multiplication and comultiplication depend on $\hbar$. This is true, in particular, for the various PBW-type bases in the literature.

We now arrive at an interesting juncture: Once $\mathcal{O}_{\hbar}(G)$ is shown to be a formal deformation, we know its irreducible representations are the same as those for $\mathcal{O}(G)$, just tensored with ${\mathbb{C}}[[\hbar]]$. We can then consider a Peter-Weyl type basis of $\mathcal{O}_{\hbar}(G)$, meaning a basis consisting of matrix elements of irreducible representations. We shall see that this provides the sought after preferred presentation of $\mathcal{O}_{\hbar}(G)$.

2.3. Standard deformation of $U(\mathfrak{g})$

Consider the standard Chevalley generators $E_{i},F_{i},H_{i}$ of $U(\mathfrak{g})$. The deformation $U_{\hbar}(\mathfrak{g})$ is usually defined to be the algebra with these same generators, but deformed relations and a deformed coproduct. The main structure we will need here is the coproduct, so we state that explicitly:

[TABLE]

The rest of the structure can be found in many places, see e.g. [CP94]. The relations for multiplication are also deformed. For instance, in $\mathfrak{sl}_{2}$ the undeformed relation $EF-FE=2H$ becomes

[TABLE]

So the deformation is certainly not preferred with respect to any PBW type basis.

2.4. Standard deformation of $\mathcal{O}(M_{n})$

Here we give the deformed relations for $\mathcal{O}(M_{n})$, as constructed by Faddeev-Reshetikhin-Takhtajan [FRT90]. We focus on the case $n=2$, and for simplicity of presentation define $q=e^{\hbar}$. Consider the coordinate functions

[TABLE]

on the space of $2\times 2$ matrices. The FRT formalism starts with a solution $R$ to the quantum Yang-Baxter equation ($R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$) and imposes the relations $RX_{1}X_{2}=X_{2}X_{1}R$ where $X_{1}=X\otimes I$ and $X_{2}=I\otimes X$. In coordinates,

[TABLE]

For $M_{2}$,

[TABLE]

which produces the relations

[TABLE]

The coproduct is defined on generators by

[TABLE]

and is extended multiplicatively to monomials of higher degree. For example

[TABLE]

This is dependent on $q$, so the deformation is not preferred, at least when using the PBW basis $\{a^{i}\,b^{j}\,c^{k}\,d^{l}\quad|\quad i,j,k,l\in\mathbb{Z}_{\geq 0}\}$ to identify $\mathcal{O}_{\hbar}(M_{n})$ with $\mathcal{O}(M_{n})[[\hbar]]$.

3. Peter-Weyl bases and preferred deformations

Another approach to deforming $\mathcal{O}(G)$ is by duality: one simply defines $\mathcal{O}_{\hbar}(G)$ as the restricted dual of $U_{\hbar}(\mathfrak{g})$. In this setting the Peter-Weyl basis arises naturally. It is with this basis that we get a preferred presentation of $\mathcal{O}_{\hbar}(G)$.

3.1. The Peter Weyl basis

Since $\mathcal{O}_{\hbar}(G)$ is cosemisimple, the restricted dual definition implies

[TABLE]

where the $\lambda$ runs over the dominant integral weights of $G$, the $V_{\lambda}$ are the corresponding representations of $U_{\hbar}(\mathfrak{g})$, and the isomorphism is as coalgebras over $k[[\hbar]]$. See e.g. [KS97, Chapter 11]. This becomes an isomorphism of Hopf algebras if one defines multiplication on $\oplus_{\lambda}\text{End}(V_{\lambda})$ as the dual of the coproduct for $U_{\hbar}(\mathfrak{g})$.

For any $\lambda$, $\text{End}(V_{\lambda})$ is naturally identified with $V_{\lambda}\otimes V_{\lambda}^{*}$. Taking duals, we identify $\text{End}(V_{\lambda})^{*}$ with $V_{\lambda}^{*}\otimes V_{\lambda}$. This gives a natural way to choose a basis for $\mathcal{O}_{\hbar}(G)$: pick dual bases $B_{\lambda}$ and $B^{*}_{\lambda}$ for each pair $V_{\lambda}$,$V_{\lambda}^{*}$. Then

[TABLE]

is a basis for $\mathcal{O}_{\hbar}(G)$, which we call a Peter-Weyl basis. The pairing of $U(\mathfrak{g})$ with $\mathcal{O}(G)$ is given by, for $Y^{*}\otimes X\in\mathcal{O}(G)$ and $u\in U(\mathfrak{g})$,

[TABLE]

Remark*.*

One often reverses order of factors when taking duals of tensor products. We have not done so, in part to match conventions in [GGS92].

3.2. Comultiplication

We are identifying $\prod_{\lambda}\text{End}V_{\lambda}$ with a completion of $U_{\hbar}(\mathfrak{g})$, and $\mathcal{O}_{\hbar}(G)$ with the dual Hopf algebra to this. Thus, in both $\mathcal{O}(G)$ and $\mathcal{O}_{\hbar}(G)$, co-multiplication is the dual of multiplication in $\prod_{\lambda}\text{End}V_{\lambda}$. In coordinates, for $X,Y\in B_{\lambda}$,

[TABLE]

3.3. Multiplication (abstract)

Multiplication is the dual to comultiplication in $U_{\hbar}(\mathfrak{g})$. In coordinates this means, for $X_{1},Y_{1}\in B_{\lambda}$, $X_{2},Y_{2}\in B_{\mu}$, and any $u\in U(\mathfrak{g})$,

[TABLE]

To be explicit we need to express this in terms of the Peter Weyl basis. The resulting structure constants are closely related to the famous 3j symbols from physics.

3.4. $3j$ symbols

These are often studied just for $\text{SL}(2)$, but we need the following more general notion. For each triple $\lambda,\mu,\nu$, choose a basis $\{\phi_{1},\cdots,\phi_{c_{\lambda,\mu}^{\nu}}\}$ for the space of embeddings of $V_{\nu}\hookrightarrow V_{\lambda}\otimes V_{\mu}$. For $X_{1}\in B_{\lambda},X_{2}\in B_{\mu},X_{3}\in B_{\nu}$, write

[TABLE]

The constants \displaystyle\left(\begin{array}[]{ccc}\lambda&\mu&\nu\\ X_{1}&X_{2}&X_{3}\end{array}\right)_{k} are called the $3j$ symbols.

Taking duals gives a basis $\{\phi^{*}_{1},\cdots,\phi^{*}_{c_{\lambda,\mu}^{\nu}}\}$ of the space of surjections $V^{*}_{\lambda}\otimes V^{*}_{\mu}\rightarrow V_{\nu}^{*}$. We then get dual $3j$ symbols defined by

[TABLE]

This can be done just as easily for representations of $U(\mathfrak{g})$ or $U_{\hbar}(\mathfrak{g})$.

Remark*.*

In the $\text{SL}(2)$ case, there is a unique (up to signs) orthonormal weight basis, so a chosen Peter-Weyl basis. The spaces of embeddings $V_{\nu}\hookrightarrow V_{\lambda}\otimes V_{\mu}$ are 1 dimensional, and the inner product can be used to normalize the embedding, fixing the $3j$ symbols. As mentioned earlier, these have been calculated extensively. Using orthonormal bases also implies that the the $3j$ symbols and dual $3j$ symbols coincide exactly.

3.5. Structure constants for multiplication

It is now immediate from definitions that the structure constants for multiplication in the Peter-Weyl basis are given by, for $X_{1},Y_{1}\in B_{\lambda}$ and $X_{2},Y_{2}\in B_{\mu}$,

[TABLE]

Multiplication as defined in (3) does not depend on the basis $\{\phi_{1},\ldots,\phi_{k}\}$, so

[TABLE]

must be independent of the choice of basis $\{\phi_{1},\ldots,\phi_{k}\}$ as well.

3.6. Preferred presentation of $\mathcal{O}_{\hbar}(G)$

The set of irreducible representations $V_{\lambda}$ of $U(\mathfrak{g})$ and $U_{\hbar}(\mathfrak{g})$ correspond exactly, and the comultiplication from §3.2 does not reference $U(\mathfrak{g})$ at all, so is unchanged under deformation. The multiplication from §3.3 does change when we move to $U_{\hbar}(\mathfrak{g})$, since it’s definition uses the coproduct of $U(\mathfrak{g})$, which is deformed in $U_{\hbar}(\mathfrak{g})$. But the presentation in §3.5 is still valid. The only difference is that the spaces of embeddings $V_{\nu}\hookrightarrow V_{\lambda}\otimes V_{\mu}$ change.

In order to see $\mathcal{O}_{\hbar}(\mathfrak{g})$ as a preferred deformation of $\mathcal{O}(\mathfrak{g})$, one must simply choose

• •

A basis for each $U_{\hbar}(\mathfrak{g})$ module $V_{\lambda}$ which specializes to a basis at $h=0$,

• •

A basis for each space of $U_{q}(\mathfrak{g})$-homomorphisms $V_{\nu}\hookrightarrow V_{\lambda}\otimes V_{\mu}$ which also specializes to a basis at $\hbar=0$. This leads to a definition of quantum 3j symbols.

Then the construction above gives a deformation where the structure constants of comultiplication are manifestly identical, and the structure constants for multiplication are given by (4), but with the 3j symbols replaces by their deformed counterparts.

Remark*.*

We have relied on the fact that we already have a (non-preferred) deformation of $U(\mathfrak{g})$ to construct our preferred presentation of $\mathcal{O}_{\hbar}(G)$. One might try to use this approach to construct a deformation from scratch, by simply deforming the spaces of embeddings $V_{\nu}\hookrightarrow V_{\lambda}\otimes V_{\mu}$. However, this deformation is not arbitrary: one needs to ensure that $\mathcal{O}(G)$ remains a Hopf algebra. Directly ensuring this seems difficult.

3.7. Preferred deformation of $U(\mathfrak{g})$

This method also gives a preferred presentation of a completion of $U_{\hbar}(\mathfrak{g})$ by working with the topological basis $\{f_{b,c}^{\lambda}=b\otimes c^{*}\in\text{End}V_{\lambda}\}$. The operations are the duals those of $\mathcal{O}_{\hbar}(G)$. However, $U_{\hbar}(\mathfrak{g})$ is a proper subalgebra of $\prod_{\lambda}\text{End}(V_{\lambda})$, and the preferred presentation does not restrict in any nice way. It is also unclear how to relate the Peter-Weyl type bases with the Chevalley generators. So this approach is not really satisfactory.

3.8. Non-simply-connected groups and matrix algebras

The condition that $G$ be simply connected is not really needed. A non-simply connected reductive Lie group $G^{\prime}$ is always the quotient of a corresponding simply-connected one, and the category of finite dimensional representations of $G^{\prime}$ is a sub-tensor-category of the category of finite dimension representations of $G$. The irreducible representations of that category are parameterized by $\lambda$ in the positive part of some sub-lattice $P^{\prime}$ of the weight lattice of $G$. The whole story then goes through by realizing $\mathcal{O}(G^{\prime})$ as $\oplus\text{End}(V_{\lambda})^{*},$ where now one restricts to $\lambda\in P^{\prime}_{+}$.

In type $A$ one can also consider $\mathcal{O}(\text{GL}_{k})$, and again the story goes through without significant changes, only now there are more representations than for $\mathcal{O}(\text{SL}_{k})$, since any irreducible representation can be tensored with any integer power of the determinant representation. In §4 we will actually work with $\mathcal{O}(M_{k})$, the function algebra on the algebra of all $k\times k$ matrices. This is isomorphic to $\oplus_{\lambda}\text{End}V_{\lambda}^{*}$, where now the $\lambda$ index the polynomial representations of $\text{GL}_{k}$. These $\lambda$’s are naturally indexed by partitions with at most $k$ parts.

4. Relation to Schur-Weyl Duality

For the case of $\rm{GL}_{k}$, or $\rm{SL_{k}}$, [Gia92, GGS92] studied another approach to finding a preferred deformation. Their approach most naturally realizes $\mathcal{O}(M_{k})$, the function algebra on the algebraic monoid of $k\times k$ matrices. We now discuss how their results naturally arise in our framework.

4.1. General categorical discussion

Identifying $\mathcal{O}(G)$ with $\oplus_{\lambda}\text{End}(V_{\lambda})^{*}$ as we have is not necessarily the most natural thing to do, since it requires choosing a representation in each isomorphism class of simples. In fact, any $f\in\text{End}(V)^{*}$, for any representation $V$, gives a function of $G$. However different elements of $\text{End}(V)^{*}$ can give identical functions on $G$, so $\mathcal{O}(G)$ should be identified with a quotient of $\oplus_{V}\text{End}(V)^{*}$. This is a badly infinite sum, but ignoring that for now, multiplication is simple: given two elements of $\mathcal{O}(G)$, $f\in\text{End}(V)^{*},g\in\text{End}(W)^{*}$,

[TABLE]

We work with $\oplus_{\lambda}(\text{End}V_{\lambda})^{*}$ essentially because every element of $\mathcal{O}(G)$ appears exactly once. Equivalently, using the restricted dual definition, every linear functional on $U(\mathfrak{g})$ is represented only once.

4.2. Using $V^{\otimes n}$, undeformed

Now we will restrict to considering $\mathcal{O}(M_{k})$. Then there is another natural space which encodes every function exactly once:

[TABLE]

where $V$ is the vector representation. To see this, recall that, by Schur-Weyl duality,

[TABLE]

where $\lambda$ ranges over all partitions of $n$ with at most $k$ rows, and $V_{\lambda},W_{\lambda}$ are the irreducible representations of $M_{k}$ and $S_{n}$ respectively. Then, by Schur’s lemma,

[TABLE]

We will need to understand this identification explicitly. Fix $\lambda,$ and choose any $w,w^{*}\in W_{\lambda},W_{\lambda}^{*}$ with $w^{*}(w)=1$. Then, for any $c^{*}\otimes b\in\text{End}(V_{\lambda})^{*}$, the element $(c^{*}\otimes w^{*}_{m})\otimes(b\otimes m_{w})\in\text{End}(V^{\otimes n})^{*}$ gives the same function on $U_{q}(\mathfrak{gl}_{n})$ as $c^{*}\otimes b$, but it is not $S_{n}$ invariant. To fix that, consider the Young symmetrizer $\displaystyle P=\frac{1}{n!}\sum_{\sigma\in S_{n}}\sigma$. Then

[TABLE]

is clearly $S_{n}$ invariant, and gives the same function on $U_{q}(\mathfrak{gl}_{n})$. Explicitly,

[TABLE]

Since there is only one $S_{n}$ invariant element corresponding to a given function, this is independent of the choice of $w,w^{*}$.

Multiplication is then given by, for $f\in(\text{End}V^{\otimes n})^{S_{n}}$ and $g\in\text{End}(V^{\otimes m})^{S_{m}}$,

[TABLE]

Comultiplication would normally be given by, for any dual bases $C$ and $C^{*}$ of $V^{\otimes n}$,

[TABLE]

However, while this is in $\text{End}(V^{\otimes n})\otimes\text{End}(V^{\otimes n})$, and corresponds to the correct element of $\mathcal{O}(M_{k})\otimes\mathcal{O}(M_{k})$, it is not in $(\text{End}V^{\otimes n})^{S_{n}}\otimes(\text{End}V^{\otimes n})^{S_{n}}$ . To fix this, apply the symmetrizer $P$ to each of the two factors to get something in the right space which gives the same function on $U_{\hbar}(\text{gl}_{n})$. The correct definition becomes

[TABLE]

4.3. Using $V^{\otimes n}$, deformed

By quantum Schur-Weyl duality $V^{\otimes n}\simeq\oplus_{\lambda}V_{\lambda}\boxtimes W_{\lambda}$, where $V$ is now the vector representation of $U_{\hbar}(\mathfrak{gl}_{n})$, the $V_{\lambda}$ are the polynomial representation of $U_{\hbar}(\mathfrak{gl}_{n})$, and the $W_{\lambda}$ are the irreducible representations of the Hecke algebra $H_{n}$ corresponding to partitions with at most $k$ rows. Then by Schur’s lemma,

[TABLE]

where the $H_{n}$ superscript means $H_{n}$ equivariant functions. The space on the left can be written as $\oplus((V^{*})^{\otimes n}\otimes V^{\otimes n})^{H_{n}},$ where the $H_{n}$ still means equivariant.

The key to understanding the operations in §4.2 was to understand the projection

[TABLE]

where $S_{n}$ acts simultaneously on the two factors. This was defined as the Young symmetrizer $P$ acting simultaneously on both factors, but to quantize we need a different characterization, since it is not clear how to have $H_{n}$ act on $(V^{*})^{n}\otimes V^{n}$.

The crucial thing in the previous section is that $P$ acts on $\oplus_{n}(\text{End}V^{n})^{*}$ as a projection so that, for any $\phi\in(V^{*})^{n}\otimes V^{n}$, $\phi$ and $P(\phi)$ define the same function on $U(\mathfrak{gl}_{k})$. In this form, there is no problem giving the deformed definition.

Definition 4.1**.**

$\pi:\oplus_{n}(V^{*})^{\otimes n}\otimes V^{\otimes n}\rightarrow\oplus_{n}((V^{*})^{\otimes n}\otimes V^{\otimes n})^{H_{n}}$ is the unique projection such that, for any $\phi\in(V^{*})^{\otimes n}\otimes V^{\otimes n}$ , $\phi$ and $\pi(\phi)$ define the same function on $U_{\hbar}(\mathfrak{g})$.

This induces a Hopf algebra structure on $\oplus((V^{*})^{\otimes n}\otimes V^{\otimes n})^{H_{n}}$ because the subset of $\oplus((V^{*})^{\otimes n}\otimes V^{\otimes n})$ consisting of elements that define the zero function on $U_{\hbar}(\mathfrak{g})$ is a Hopf ideal. Multiplication and comultiplication on $\oplus((V^{*})^{\otimes n}\otimes V^{\otimes n})^{H_{n}}$ are given by:

[TABLE]

[TABLE]

Remark*.*

It would be nice to have a more explicit formula for $\pi$. In the case $n=2$ such a formula is known. As we shall see in secton 4.5, the $H_{2}$-equivariant endomorphisms of $V^{*}\otimes V^{*}$ are determined by an involution $Q$. It follows that $\pi=\frac{1+Q}{2}$, see [Gia92, GGS92].

In general one might try to replace $P$ with the $q$-symmetrizer from [Gyo86]. This does give a natural analogue of $P$ acting on $V^{\otimes n}$, but we would need it to act on $(V^{*})^{\otimes n}\otimes V^{\otimes n}$. If the $T_{i}$ are the generators of the Hecke algebra, the appropriate action on $(V^{*})^{n}$ should replace $T_{i}$ with $T_{i}^{-1}$, and these satisfy a different set of Hecke-algebra relations. So the Hecke algebra does not even naturally act on $(V^{*})^{\otimes n}\otimes V^{\otimes n}$. In fact no symmetrizer that acts by simultaneous permutations in the $(V^{*})^{\otimes n}$ and $V^{\otimes n}$ can work, as then the relation $ad-da=(q-q^{-1})bc$ in (1) would not be possible.

4.4. Preferred presentation

We can now construct a preferred presentation of $\mathcal{O}_{\hbar}(M_{n})$.

• •

Fix the Schur-Weyl duality isomorphism $V^{\otimes n}\simeq\oplus_{\lambda}V_{\lambda}\boxtimes W_{\lambda}$. Of course one then gets a corresponding isomorphism $(V^{\otimes n})^{*}\simeq\oplus_{\lambda}V_{\lambda}^{*}\boxtimes W_{\lambda}^{*}$.

• •

Fix bases $B_{\lambda}$ for each $V_{\lambda}$, and $P_{\lambda}$ for each $W_{\lambda}$, and their dual bases $B_{\lambda}^{*}$ for each $V_{\lambda}^{*}$, and $P_{\lambda}^{*}$ for each $W_{\lambda}^{*}$, in such a way that all specialize at $h=0$.

• •

Then $\displaystyle\quad\bigcup_{\lambda}\left\{X^{\lambda}_{b,c^{*}}:=\frac{1}{\dim W_{\lambda}}\sum_{p\in P_{\lambda}}(c^{*}\boxtimes p^{*})\otimes(b\boxtimes p):b\in B_{\lambda},c^{*}\in B_{\lambda}^{*}\right\}$

is a basis for $\mathcal{O}_{\hbar}(M_{n})$. As a function on $U_{\hbar}(\mathfrak{g})$, the element $X^{\lambda}_{b,c^{*}}$ agrees with $c^{*}\otimes b\in(\text{End}V_{\lambda})^{*}$.

Since $X^{\lambda}_{b,c^{*}}$ agrees with $c^{*}\otimes b\in(\text{End}V_{\lambda})^{*}$, the structure constants of multiplication and comultiplication in this basis must agree with (2) and (4). It is an interesting exercise to directly obtain these formulae from the new definitions of comultiplication (5) and multiplication (6).

4.5. Comparing with previous work

We now compare the current approach with the “method of quantum symmetry” from [Gia92, GGS92]. The starting point there is to view $\mathcal{O}(M_{k})$ as the symmetric algebra $SX=\bigoplus_{n\geq 0}(X^{\otimes n})^{S_{n}}$, where $X=V^{*}\otimes V$. To quantize, $S_{n}$ is replaced by a “quantum symmetric group” $qS_{n}$ with generators $\tau_{1},\ldots,\tau_{n-1}$ and relations $\tau_{i}^{2}=\operatorname{Id}$ and $\tau_{i}\tau_{j}=\tau_{j}\tau_{i}$ if $|i-j|>1$. Note that if the braid relations $\tau_{i}\tau_{i+1}\tau_{i}=\tau_{i+1}\tau_{i}\tau_{i+1}$ are added then we have the Artin presentation of $S_{n}$. As mentioned in the introduction, $qS_{n}$ is a subgroup of the cactus group.

To describe the $qS_{n}$-action on $X^{\otimes n}$ we first deform the flip operator $\sigma:V^{*}\otimes V^{*}\rightarrow V^{*}\otimes V^{*}$ where $\sigma(\alpha\otimes\beta)=\beta\otimes\alpha$. Let $r=\sum_{i<j}e_{ij}\wedge e_{ji}=\frac{1}{2}\,(e_{ij}\otimes e_{ji}-e_{ji}\otimes e_{ij})$. This is the standard unitary solution to the modified classical Yang-Baxter equation associated to $\mathcal{O}_{\hbar}(M_{k})$. Define an involution of $V^{*}\otimes V^{*}$ by $Q=(\exp(-\hbar r))\sigma(\exp{\hbar r})$. With this there is an action of $qS_{n}$ on $(V^{*})^{\otimes n}$ where $\tau_{i}$ acts as $Q$ in factors $i$ and $i+1$ and the identity elsewhere. Taking duals there is a corresponding action on $(V^{*})^{\otimes d}$ and hence $qS_{n}$ acts diagonally on $(V^{*})^{\otimes n}\otimes V^{\otimes n}=X^{\otimes n}$.

One of the main results of [Gia92, GGS92] is that the set of invariant elements of the tensor algebra $TX$ is a bialgebra which is isomorphic to $\mathcal{O}_{\hbar}(M_{k})$. Moreover, the comultiplication in $\bigoplus_{n\geq 0}(X^{\otimes n})^{qS_{n}}$ is independent of $\hbar$ and coincides with the usual comultiplication in $\mathcal{O}(M_{k})=\bigoplus_{n\geq 0}(X^{\otimes n})^{S_{n}}$. Thus this construction yields the desired preferred presentation of $\mathcal{O}_{\hbar}(M_{k})$.

This essentially coincides with our construction. Using the notation of [GGS92, §10], $(k\langle M(n)\rangle^{*},\otimes)$ is naturally the tensor algebra of $V^{*}\otimes V$, which we identify with $\oplus_{n}(\text{End}V^{\otimes n})^{*}$, and think of as functions on $U_{\hbar}(\mathfrak{g})$. The space $\text{sk}_{q}\langle M(n)^{*}\rangle$ is generated by the images of the operators $\frac{1}{2}(Id-\tau_{i})$ acting on $(V^{*}\otimes V)^{\otimes n}$, and these images are easily seen to define the zero function on $U_{\hbar}(\mathfrak{g})$. So, the quotient in the top line of the diagram in [GGS92, Theorem 10.8] is by a set of elements which are all the zero function on $U_{\hbar}(\mathfrak{g})$, and by comparing dimensions it agrees with our $\pi$. Thus the comultiplication given in [Gia92, GGS92] coincides exactly with (2) and (5), and the multiplication is described using the projection formula (6).

The expression for the multiplicative structure constants in terms of $3j$ symbols is largely new to this paper, although the multiplication formulas for quantum linear spaces given in [Gia92, GGS92] can easily be expressed in the $3j$ symbol notation, and this in turn gives some of the structure constants for $\mathcal{O}(M_{n})$. So this idea really dates to those papers as well.

4.6. Deriving the $R$-matrix relations in $\mathcal{O}_{q}(M_{2})$

We now derive the last two relations in the FRT construction of $\mathcal{O}_{q}(M_{2})$ (see §2.4) in our language (the others are simpler). One could also see that the constructions agree by directly showing that the $\frac{1}{2}(\operatorname{Id}-\tau)$ action on $X\otimes X$ gives the FRT relations.

The variables $a,b,c,d$ in our language are

[TABLE]

As a representation of $U_{\hbar}(\mathfrak{gl}_{2})$, $V\otimes V\simeq W\oplus T$, where $W$ is a three dimensional representation and $T$ is one dimensional. These have basis

[TABLE]

Let $\quad s=e_{2}\otimes e_{1}+qe_{1}\otimes e_{2},\quad t=e_{2}\otimes e_{1}-q^{-1}e_{1}\otimes e_{2}.$ Then $\{s,t\}$ spans the 0 weight space of $V\otimes V$. Let $\{s^{*},t^{*}\}$ be the dual basis of this weight space. Then

[TABLE]

The Hecke algebra is the algebra of operators commuting with the action of $U_{\hbar}(\mathfrak{gl}_{2})$, so it is spanned by the projections onto $W$ and $T$. Thus $s^{*}\otimes s$ and $t^{*}\otimes t$ are both $H_{2}$ equivariant. Both $s^{*}\otimes t$ and $t^{*}\otimes s$ are zero as functions on $U_{\hbar}(\mathfrak{gl}_{2})$ by Schur’s lemma, so these are both killed by $\pi$. Thus

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Now the relation $bc=cb$ is obvious, and $ad-da=(q-q^{-1})bc$ is a simple calculation.