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

Last updated: 22 November 2014


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 k of type SO2k.

Generators:Ti= i i+1 , for1ik, X0= andXk= and relations X0 T1 T2 Tk-2 Tk-1 Xk where gi = gi, if gi gj gigjgi = gjgigj, if gi gj gigjgigj = gjgigjgi, if gi gj

Let Xεi= i andXλ= (Xε1)λ1 (Xε2)λ2 (Xεk)λk for λ=λ1ε1++λkεk. Then X={Xλ|λL}, whereL=i=1k εi, is an abelian subgroup of k.

  Xεi= i i and g1= andg2= generate a free group on two generators.


(1) The affine braid group of type GLk is k with Xk=1.
(2) The braid group is k with X0=1 and Xk=1.
(3) The two pole Hecke algebra Hk is k with - =(q-q-1)
(4) The two pole Temperley-Lieb algebra is k with defined by =q- and relations (3) and = and =
(5) The two pole BMW algebra is 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 𝒵k is k with relations (5) and = = =z-1· and b =ε1(b)· ,whereε1(b) .
(7) The two boundary Temperley-Lieb algebra 𝒯k is k with relations X02=(s-s-1) X0+1, Xk2=(t-t-1) Xk+1, and (3), (4) and (6).

Cyclotomic algebras

Let I1 be an ideal of 1 such that C1=1I1 is finite dimensional. The cyclotomic BMW algebra and the cyclotomic Hecke algebra are CBk= BkI1 andCHk= HkI1, respectively. Let I=ideal ofCBk generated by Then CBkI CHkand CBk-1CBk-2CBk-1 I b1b2 b1 b2 1 k Hence any simple CBk-module is

(a) an CHk-module or
(b) an inflation of a CBk-2-module.
All finite dimensional simple Bk-modules appear this way.


Let U be a quasitriangular Hopf algebra:

(1) If M and N are U-modules then MN is a U-module,
(2) There are natural U-module isomorphisms ŘMN: MNNM M N N M such that M(NP) (NP)M = M N P P N M and (MN)P P(MN) = M N P P N M Let M1,M2 and V be U-modules. Then kacts onM1 VkM2 since M1 V V V V V V V M2 EndU (M1VkM2).

Schur functors

Let U be a quantum group: U=U<0U0U>0 with U>0generated by raising operatorsE1,,En U<0generated by lowering operatorsF1,,Fn 𝒰0= [K1±1,,Kn±1]. Let M be a U-module and λ:𝒰0,a character of𝒰0. A highest weight vector of weight λ is mM with Kim=λ(Ki)m andEim=0. Then (M1VkM2)λ+= {highest weight vectors of weightλinM1VkM2} is a k-module.

(M1VkM2)λ+= HomU(M(λ),M1VkM2) where M(λ) is a Verma module, and the general Schur functors are Exti(M(λ),M1VkM2).


(1) If U=Uq𝔤𝔩n and V=L(ε1) (V simple, dimV=n) then M1VkM2 is an Hk-module.
(2) If U=U1𝔰𝔬n or U=Uq𝔰𝔭n and V=L(ω1) (V simple, dimV=n), M1𝒪[λ], M2𝒪[μ] (Z(U) acts by constants) then M1VkM2 is a 𝒵k module.
(3) If U=U1𝔰𝔩2, V=L(ω1) (V simple, dimV=2), M1𝒪[λ], M2𝒪[μ] (Z(U) acts by constants) then M1VkM2 is a 𝒯k 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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