Boundary diagram algebrasSeminar on Transformation groups & mathematical physicsA joint seminar of the Universities of Köln, Hamburg, Bochum, Bremen and DarmstadtUniversity of KölnNovember 19, 2005.

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 ${SO}_{2k}\text{.}$

$Generators:Ti= i i+1 , for 1≤i≤k,$ $X0= andXk=$ and relations $X0 T1 T2 Tk-2 Tk-1 Xk$ where $gi = gi, if {g}_{i} {g}_{j} gigjgi = gjgigj, if {g}_{i} {g}_{j} gigjgigj = gjgigjgi, if {g}_{i} {g}_{j}$

Let $Xεi= i andXλ= (Xε1)λ1 (Xε2)λ2⋯ (Xεk)λk$ for $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{k}{\epsilon }_{k}\text{.}$ Then $X={Xλ | λ∈L}, whereL=∑i=1k ℤεi,$ is an abelian subgroup of ${\stackrel{\sim }{ℬ}}_{k}\text{.}$

$Xεi= i ∼ i$ and $g1= andg2=$ 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 $- =(q-q-1)$ (4) The two pole Temperley-Lieb algebra is $ℂ{\stackrel{\sim }{ℬ}}_{k}$ with $defined by =q-$ and relations (3) and $= and =$ (5) The two pole BMW algebra is $ℂ{\stackrel{\sim }{ℬ}}_{k}$ with $defined by - =(q-q-1) ( - )$ and relations (4) and $= z-z-1q-q-1 +1, =z , =z-1$ (6) The two boundary BMW algebra ${\stackrel{\sim }{𝒵}}_{k}$ is $ℂ{ℬ}_{k}$ with relations (5) and $= = =z-1·$ and $b =ε1(b)· ,whereε1(b) ∈ℂ.$ (7) The two boundary Temperley-Lieb algebra ${\stackrel{\sim }{𝒯}}_{k}$ is $ℂ{\stackrel{\sim }{ℬ}}_{k}$ with relations $X02=(s-s-1) X0+1, Xk2=(t-t-1) Xk+1,$ and (3), (4) and (6).

Cyclotomic algebras

Let ${I}_{1}$ be an ideal of $ℂ{\stackrel{\sim }{ℬ}}_{1}$ such that $C1=ℂℬ1I1 is finite dimensional.$ The cyclotomic BMW algebra and the cyclotomic Hecke algebra are $CB∼k= B∼k⟨I1⟩ andCH∼k= H∼k⟨I1⟩,$ respectively. Let $I=ideal of CB∼k generated by$ Then $CB∼kI≃ CH∼kand CB∼k-1⊗CB∼k-2CB∼k-1 ⟶∼ I b1⊗b2 ⟼ {b}_{1} {b}_{2} 1 \cdots k$ 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 $ŘMN: M⊗N⟶N⊗M M ⊗ N N ⊗ M$ such that $M⊗(N⊗P) (N⊗P)⊗M = M ⊗ N ⊗ P P ⊗ N ⊗ M and (M⊗N)⊗P P⊗(M⊗N) = M ⊗ N ⊗ P P ⊗ N ⊗ M$ Let ${M}_{1},{M}_{2}$ and $V$ be $U\text{-modules.}$ Then $ℬ∼k acts on M1 ⊗V⊗k⊗M2$ since ${M}_{1} \otimes V \otimes V \otimes V \otimes V \otimes V \otimes V \otimes V \otimes {M}_{2} ∈EndU (M1⊗V⊗k⊗M2).$

Schur functors

Let $U$ be a quantum group: $U=U<0U0U>0$ with $U>0 generated by raising operators E1,…,En U<0 generated by lowering operators F1,…,Fn$ $𝒰0=ℂ [K1±1,…,Kn±1].$ Let $M$ be a $U\text{-module}$ and $λ:𝒰0⟶ℂ,a character of 𝒰0.$ A highest weight vector of weight $\lambda$ is $m\in M$ with $Kim=λ(Ki)m andEim=0.$ Then $(M1⊗V⊗k⊗M2)λ+= {highest weight vectors of weight λ in M1⊗V⊗k⊗M2}$ is a ${\stackrel{\sim }{ℬ}}_{k}\text{-module.}$

 $(M1⊗V⊗k⊗M2)λ+= HomU(M(λ),M1⊗V⊗k⊗M2)$ where $M\left(\lambda \right)$ is a Verma module, and the general Schur functors are $Exti(M(λ),M1⊗V⊗k⊗M2).$

Examples

 (1) If $U={U}_{q}{𝔤𝔩}_{n}$ and $V=L\left({\epsilon }_{1}\right)$ $\text{(}V$ simple, $\text{dim} V=n\text{)}$ 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 $V=L\left({\omega }_{1}\right)$ $\left(V$ simple, $\text{dim} V=n\text{),}$ ${M}_{1}\in {𝒪}^{\left[\lambda \right]},$ ${M}_{2}\in {𝒪}^{\left[\mu \right]}$ $\text{(}Z\left(U\right)$ 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, $\text{dim} V=2\text{),}$ ${M}_{1}\in {𝒪}^{\left[\lambda \right]},$ ${M}_{2}\in {𝒪}^{\left[\mu \right]}$ $\text{(}Z\left(U\right)$ 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.