Poles of finite-dimensional representations of Yangians

Abstract

Let ${\mathfrak {g}}$ be a finite-dimensional simple Lie algebra over ${\mathbb {C}}$, and let $Y_\hbar ({\mathfrak {g}})$ be the Yangian of ${\mathfrak {g}}$. In this paper, we study the sets of poles of the rational currents defining the action of $Y_\hbar ({\mathfrak {g}})$ on an arbitrary finite-dimensional vector space V. Using a weak, rational version of Frenkel and Hernandez’ Baxter polynomiality, we obtain a uniform description of these sets in terms of the Drinfeld polynomials encoding the composition factors of V and the inverse of the q-Cartan matrix of ${\mathfrak {g}}$. We then apply this description to obtain a concrete set of sufficient conditions for the cyclicity and simplicity of the tensor product of any two irreducible representations, and to classify the finite-dimensional irreducible representations of the Yangian double.

Appendix A: Tensor products of fundamental modules

Let ${\mathfrak {g}}$ be an arbitrary simple Lie algebra, and let ${\textsf{d}}_{ij}(u)\in {\mathbb {C}}[u]$ denote the denominator of the normalized R-matrix ${\textsf{R}}_{L_{\varpi _i},L_{\varpi _j}}(u)\in {\text {End}}(L_{\varpi _i}\otimes L_{\varpi _j})(u)$, as defined in Sects. 7.5 and 7.8. The goal of this appendix is to prove Proposition A.4, which in particular asserts that the denominator ${\textsf{d}}_{ij}(u+\hbar d_j)$ is symmetric in the indices i and j. As indicated in the proof of Proposition 7.8, the $U_q(L{\mathfrak {g}})$-analogue of this result is well-known, having been established in “Appendix A” of the foundational paper [1] using Conjecture 2 therein, which is now a theorem. We begin with three lemmas pertinent to our proof of this proposition.

A.1 For each $i\in {{{\textbf {I}}}}$, define $\nu _i\in Y_\hbar ({\mathfrak {g}})$ by

$$\begin{aligned} \nu _i=\kappa \xi _{i,0}-\xi _{i,0}^2 +\sum _{\alpha \in R_+}(\alpha _i,\alpha )x_\alpha ^- x_\alpha ^+, \end{aligned}$$

where $R_+$ is the set of positive roots of ${\mathfrak {g}}$ and $x^{\pm }_{\alpha } \in {\mathfrak {g}}_{\pm \alpha }\subset Y_\hbar ({\mathfrak {g}})$ are chosen so as to have $(x^+_{\alpha },x^-_{\alpha })=1$, as in Sect. 2.9.

Lemma

There is an algebra isomorphism uniquely determined by ${\vartheta _\hbar }|_{{\mathfrak {g}}}={\textbf{1}}_{{\mathfrak {g}}}$ and

$$\begin{aligned} \vartheta _\hbar (\xi _{i1})=\xi _{i1}+\hbar \nu _i \quad \forall \quad i\in {{{\textbf {I}}}}. \end{aligned}$$

Moreover, for each $a\in {\mathbb {C}}$, $\vartheta _\hbar$ satisfies the relations

$$\begin{aligned} \vartheta _{\hbar }^{-1}=\vartheta _{-\hbar },\quad (\vartheta _\hbar \otimes \vartheta _\hbar )\circ \Delta =\Delta ^{\textrm{op}}\circ \vartheta _\hbar \quad \text { and }\quad \tau _a\circ \vartheta _{\hbar }=\vartheta _{\hbar }\circ \tau _a. \end{aligned}$$

Proof

For each fixed $i\in {{{\textbf {I}}}}$, define $J(d_ih_i)=\xi _{i1}+\frac{\hbar }{2}\nu _i\in Y_\hbar ({\mathfrak {g}})$. Then the set of elements $\{x,J(d_i h_i)\}_{x\in {\mathfrak {g}}, i\in {{{\textbf {I}}}}}$ generates $Y_\hbar ({\mathfrak {g}})$, and by [13, Thm. 1] and [27, §3.2] (see also [28, Thm. 2.6]), it is a subset of the generators $\{x,J(x)\}_{x\in {\mathfrak {g}}}$ in Drinfeld’s original presentation of $Y_\hbar ({\mathfrak {g}})$: see [12] and [28, Def. 2.1] (here we follow the notation of [28]). Since the defining relations in this presentation, as given in [28, Def. 2.1], depend only on $\hbar ^2$, the assignment

$$\begin{aligned} x\mapsto x \qquad \text {and}\qquad J(d_i h_i)\mapsto J(d_i h_i) \quad \forall \quad i\in {{{\textbf {I}}}}\; \text { and }\; x\in {\mathfrak {g}}\end{aligned}$$

uniquely extends to a homomorphism of algebras $\vartheta _\hbar :Y_{-\hbar }({\mathfrak {g}})\rightarrow Y_\hbar ({\mathfrak {g}})$. By definition of $\{J(d_i h_i)\}_{i\in {{{\textbf {I}}}}}$, $\vartheta _\hbar$ is uniquely determined by the requirement that $\vartheta |_{\mathfrak {g}}={\textbf{1}}_{\mathfrak {g}}$ and $\vartheta _\hbar (\xi _{i1})=\xi _{i1}+\hbar \nu _i$ for all $i\in {{{\textbf {I}}}}$, as claimed. Moreover, $\vartheta _\hbar$ is clearly invertible with $\vartheta _\hbar ^{-1}=\vartheta _{-\hbar }$ and $\tau _a\circ \vartheta _{\hbar }=\vartheta _{\hbar }\circ \tau _a$. We are thus left to prove that $(\vartheta _\hbar \otimes \vartheta _\hbar )\circ \Delta =\Delta ^{\textrm{op}}\circ \vartheta _\hbar$. This is a simple consequence of the fact that the coproduct $\Delta$ of $Y_\hbar ({\mathfrak {g}})$ is given on $J(d_i h_i)$ by

$$\begin{aligned} \Delta (J(d_i h_i))=J(d_i h_i)\otimes 1 + 1\otimes J(d_i h_i) +\frac{\hbar }{2}[d_i h_i\otimes 1, \Omega ], \end{aligned}$$

where $\Omega \in ({\mathfrak {g}}\otimes {\mathfrak {g}})^{\mathfrak {g}}\subset Y_\hbar ({\mathfrak {g}})\otimes Y_\hbar ({\mathfrak {g}})$ is the Casimir tensor. $\square$

A.2 Now let ${\mathcal {R}}(u)\in Y_\hbar ({\mathfrak {g}})^{\otimes 2}[\![u^{-1}]\!]$ be the universal R-matrix of $Y_\hbar ({\mathfrak {g}})$, as in Sect. 7.5. More precisely, ${\mathcal {R}}(u)$ is the unique formal series in $1+u^{-1}Y_\hbar ({\mathfrak {g}})^{\otimes 2}[\![u^{-1}]\!]$ satisfying the intertwiner equation

$$\begin{aligned} \tau _u\otimes {\textbf{1}}\circ \Delta ^{\textrm{op}}(x)= {\mathcal {R}}(u) \cdot \tau _u\otimes {\textbf{1}}\circ \Delta (x) \cdot {\mathcal {R}}(u)^{-1} \quad \forall \; x\in Y_\hbar ({\mathfrak {g}})\end{aligned}$$

(A.1)

in $Y_\hbar ({\mathfrak {g}})^{\otimes 2}[u;u^{-1}]\!]$, in addition to the cabling identities

$$\begin{aligned} \begin{aligned} \Delta \otimes {\textbf{1}}({\mathcal {R}}(u))&= {\mathcal {R}}_{13}(u){\mathcal {R}}_{23}(u)\\ {\textbf{1}}\otimes \Delta ({\mathcal {R}}(u))&= {\mathcal {R}}_{13}(u){\mathcal {R}}_{12}(u) \end{aligned} \end{aligned}$$

(A.2)

in $Y_\hbar ({\mathfrak {g}})^{\otimes 3}[\![u^{-1}]\!]$. Here $\tau _u:Y_\hbar ({\mathfrak {g}})\rightarrow Y_\hbar ({\mathfrak {g}})[u]$ is the algebra homomorphism obtained replacing a by a formal variable u in the definition of the shift automorphism $\tau _a$ from Sect. 2.5. The existence and uniqueness of ${\mathcal {R}}(u)$ was established by Drinfeld in [12, Thm. 3] using cohomological techniques, though the proof was not published. A constructive proof based on the Gauss decomposition of ${\mathcal {R}}(u)$ may be found in Theorem 7.4 and “Appendix B” of [25].

In addition ${\mathcal {R}}(u)$ satisfies

$$\begin{aligned} {\mathcal {R}}(u)^{-1}={\mathcal {R}}_{21}(-u) \quad \text { and }\quad (\tau _a\otimes \tau _b){\mathcal {R}}(u)={\mathcal {R}}(u+a-b) \end{aligned}$$

(A.3)

for all $a,b\in {\mathbb {C}}$. In what follows, we write ${\mathcal {R}}^\hbar (u)$ for the universal R-matrix ${\mathcal {R}}(u)$ of $Y_\hbar ({\mathfrak {g}})$ to emphasize its dependence on $\hbar$.

Lemma

The universal R-matrices of $Y_\hbar ({\mathfrak {g}})$ and $Y_{-\hbar }({\mathfrak {g}})$ are related by

$$\begin{aligned} {\mathcal {R}}^\hbar (u)=(\vartheta _{\hbar }\otimes \vartheta _\hbar )\left( {\mathcal {R}}^{-\hbar }_{21}(-u)\right) . \end{aligned}$$

Proof

By the first relation of (A.3), this is equivalent to the identity

$$\begin{aligned} {\mathcal {R}}^\hbar (u)=(\vartheta _{\hbar }\otimes \vartheta _\hbar )\left( {\mathcal {R}}^{-\hbar }(u)\right) ^{-1}. \end{aligned}$$

It follows readily from the relations $(\vartheta _\hbar \otimes \vartheta _\hbar )\circ \Delta =\Delta ^{\textrm{op}}\circ \vartheta _\hbar$ and $\tau _a\circ \vartheta _{\hbar }=\vartheta _{\hbar }\circ \tau _a$ of Lemma A.1 that $(\vartheta _{\hbar }\otimes \vartheta _\hbar )\left( {\mathcal {R}}^{-\hbar }(u)\right) ^{-1}$ satisfies the intertwiner equation (A.1) and the cabling identities (A.2) satisfied by ${\mathcal {R}}^\hbar (u)$. As it also lies in $1+u^{-1}Y_\hbar ({\mathfrak {g}})^{\otimes 2}[\![u^{-1}]\!]$, it coincides with ${\mathcal {R}}^\hbar (u)$ by uniqueness. $\square$

A.3 Given a representation V of $Y_\hbar ({\mathfrak {g}})$, we shall denote the associated representation $\vartheta _\hbar ^*(V)$ of $Y_{-\hbar }({\mathfrak {g}})$ by $V^{\vartheta }$.

Lemma

Let $V\in {\text {Rep}}_{fd}(Y_\hbar ({\mathfrak {g}}))$ and fix $i\in {{{\textbf {I}}}}$ and $s\in {\mathbb {C}}$. Then:

• (1) V is a highest weight module if and only if $V^{\vartheta }$ is.

• (2) The $Y_\hbar ({\mathfrak {g}})$-module $L_{\varpi _i}(s)$ satisfies

  $$\begin{aligned} L_{\varpi _i}(s)^{\vartheta }\cong L_{\varpi _i}(s+\hbar \kappa -\hbar d_i) \end{aligned}$$

Proof

Since $\vartheta _\hbar ^{-1}=\vartheta _{-\hbar }$, to prove Part (1) it suffices to show that $V^\vartheta$ is a highest weight module if V is. To this end, let us suppose V is a highest weight module with highest weight vector $v\in V$. Let $\lambda \in {\mathfrak {h}}^*$ be the ${\mathfrak {g}}$-weight of v. Since the weight space $V_\lambda =(V^\vartheta )_\lambda$ is one-dimensional and preserved by each operator $\vartheta _\hbar (\xi _i(u))$, there is an ${{{\textbf {I}}}}$-tuple $(\mu _i(u))_{i\in {{{\textbf {I}}}}}$, with $\mu _i(u)\in 1+u^{-1}{\mathbb {C}}[\![u^{-1}]\!]$ for each i, such that

$$\begin{aligned} \vartheta _\hbar (\xi _i(u)) v=\mu _i(u) v \quad \forall \quad i\in {{{\textbf {I}}}}. \end{aligned}$$

Since $\vartheta _\hbar$ is an isomorphism, the $Y_{-\hbar }({\mathfrak {g}})$-module $V^\vartheta$ is generated by v. We are thus left to show that $\vartheta _\hbar (x_i^+(u))v=0$ for all $i\in {{{\textbf {I}}}}$. This follows from (8.2), for instance, and that $\vartheta _\hbar (\xi _i(u)) v=\mu _i(u) v$ and $\vartheta _\hbar (x_{i,0}^+)v=x_{i,0}^+v=0$ for all $i\in {{{\textbf {I}}}}$.

Consider now Part (2). Since $L_{\varpi _i}(s)^{\vartheta }$ is a finite-dimensional irreducible $Y_{-\hbar }({\mathfrak {g}})$-module, there is an ${{{\textbf {I}}}}$-tuple of monic polynomials $\underline{P}=(P_i(u))_{i\in {{{\textbf {I}}}}}$ such that $L(\underline{P})\cong L_{\varpi _i}(s)^{\vartheta }$. Letting $v\in L_{\varpi _i}(s)^{\vartheta }$ be a highest weight vector, we have $v\in L_{\varpi _i}(s)^{\vartheta }_{\lambda }$ with $\lambda =\sum _{j\in {{{\textbf {I}}}}} \deg P_j(u)\varpi _j$. On the other hand, the proof of Part (1) shows that v is a highest weight vector for $L_{\varpi _i}(s)$, and therefore has weight $\varpi _i$. It follows that $\deg P_j(u)=\delta _{ij}$, and thus that $L_{\varpi _i}(s)^{\vartheta }\cong L_{\varpi _i}(b)$ for some $b\in {\mathbb {C}}$. Moreover, since

$$\begin{aligned} \vartheta _h(\xi _i(u))v=\frac{u-b-\hbar d_i}{u-b}=1-\hbar d_i\sum _{r\ge 0} b^r u^{-r-1}, \end{aligned}$$

the value of b is determined by $\vartheta _h(\xi _{i1})v=d_i bv$. Since $\vartheta _h(\xi _{i1})=\xi _{i1}+\hbar \nu _i$, $\xi _{i1} v=d_i sv$ and $\nu _i v=d_i \kappa - d_i^2$, we can conclude that $b=s+\hbar \kappa -\hbar d_i$. $\square$

A.4 With the above machinery at our disposal, we are now prepared to prove the main result of this appendix.

Proposition

For each $i,j\in {{{\textbf {I}}}}$ and diagram automorphism $\omega$ of ${\mathfrak {g}}$, one has

$$\begin{aligned} {\textsf{d}}_{ij}(u)={\textsf{d}}_{\omega (i)\omega (j)}(u)\quad \text { and }\quad {\textsf{d}}_{ji}(u+\hbar d_i)={\textsf{d}}_{ij}(u+\hbar d_j). \end{aligned}$$

Proof

Recall from Sect. 2.6 that any diagram automorphism $\omega$ defines an automorphism of $Y_\hbar ({\mathfrak {g}})$, which is readily seen to be a homomorphism of coalgebras (see §2.9). As $\omega$ comutes with the shift automorphism $\tau _a$ for each $a\in {\mathbb {C}}$, it follows that $(\omega \otimes \omega ){\mathcal {R}}(u)$ satisfies the defining relations (A.1) and (A.2) of ${\mathcal {R}}(u)$, and so coincides with it by uniqueness. The first equality of the proposition now follows readily from the definition of ${\textsf{d}}_{ij}(u)$ (see Sects. 7.5 and 7.8) and the identification

$$\begin{aligned} (L_{\varpi _i}\otimes L_{\varpi _j}(s))^\omega \cong L_{\varpi _i}^\omega \otimes L_{\varpi _j}(s)^\omega \cong L_{\varpi _{\omega (i)}}\otimes L_{\varpi _{\omega (j)}}(s). \end{aligned}$$

Let us now turn to the second identity of the proposition. To make the dependencies on $\hbar$ clear, let us write ${\textsf{R}}_{L_{\varpi _j},L_{\varpi _i}}^\hbar (u)$ for the normalized R-matrix ${\textsf{R}}_{L_{\varpi _j},L_{\varpi _i}}(u)$ of $Y_\hbar ({\mathfrak {g}})$ defined in Sects. 7.5 and ${\textsf{d}}_{ji}^\hbar (u)$ for its denominator. It follows from the second identity of (A.3) and Lemmas A.1 and A.3 that one has the equality of normalized R-matrices

$$\begin{aligned} {\textsf{R}}_{L_{\varpi _j},L_{\varpi _i}}^{-\hbar }(u+\hbar d_i-\hbar d_j)= (1\,2)\circ {\textsf{R}}_{L_{\varpi _i},L_{\varpi _j}}^{\hbar }(-u) \circ (1\,2), \end{aligned}$$

and therefore ${\textsf{d}}_{ji}^{-\hbar }(u+\hbar d_i-\hbar d_j)=(-1)^{\deg {\textsf{d}}_{ij}^\hbar }{\textsf{d}}_{ij}^\hbar (-u)$. The desired equality now follows from Theorem 5.2 and Part (1) of Proposition 7.7, which give

$$\begin{aligned} {\textsf{Z}}({\textsf{d}}_{ji}^\hbar (u))\subset \sigma _i(L_{\varpi _j})+\hbar d_i\subset \frac{\hbar }{2}{\mathbb {Z}}\end{aligned}$$

and hence ${\textsf{d}}_{ji}^{-\hbar }(u)=(-1)^{\deg {\textsf{d}}_{ij}^\hbar }{\textsf{d}}_{ji}^{\hbar }(-u)$. $\square$