Boundary diagram algebras

Seminar on Transformation groups & mathematical physics

A joint seminar of the Universities of Köln, Hamburg, Bochum, Bremen and Darmstadt

University of Köln
November 19, 2005.

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

Last updated: 22 November 2014

Abstract

Abstract. This talk is about diagram algebras which come from the two-boundary braid group (braids with two poles). This is a generalization of recent work (from statistical mechanics) on two-boundary Temperley-Lieb algebras. The generalized setting naturally includes two boundary Hecke algebras and two-boundary BMW algebras. These algebras are like affine Hecke algebras (of type A) and affine BMW algebras except with two poles.

The affine braid group ${\stackrel{\sim }{ℬ}}_k$ of type ${SO}_{2k}\text{.}$

$$\text{Generators:}{T}_i=\begin{array}{c} i i+1 \end{array},\text{for}\hspace{0.17em}1\le i\le k,$$ $$X_0=\begin{array}{c} \end{array}\text{and}{X}_k=\begin{array}{c} \end{array}$$ and relations $$\begin{array}{c} X0 T1 T2 Tk-2 Tk-1 Xk \end{array}$$ where $$\begin{array}{rcll}{\displaystyle g_i}& =& {\displaystyle g_i,}& \text{if}\begin{array}{c} gi gj \end{array}\\ {\displaystyle g_i{g}_j{g}_i}& =& {\displaystyle g_j{g}_i{g}_j,}& \text{if}\begin{array}{c} gi gj \end{array}\\ {\displaystyle g_i{g}_j{g}_i{g}_j}& =& {\displaystyle g_j{g}_i{g}_j{g}_i,}& \text{if}\begin{array}{c} gi gj \end{array}\end{array}$$

Let $$X^{{\epsilon }_i}=\begin{array}{c} i \end{array}\text{and}{X}^{\lambda }={\left(X^{{\epsilon }_1}\right)}^{{\lambda }_1}{\left(X^{{\epsilon }_2}\right)}^{{\lambda }_2}\cdots {\left(X^{{\epsilon }_k}\right)}^{{\lambda }_k}$$ for $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_k{\epsilon }_k\text{.}$ Then $$X=\left\{X^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in L\},\text{where}L=\sum _{i=1}^k\mathbb{Z}{\epsilon }_i,$$ is an abelian subgroup of ${\stackrel{\sim }{ℬ}}_k\text{.}$

  $$X^{{\epsilon }_i}=\begin{array}{c} i \end{array}\sim \begin{array}{c} i \end{array}$$ and $$g_1=\begin{array}{c} \end{array}\text{and}{g}_2=\begin{array}{c} \end{array}$$ generate a free group on two generators.

Quotients

(1)	The affine braid group of type ${GL}_k$ is ${\stackrel{\sim }{ℬ}}_k$ with $X_k=1\text{.}$
(2)	The braid group is ${\stackrel{\sim }{ℬ}}_k$ with $X_0=1$ and $X_k=1\text{.}$
(3)	The two pole Hecke algebra ${\stackrel{\sim }{H}}_k$ is $ℂ{\stackrel{\sim }{ℬ}}_k$ with $$\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})\begin{array}{c} \end{array}$$
(4)	The two pole Temperley-Lieb algebra is $ℂ{\stackrel{\sim }{ℬ}}_k$ with $$\begin{array}{c} \end{array}\text{defined by}\begin{array}{c} \end{array}=q-\begin{array}{c} \end{array}$$ and relations (3) and $$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}\text{and}\begin{array}{c} \end{array}=\begin{array}{c} \end{array}$$
(5)	The two pole BMW algebra is $ℂ{\stackrel{\sim }{ℬ}}_k$ with $$\begin{array}{c} \end{array}\text{defined by}\begin{array}{c} \end{array}-\begin{array}{c} \end{array}=(q-q^{-1})(\begin{array}{c} \end{array}-\begin{array}{c} \end{array})$$ and relations (4) and $$\begin{array}{c} \end{array}=\frac{z-z^{-1}}{q-q^{-1}}+1,\begin{array}{c} \end{array}=z\begin{array}{c} \end{array},\begin{array}{c} \end{array}=z^{-1}\begin{array}{c} \end{array}$$
(6)	The two boundary BMW algebra ${\stackrel{\sim }{𝒵}}_k$ is $ℂ{ℬ}_k$ with relations (5) and $$\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=\begin{array}{c} \end{array}=z^{-1}·\begin{array}{c} \end{array}$$ and $$\begin{array}{c} b \end{array}={\epsilon }_1\left(b\right)·\begin{array}{c} \end{array},\text{where}{\epsilon }_1\left(b\right)\in ℂ\text{.}$$
(7)	The two boundary Temperley-Lieb algebra ${\stackrel{\sim }{𝒯}}_k$ is $ℂ{\stackrel{\sim }{ℬ}}_k$ with relations $$X_0^2=(s-s-1)X_0+1,X_k^2=(t-t^{-1})X_k+1,$$ and (3), (4) and (6).

Cyclotomic algebras

Let $I_1$ be an ideal of $ℂ{\stackrel{\sim }{ℬ}}_1$ such that $$C_1=\frac{ℂ{ℬ}_1}{I_1}\text{is finite dimensional.}$$ The cyclotomic BMW algebra and the cyclotomic Hecke algebra are $$C{\stackrel{\sim }{B}}_k=\frac{{\stackrel{\sim }{B}}_k}{⟨I_1⟩}\text{and}C{\stackrel{\sim }{H}}_k=\frac{{\stackrel{\sim }{H}}_k}{⟨I_1⟩},$$ respectively. Let $$I=\text{ideal of}\hspace{0.17em}C{\stackrel{\sim }{B}}_k\hspace{0.17em}\text{generated by}\begin{array}{c} \end{array}$$ Then $$\frac{C{\stackrel{\sim }{B}}_k}{I}\simeq C{\stackrel{\sim }{H}}_k\text{and}\begin{array}{ccc}C{\stackrel{\sim }{B}}_{k-1}{\otimes }_{C{\stackrel{\sim }{B}}_{k-2}}C{\stackrel{\sim }{B}}_{k-1}& \stackrel{\sim }{⟶}& I\\ b_1\otimes b_2& ⟼& \begin{array}{c} b1 b2 1 ⋯ k \end{array}\end{array}$$ Hence any simple $C{\stackrel{\sim }{B}}_k\text{-module}$ is

(a)	an $C{\stackrel{\sim }{H}}_k\text{-module}$ or
(b)	an inflation of a $C{\stackrel{\sim }{B}}_{k-2}\text{-module.}$

All finite dimensional simple ${\stackrel{\sim }{B}}_k\text{-modules}$ appear this way.

$ℛ\text{-matrices}$

Let $U$ be a quasitriangular Hopf algebra:

(1)

If $M$ and $N$ are $U\text{-modules}$ then $M\otimes N$ is a $U\text{-module,}$

(2)

There are natural $U\text{-module}$ isomorphisms $${Ř}_{MN}:M\otimes N⟶N\otimes M\begin{array}{c} M ⊗ N N ⊗ M \end{array}$$ such that $$\begin{array}{c} M⊗(N⊗P) (N⊗P)⊗M \end{array}=\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}\text{and}\begin{array}{c} (M⊗N)⊗P P⊗(M⊗N) \end{array}=\begin{array}{c} M ⊗ N ⊗ P P ⊗ N ⊗ M \end{array}$$ Let $M_1,M_2$ and $V$ be $U\text{-modules.}$ Then $${\stackrel{\sim }{ℬ}}_k\hspace{0.17em}\text{acts on}\hspace{0.17em}{M}_1\otimes V^{\otimes k}\otimes M_2$$ since $$\begin{array}{c} M1 ⊗ V ⊗ V ⊗ V ⊗ V ⊗ V ⊗ V ⊗ V ⊗ M2 \end{array}\in {\text{End}}_U(M_1\otimes V^{\otimes k}\otimes M_2)\text{.}$$

Schur functors

Let $U$ be a quantum group: $$U=U_{<0}{U}_0{U}_{>0}$$ with $$\begin{array}{c}{U}_{>0}\hspace{0.17em}\text{generated by raising operators}\hspace{0.17em}{E}_1,\dots ,E_n\\ U_{<0}\hspace{0.17em}\text{generated by lowering operators}\hspace{0.17em}{F}_1,\dots ,F_n\end{array}$$ $${𝒰}^0=ℂ[K_1^{\pm 1},\dots ,K_n^{\pm 1}]\text{.}$$ Let $M$ be a $U\text{-module}$ and $$\lambda :{𝒰}^0⟶ℂ,\text{a character of}\hspace{0.17em}{𝒰}^0\text{.}$$ A highest weight vector of weight $\lambda$ is $m\in M$ with $$K_{i}m=\lambda \left(K_i\right)m\text{and}{E}_{i}m=0\text{.}$$ Then is a ${\stackrel{\sim }{ℬ}}_k\text{-module.}$

$\hspace{0.17em}$ $${(M_1\otimes V^{\otimes k}\otimes M_2)}_{\lambda }^{+}={\text{Hom}}_U(M\left(\lambda \right),M_1\otimes V^{\otimes k}\otimes M_2)$$ where $M\left(\lambda \right)$ is a Verma module, and the general Schur functors are $${\text{Ext}}^i(M\left(\lambda \right),M_1\otimes V^{\otimes k}\otimes M_2)\text{.}$$

Examples

(1)	If $U=U_q{𝔤𝔩}_n$ and $V=L\left({\epsilon }_1\right)$ $\text{(}V$ simple, $\text{dim}\hspace{0.17em}V=n\text{)}$ then $M_1\otimes V^{\otimes k}\otimes M_2$ is an ${\stackrel{\sim }{H}}_k\text{-module.}$
(2)	If $U=U_1{𝔰𝔬}_n$ or $U=U_q{𝔰𝔭}_n$ and $V=L\left({\omega }_1\right)$ $(V$ simple, $\text{dim}\hspace{0.17em}V=n\text{),}$ $M_1\in {𝒪}^{\left[\lambda \right]},$ $M_2\in {𝒪}^{\left[\mu \right]}$ $\text{(}Z\left(U\right)$ acts by constants) then $M_1\otimes V^{\otimes k}\otimes M_2$ is a ${\stackrel{\sim }{𝒵}}_k$ module.
(3)	If $U=U_1{𝔰𝔩}_2,$ $V=L\left({\omega }_1\right)$ $\text{(}V$ simple, $\text{dim}\hspace{0.17em}V=2\text{),}$ $M_1\in {𝒪}^{\left[\lambda \right]},$ $M_2\in {𝒪}^{\left[\mu \right]}$ $\text{(}Z\left(U\right)$ acts by constants) then $M_1\otimes V^{\otimes k}\otimes M_2$ is a ${\stackrel{\sim }{𝒯}}_k$ module.

Notes and References

These are a typed copy of Boundary diagram algebras given at the Seminar on Transformation groups & mathematical physics, A joint seminar of the Universities of Köln, Hamburg, Bochum, Bremen and Darmstadt, University of Köln, November 19, 2005.

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