Affine Braids, Markov Traces and the Category $𝒪$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 21 December 2013

Preliminaries on Quantum Groups

Let $U_h𝔤$ be the Drinfel’d-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ Let us fix some notations. In particular, fix a triangular decomposition $$𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+},{𝔫}^{+}=\underset{\alpha >0}{⨁}{𝔤}_{\alpha },{𝔟}^{+}=𝔥\oplus {𝔫}^{+},$$ and let $W$ be the Weyl group of $𝔤\text{.}$ Let $⟨,⟩$ be the usual inner product on ${𝔥}^{*}$ so that, if $\alpha$ is a root, the corresponding reflection $s_{\alpha }$ in $W$ is given by $$s_{\alpha }\lambda =\lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha ,$$ where ${\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩}\text{.}$ The element $\rho =\frac{1}{2}\sum _{\alpha >0}\alpha$ is often viewed as an element of $𝔥$ by using the form $⟨,⟩$ to identify $𝔥$ and ${𝔥}^{*}\text{.}$ We shall use the conventions for quantum groups as in [Dri1990] and [LRa1977] so that $q=e^{h/2},$ $𝔥\subseteq U_h𝔤,$ and $U_h𝔤\cong U𝔤\left[\left[h\right]\right],$ as algebras. The quantum group has a triangular decomposition corresponding to that of $𝔤,$ $$U_h𝔤=U_h{𝔫}^{-}\otimes U_h𝔥\otimes U_h{𝔫}^{+}\text{and}{U}_h{𝔟}^{+}=U_h𝔥\otimes U_h{𝔫}^{+}\text{.}$$

The category $𝒪$

If $M$ is a $U_h𝔤$ module and $\lambda \in {𝔥}^{*}$ the $\lambda$ weight space of $M$ is $$\begin{array}{ccc}{M}_{\lambda }& =& \{m\in M\hspace{0.17em}|\hspace{0.17em}am=\lambda \left(a\right)m,\hspace{0.17em}\text{for all}\hspace{0.17em}a\in 𝔥\}\text{.}\end{array}$$ The category $𝒪$ is the category of $U_h𝔤$ modules $M$ such that

(a)	$M=\underset{\lambda \in {𝔥}^{*}}{⨁}{M}_{\lambda },$
(b)	For all $m\in M,$ $\text{dim}\left(U_h{𝔫}^{+}m\right)$ is finite,
(c)	$M$ is finitely generated as a $U_h𝔤$ module.

For $\mu \in {𝔥}^{*}$ let $M\left(\mu \right)$ be the Verma module of highest weight $\mu ,$ and let
$L\left(\mu \right)$ be the irreducible module of highest weight $\mu \text{.}$ The irreducible module $L\left(\mu \right)$ is the quotient of $M\left(\mu \right)$ by a maximal proper submodule and $M\left(\mu \right)=U_h𝔤{\otimes }_{U_h{𝔟}^{+}}ℂv_{\mu }^{+}$ where $ℂv_{\mu }^{+}$ is the one dimensional $U_h{𝔟}^{+}$ module spanned by a vector $v_{\mu }^{+}$ such that $a{v}_{\mu }^{+}=\mu \left(a\right)v_{\mu }^{+}$ for $a\in 𝔥$ and $U_h{𝔫}^{+}{v}_{\mu }^{+}=0\text{.}$ Every module $M\in 𝒪$ has a finite composition series with factors $L\left(\mu \right),$ $\mu \in {𝔥}^{*}\text{.}$ Each of the sets $$\left\{\left[L\left(\lambda \right)\right]\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {𝔥}^{*}\}\text{and}\left\{\left[M\left(\lambda \right)\right]\hspace{0.17em}\right|\hspace{0.17em}\lambda \in {𝔥}^{*}\}$$ (where $\left[M\right]$ denotes the isomorphism class of the module $M\text{)}$ are bases of the Grothendieck group of the category $𝒪\text{.}$

If $M$ is a $U_h𝔤$ module generated by a highest weight vector of weight $\lambda$ (i.e., a vector $v^{+}$ such that $a{v}^{+}=\lambda \left(a\right)v^{+}$ for $a\in 𝔥$ and $U_h{𝔫}^{+}{v}^{+}=0\text{)}$ then any element of the center $Z\left(U_h𝔤\right)$ acts on $M$ by a constant, $$zm={\chi }^{\lambda }\left(z\right)m,\text{for}\hspace{0.17em}z\in Z\left(U_h𝔤\right),\hspace{0.17em}m\in M,\hspace{0.17em}{\chi }^{\lambda }\left(z\right)\in ℂ\text{.}$$ For each $U_h𝔤$ module $M\in 𝒪$ let $$M^{\left[\lambda \right]}=\underset{\nu \in Q}{⨁}{M}_{\lambda +\nu }^{\left[\lambda \right]},\text{where}Q=\sum _{i=1}^n\mathbb{Z}{\alpha }_i,$$ ${\alpha }_1,\dots ,{\alpha }_n$ are the simple roots and $$\begin{array}{ccc}{M}_{\lambda +\nu }^{\left[\lambda \right]}& =& \{m\in M_{\lambda +\nu }\hspace{0.17em}|\hspace{0.17em}\text{there is}\hspace{0.17em}k\in {\mathbb{Z}}_{>0}\hspace{0.17em}\text{such that}\hspace{0.17em}{(z-{\chi }^{\lambda }\left(z\right))}^{k}m=0\hspace{0.17em}\text{for all}\hspace{0.17em}z\in Z\left(U_h𝔤\right)\}\text{.}\end{array}$$ Then $$M=\underset{\lambda }{⨁}{M}^{\left[\lambda \right]},$$ where the sum is over all integrally dominant weights $\lambda \in {𝔥}^{*}$ i.e., $\lambda \in {𝔥}^{*}$ such that $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\notin {\mathbb{Z}}_{<0}$ for all $\alpha \in R^{+}\text{.}$ To summarize, there is a decomposition of the category $𝒪,$ $$\begin{array}{cc}𝒪=\underset{\lambda }{⨁}{𝒪}^{\left[\lambda \right]},& \text{(2.1)}\end{array}$$ where the sum is over all integrally dominant weights $\lambda \in {𝔥}^{*}$ and ${𝒪}^{\left[\lambda \right]}$ is the full subcategory of modules $M\in 𝒪$ such that $M=M^{\left[\lambda \right]}\text{.}$ The Grothendieck group of the category ${𝒪}^{\left[\lambda \right]}$ has bases $$\begin{array}{cc}\left\{\left[L\left(\mu \right)\right]\hspace{0.17em}\right|\hspace{0.17em}\mu \in W^{\lambda }\circ \lambda \}\text{and}\{\left[M\left(\mu \right)\right]\mu \in W^{\lambda }\circ \lambda \}\text{.}& \text{(2.2)}\end{array}$$ where the integral Weyl group corresponding to $\lambda$ is $$\begin{array}{cc}{W}^{\lambda }=⟨s_{\alpha }\hspace{0.17em}|\hspace{0.17em}⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathbb{Z}⟩\text{and}w\circ \lambda =w(\lambda +\rho )-\rho ,\hspace{0.17em}w\in W,\lambda \in {𝔥}^{*}\text{.}& \text{(2.3)}\end{array}$$ defines the dot action of $W$ on ${𝔥}^{*}\text{.}$

Jantzen filtrations

Following the notations for the quantum group used in [LRa1977, §2], let $𝔥,$ $X_1,\dots ,X_r$ and $Y_1,\dots ,Y_r$ be the standard generators of the quantum group $U_h𝔤$ which satisfy the quantum Serre relations. The Cartan involution $\theta :U_h𝔤\to U_h𝔤$ is the algebra anti-involution defined by $$\begin{array}{cc}\theta \left(X_i\right)=Y_i,\theta \left(Y_i\right)=X_i,\text{and}\theta \left(a\right)=a,\text{for}\hspace{0.17em}a\in 𝔥\text{.}& \text{(2.4)}\end{array}$$ The Cartan involution $\theta$ is a coalgebra homomomorphism. A contravariant form on a $U_h𝔤$ module $M$ is a symmetric bilinear form $⟨,⟩:M\times M\to ℂ$ such that $$⟨u{m}_1,m_2⟩=⟨m_1,\theta \left(u\right)m_2⟩,\hspace{0.17em}u\in U_h𝔤,\hspace{0.17em}{m}_1,m_2\in M\text{.}$$

Fix $\lambda \in {𝔥}^{*}$ and $\delta \in {𝔥}^{*}$ such that $\lambda +t\delta$ is integrally dominant for all small positive real numbers $t\text{.}$ Consider $t$ as an indeterminate and consider the Verma module $$M(\lambda +t\delta )=U_h𝔤\left[t\right]{\otimes }_{U_h{𝔟}^{+}\left[t\right]}ℂv_{\lambda +t\delta }$$ as the module for $U_h𝔤\left[t\right]=ℂ\left[t\right]{\otimes }_{ℂ}{U}_h𝔤$ generated by a vector $v^{+}$ such that $a{v}^{+}=(\lambda +t\delta )\left(a\right)v^{+}$ for $a\in 𝔥$ and $U_h{𝔫}^{+}\left[t\right]v^{+}=0\text{.}$ There is a unique contravariant form ${⟨,⟩}_t:M(\lambda +t\delta )\times M(\lambda +t\delta )\to ℂ\left[t\right]$ such that ${⟨v^{+},v^{+}⟩}_t=1\text{.}$ Define $$M(\lambda +t\delta )\left(j\right)=\{m\in M(\lambda +t\delta )\hspace{0.17em}|\hspace{0.17em}{⟨m,n⟩}_t\in t^jℂ\left[t\right]\hspace{0.17em}\text{for all}\hspace{0.17em}n\in M(\lambda +t\delta )\}\text{.}$$ The “specialization of $M(\lambda +t\delta )\left(j\right)$ at $t=0\text{”}$ is $$M{\left(\lambda \right)}^{\left(j\right)}=\text{image of}\hspace{0.17em}M(\lambda +t\delta )\left(j\right)\hspace{0.17em}\text{in}\hspace{0.17em}M(\lambda +t\delta ){\oplus }_{\text{<mi>ℂ</mi><mrow><mo>[</mo><mi>t</mi><mo>]</mo></mrow>}}ℂ\left[t\right]/tℂ\left[t\right]$$ and the Jantzen filtration of $M\left(\lambda \right)$ is $$\begin{array}{cc}M\left(\lambda \right)=M{\left(\lambda \right)}^{\left(0\right)}\supseteq M{\left(\lambda \right)}^{\left(1\right)}\supseteq \cdots \text{.}& \text{(2.5)}\end{array}$$ By [Jan1980, Theorem 5.3], the Jantzen filtration is a filtration of $M\left(\lambda \right)$ by $U_h𝔤$ modules, the module $M{\left(\lambda \right)}^{\left(1\right)}$ is a maximal proper submodule of $M\left(\lambda \right)$ and each quotient $M{\left(\lambda \right)}^{\left(i\right)}M{\left(\lambda \right)}^{(i+1)}$ has a nondegenerate contravariant form. It is known [Bar1983] that the Jantzen filtration does not depend on the choice of $\delta \text{.}$ It is a deep theorem [BBe1993] that the quotients $M{\left(\lambda \right)}^{\left(i\right)}/M{\left(\lambda \right)}^{(i+1)}$ are semisimple and that if $w\in W^{\mu }$ and $y\in W^{\mu }$ are maximal length in their cosets $w{W}_{\mu +\rho }$ and $y{W}_{\mu +\rho },$ respectively, then the Kazhdan-Lusztig polynomial for $W^{\mu }$ is $$\begin{array}{cc}\sum _{j\ge 0}[M{(w\circ \mu )}^{\left(j\right)}/M{(w\circ \mu )}^{(j+1)}:L(y\circ \mu )]v^{\frac{1}{2}(\ell \left(y\right)-\ell \left(w\right)-j)}=P_{wy}\left(v\right),& \text{(2.6)}\end{array}$$ where $\ell$ is the length function on $W^{\mu }$ and $[M{(w\circ \mu )}^{\left(j\right)}/M{(w\circ \mu )}^{(j+1)}:L(y\circ \mu )]$ is the multiplicity of the simple module $L(y\circ \mu )$ in the $j\text{th}$ factor of the Jantzen filtration of $M\left(\lambda \right)\text{.}$

The BGG resolution

Not all simple modules $L\left(\lambda \right)$ in the category $𝒪$ have a BGG resolution. The general form of the BGG resolution given by Gabber and Joseph [GJo1981] is as follows.

Let $\mu \in {𝔥}^{*}$ be such that $-(\mu +\rho )$ is dominant and regular and let $W_J^{\mu }$ be a parabolic subgroup of the integral Weyl group $W^{\mu }\text{.}$ Let $w_0$ be the longest element of $W_J^{\mu }$ and fix $\nu =w_0\circ \mu \text{.}$ Define a resolution $$\begin{array}{cc}0⟶C_{\ell \left(w_0\right)}⟶\cdots ⟶C_2\stackrel{d_2}{⟶}{C}_1\stackrel{d_1}{⟶}{C}_0⟶L\left(\nu \right)⟶0& \text{(2.7)}\end{array}$$ of the simple module $L\left(\nu \right)$ by Verma modules by setting $$C_j=\underset{\ell \left(w\right)=j}{⨁}M(w\circ \nu ),$$ where the sum is over all $w\in W_J^{\mu }$ of length $j,$ and defining the map $$d_j:C_j\to C_{j-1},\text{by the matrix}{\left(d_j\right)}_{v,w}=\{\begin{array}{cc}{\epsilon }_{v,w}{\iota }_{v,w},& \text{if}\hspace{0.17em}v\to w,\\ 0,& \text{otherwise,}\end{array}$$ where $v\to w$ means that there is a (not necessarily simple) root $\alpha$ such that $w=s_{\alpha }v$ and $\ell \left(w\right)=\ell \left(v\right)-1,$ the maps ${\iota }_{v,w}$ are fixed choices of inclusions $${\iota }_{v,w}:M(v\circ \nu )\hookrightarrow M(w\circ \nu ),\text{and}{\epsilon }_{v,w}=\pm 1,$$ are fixed choices of signs such that $${\epsilon }_{u,v}{\epsilon }_{v,w}=-{\epsilon }_{u,v\prime }{\epsilon }_{v\prime ,w}\text{if}u\to v\to w,u\to v\prime \to w\text{and}v\ne v\prime \text{.}$$ Gabber and Joseph [GJo1981] prove that the sequence (2.7) is exact in this general setting. See [BGG1975] and [Dix1996, 7.8.14], for the original form of the BGG resolution. From the exactness of (2.7) it follows that if $-(\mu +\rho )$ is dominant and regular then, in the Grothendieck group of the category $𝒪,$ $$\begin{array}{cc}\left[L\left(\nu \right)\right]=\sum _{w\in W_J^{\mu }}{(-1)}^{\ell \left(w\right)}\left[M(w\circ \nu )\right],& \text{(2.8)}\end{array}$$ where $\nu =w_0\circ \mu$ and $w_0$ is the longest element of $W_J^{\mu }\text{.}$

${Ř}_{MN}$ matrices and the quantum Casimir $C_M$

Let $U_h𝔤$ be the Drinfeld-Jimbo quantum group corresponding to a finite dimensional complex semisimple Lie algebra $𝔤\text{.}$ There is an invertible element $ℛ=\sum a_i\otimes b_i$ in (a suitable completion of) $U_h𝔤\otimes U_h𝔤$ such that, for any two $U_h𝔤$ modules $M$ and $N,$ the map $$\begin{array}{cccc}{Ř}_{MN}:& M\otimes N& ⟶& N\otimes M\\ & m\otimes n& ⟼& \sum b_{i}n\otimes a_{i}m\end{array}\begin{array}{c} M ⊗ N N ⊗ M \end{array}$$ is a $U_h𝔤$ module isomorphism. There is also a quantum Casimir element $e^{-h\rho }u$ in the center of $U_h𝔤$ and, for a $U_h𝔤$ module $M$ we define $$\begin{array}{cccc}{C}_M:& M& ⟶& M\\ & m& ⟼& \left(e^{-h\rho }u\right)m\end{array}\begin{array}{c} M M CM \end{array}$$ In order to be consistent with the graphical calculus these operators should be written on the right. The elements $ℛ$ and $e^{-h\rho }u$ satisfy relations (see [LRa1977, (2.1-2.12)]), which imply that, for $U_h𝔤$ modules $M,N,P$ and a $U_h𝔤$ module isomorphism ${\tau }_M:M\to M,$ $$\begin{array}{cc}\begin{array}{ccc}\begin{array}{c} M ⊗ N N ⊗ M τM \end{array}& =& \begin{array}{c} M ⊗ N N ⊗ M τM \end{array}\\ {Ř}_{MN}({\text{id}}_N\otimes {\tau }_M)& =& ({\tau }_M\otimes {\text{id}}_N){Ř}_{MN},\end{array}& \text{(2.9)}\\ \begin{array}{ccccccc}\begin{array}{c} M⊗(N⊗P) (N⊗P)⊗M \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}& & \begin{array}{c} (M⊗N)⊗P P⊗(M⊗N) \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\\ {Ř}_{M,N\otimes P}& =& ({Ř}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {Ř}_{MP})& & {Ř}_{M\otimes N,P}& =& ({\text{id}}_M\otimes {Ř}_{NP})({Ř}_{MP}\otimes {\text{id}}_N),\end{array}& \text{(2.10)}\\ \begin{array}{ccc}{C}_{M\otimes N}& =& {\left({Ř}_{MN}{Ř}_{NM}\right)}^{-1}(C_M\otimes C_N)\text{.}\end{array}& \text{(2.11)}\end{array}$$ The relations (2.9) and (2.10) together imply the braid relation $$\begin{array}{cc}\begin{array}{ccc}\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}& =& \begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\\ ({Ř}_{MN}\otimes {\text{id}}_P)({\text{id}}_N\otimes {Ř}_{MP})({Ř}_{NP}\otimes {\text{id}}_M)& =& ({\text{id}}_M\otimes {Ř}_{NP})({Ř}_{MP}\otimes {\text{id}}_N)({\text{id}}_P\otimes {Ř}_{MN}),\end{array}& \text{(2.12)}\end{array}$$ If $M$ is a $U_h𝔤$ module generated by a highest weight vector $v^{+}$ of weight $\lambda$ then, by [Dri1990, Prop. 3.2], $$\begin{array}{cc}{C}_M=q^{-⟨\lambda ,\lambda +2\rho ⟩}{\text{id}}_M\text{.}& \text{(2.13)}\end{array}$$ Note that $⟨\lambda ,\lambda +2\rho ⟩=⟨\lambda +\rho ,\lambda +\rho ⟩-⟨\rho ,\rho ⟩$ are the eigenvalues of the classical Casimir operator [Dix1996, 7.8.5]. If $M$ is a finite dimensional $U_h𝔤$ module then $M$ is a direct sum of the irreducible modules $L\left(\lambda \right),$ $\lambda \in P^{+},$ and $$C_M=\underset{\lambda \in P^{+}}{⨁}{q}^{-⟨\lambda ,\lambda +2\rho ⟩}{P}_{\lambda },$$ where $P_{\lambda }:M\to M$ is the projection onto $M^{\left[\lambda \right]}$ in $M\text{.}$ From the relation (2.11) it follows that if $M=L\left(\mu \right),$ $N=L\left(\nu \right)$ are finite dimensional irreducible $U_h𝔤$ modules then ${Ř}_{MN}{Ř}_{NM}$ acts on the $\lambda$ isotypic component ${L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}$ of the decomposition $$\begin{array}{cc}L\left(\mu \right)\otimes L\left(\nu \right)=\underset{\lambda }{⨁}{L\left(\lambda \right)}^{\oplus c_{\mu \nu }^{\lambda }}& \text{(2.14)}\end{array}$$ by the constant $q^{⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩-⟨\nu ,\nu +2\rho ⟩}$ where $c_{\mu \nu }^{\lambda }$ are positive integers. Suppose that $M$ and $N$ are $U_h𝔤$ modules with contravariant forms ${⟨,⟩}_M$ and ${⟨,⟩}_N,$ respectively. Since the Cartan involution is a coalgebra homomorphism the form on $M\otimes N$ defined by $$\begin{array}{cc}⟨m_1\otimes n_1,m_2\otimes n_2⟩={⟨m_1,m_2⟩}_M{⟨n_1,n_2⟩}_N,& \text{(2.15)}\end{array}$$ for $m_1,m_2\in M,$ $n_1,n_2\in N,$ is also contravariant. If $\theta$ is the Cartan involution defined in (2.4) then a formula of Drinfeld [Dri1990, Prop. 4.2], states $$(\theta \otimes \theta )\left(ℛ\right)=\sum _i{b}_i\otimes a_i,$$ from which it follows that $$\begin{array}{ccc}⟨(m_1\otimes n_1){Ř}_{MN},n_2\otimes m_2⟩& =& \sum _i⟨(b_i\otimes a_i)(n_1\otimes m_1),n_2\otimes m_2⟩\\ & =& \sum _i⟨n_1\otimes m_1,(\theta \left(b_i\right)\otimes \theta \left(a_i\right))(n_2\otimes m_2)⟩\\ & =& \sum _i⟨n_1\otimes m_1,(a_i\otimes b_i)(n_2\otimes m_2)⟩\\ & =& \sum _i⟨m_1\otimes n_1,b_i{m}_2\otimes a_i{n}_2⟩\text{.}\end{array}$$ Thus $$\begin{array}{cc}⟨(m_1\otimes n_1){Ř}_{MN},n_2\otimes m_2⟩=⟨m_1\otimes n_1,(n_2\otimes m_2){Ř}_{NM}⟩\text{.}& \text{(2.16)}\end{array}$$

Notes and references

This is a typed version of the paper Affine Braids, Markov Traces and the Category $𝒪$ by Rosa Orellana and Arun Ram*.

*Research supported in part by National Science Foundation grant DMS-9971099, the National Security Agency and EPSRC grant GR K99015.

This paper is a slightly revised version of a preprint of 2001. We thank F. Goodman, A. Henderson and an anonymous referee for their very helpful comments on the original preprint.

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