## The affine BMW algebra ${W}_{k}$

Last updates: 15 April 2011

## The affine BMW algebra ${W}_{k}$

Let $C$ be a commutative ring and let $C{B}_{k}$ be the group algebra of the affine braid group. Fix constants

 $q,z\in C$     and     ${Z}_{0}^{\left(\ell \right)}\in C\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{.5em}{0ex}}\ell \in ℤ$
with $q$ and $z$ invertible. Let ${Y}_{i}=z{X}^{{\epsilon }_{i}}$ so that
 . (Ydf)
In the affine braid group
 ${T}_{i}{Y}_{i}{Y}_{i+1}={Y}_{i}{Y}_{i+1}{T}_{i}$. (YTc)

Assume $\left(q-{q}^{-1}\right)$ is invertible in $C$ and define ${E}_{i}$ in the group algebra of the affine braid group by

 ${T}_{i}{Y}_{i}={Y}_{i+1}{T}_{i}-\left(q-{q}^{-1}\right){Y}_{i+1}\left(1-{E}_{i}\right)$. (Edb)

The affine BMW algebra ${W}_{k}$ is the quotient of the group algebra $C{B}_{k}$ of the affine braid group ${B}_{k}$ by the relations

 ${E}_{i}{T}_{i}^{±1}={T}_{i}^{±1}{E}_{i}={z}^{\mp 1}{E}_{i},\phantom{\rule{2em}{0ex}}{E}_{i}{T}_{i-1}^{±1}{E}_{i}={E}_{i}{T}_{i+1}^{±1}{E}_{i}={z}^{±1}{E}_{i}$, (BW1)
 ${E}_{1}{Y}_{1}^{\ell }{E}_{1}={Z}_{0}^{\left(\ell \right)}{E}_{1},\phantom{\rule{2em}{0ex}}{E}_{i}{Y}_{i}{Y}_{i+1}={E}_{i}={Y}_{i}{Y}_{i+1}{E}_{i}$. (BW2)

Since ${Y}_{i+1}^{-1}\left({T}_{i}{Y}_{i}\right){Y}_{i+1}={Y}_{i+1}^{-1}{Y}_{i}{Y}_{i+1}{T}_{i}={Y}_{i}{T}_{i}$, conjugating (Edb) by ${Y}_{i+1}^{-1}$ gives

 ${Y}_{i}{T}_{i}={T}_{i}{Y}_{i+1}-\left(q-{q}^{-1}\right)\left(1-{E}_{i}\right){Y}_{i+1}$.
Left multiplying (Edb) by ${Y}_{i+1}^{-1}$ and using the second identity in (Ydf) shows that (Edb) is equivalent to ${T}_{i}-{T}_{i}^{-1}=\left(q-{q}^{-1}\right)\left(1-{E}_{i}\right)$, so that
 ${E}_{i}=1-\frac{{T}_{i}-{T}_{i}^{-1}}{q-{q}^{-1}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{T}_{i}{T}_{i+1}{E}_{i}{T}_{i+1}^{-1}{T}_{i}^{-1}={E}_{i+1}$. (BW4)
Multiply the second relation in (BW4) on the left and the right by ${E}_{i}$, and then use the relations in (BW1) to get $Ei Ei+1 Ei = Ei Ti Ti+1 Ei Ti+1 -1 Ti -1 Ei = Ei Ti+1 Ei Ti+1 -1 Ei = zEi Ti+1 -1 Ei = Ei ,$ so that
 ${E}_{i}{E}_{i±1}{E}_{i}={E}_{i}.\phantom{\rule{2em}{0ex}}\text{Note that}\phantom{\rule{2em}{0ex}}{E}_{i}^{2}=\left(1+\frac{z-{z}^{-1}}{q-{q}^{-1}}\right){E}_{i}$ (BW5)
is obtained by multiplying the first equation in (BW4) by ${E}_{i}$ and using (BW2). Thus, from the first relation in (BW2),
 ${Z}_{0}^{\left(0\right)}=1+\frac{z-{z}^{-1}}{q-{q}^{-1}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}+{q}^{-1}\right)\left({T}_{i}-q\right)=0$, (BWF)
since $\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}+{q}^{-1}\right)\left({T}_{i}-q\right){T}_{i}^{-1}=\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}^{2}-\left(q-{q}^{-1}\right){T}_{i}-1\right){T}_{i}^{-1}=\left({T}_{i}-{z}^{-1}\right)\left({T}_{i}-{T}_{i}^{-1}-\left(q-{q}^{-1}\right)\right)=\left({T}_{i}-{z}^{-1}\right)\left(q-{q}^{-1}\right)\left(-{E}_{i}\right)=-\left({z}^{-1}-{z}^{-1}\right)\left(q-{q}^{-1}\right)=0$. The relations
 ${E}_{i+1}{E}_{i}={E}_{i+1}{T}_{i}{T}_{i+1},\phantom{\rule{2em}{0ex}}{E}_{i}{E}_{i+1}={T}_{i+1}^{-1}{T}_{i}^{-1}{E}_{i+1}$, (BW6) ${T}_{i}{E}_{i+1}{E}_{i}={T}_{i+1}^{-1}{E}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{E}_{i+1}{E}_{i}{T}_{i+1}={E}_{i+1}{T}_{i}^{-1}$, (BW7)
follow from the computations $Ei+1 Ti Ti+1 = z ( Ei+1 Ti-1 Ei+1 ) Ti Ti+1 = z ( z-1 Ei+1 Ti+1 -1 ) Ti-1 Ei+1 Ti Ti+1 = Ei+1 Ei ,$ $Ti+1 -1 Ti -1 Ei+1 = Ti+1 -1 Ti -1 ( z-1 Ei+1 Ti Ei+1 ) = Ti+1 -1 Ti-1 z-1 Ei+1 Ti z Ti+1 Ei+1 = Ei Ei+1 ,$ $Ei+1 Ei Ti+1 = Ei+1 Ti+1 -1 zEi Ti+1 = z Ei+1 Ti Ei+1 Ti -1 = z z-1 Ei+1 Ti-1 = Ei+1 Ti-1 .$

A consequence (see (???)) of the defining relations of ${W}_{k}$ is the equation

 $\left({Z}_{0}^{-}-\frac{z}{q-{q}^{-1}}-\frac{{u}^{2}}{{u}^{2}-1}\right)\left({Z}_{0}^{+}+\frac{{z}^{-1}}{q-{q}^{-1}}-\frac{{u}^{2}}{{u}^{2}-1}\right){E}_{1}=\frac{-\left({u}^{2}-{q}^{2}\right)\left({u}^{2}-{q}^{-2}\right)}{\left({u}^{2}-1\right){\left(q-{q}^{-1}\right)}^{2}}{E}_{1},$
where ${Z}_{0}^{+}$ and ${Z}_{0}^{-}$ are the generating functions $Z0+ = ∑ℓ∈ℤ ≥0 Z0(ℓ) u-ℓ and Z0+ = ∑ℓ∈ℤ ≤0 Z0(ℓ) u-ℓ .$ This means that, unless the parameters ${Z}_{0}^{\left(\ell \right)}are chosen carefully, it is likely that{E}_{1}=0in{W}_{k}.$

From the point of view of the Schur-Weyl duality for the degenerate affine BMW algebra (see [OR]) the natural choice of base ring is the center of the quantum group corresponding to the orthogonal or symplectic Lie algebra, which, by the (quantum version) of the Harish-Chandra isomorphism, is isomorphic to the subring of symmetric Laurent polynomials given by $C= { z∈ ℂ[ L1±1 ,…, Lr±1 ] Sr | z( L1,…,Lr ) = z( L1-1, L2,…,Lr ) },$ where the symmetric group ${S}_{r}$ acts by permuting the variables ${L}_{1},\dots ,{L}_{r}$. Here the constants ${Z}_{0}^{\left(\ell \right)}\in C$ are given, explicitly, by setting the generating functions ${Z}_{0}^{+}$ and ${Z}_{0}^{-}$ equal, up to a normalization, to

 $\prod _{i=1}^{r}\frac{\left(u-q{L}_{i}\right)}{\left(u-{q}^{-1}{L}_{i}\right)}\cdot \frac{\left(u-q{L}_{i}^{-1}\right)}{\left(u-{q}^{-1}{L}_{i}^{-1}\right)}$     and    $\prod _{i=1}^{r}\frac{\left(u-{q}^{-1}{L}_{i}\right)}{\left(u-q{L}_{i}\right)}\cdot \frac{\left(u-{q}^{-1}{L}_{i}^{-1}\right)}{\left(u-q{L}_{i}^{-1}\right)}$,
This choice of $C$ and the ${Z}_{0}^{\left(\ell \right)}$ are the universal admissible parameters for ${W}_{k}$.

### Quotients of ${W}_{k}$

The affine Hecke algebra ${H}_{k}$ is the affine BMW algebra ${W}_{k}$ with the additional relations

 . (Ah)
Fix ${b}_{1},\dots ,{b}_{r}\in ℂ$. The cyclotomic BMW algebra ${W}_{r,k}\left({b}_{1}\dots {b}_{r}\right)$ is the affine BMW algebra ${W}_{k}$ with the additional relation
 $\left({Y}_{1}-{b}_{1}\right)\dots \left({Y}_{1}-{b}_{r}\right)=0.$ (Cyc)
The cyclotomic Hecke algebra ${H}_{r,k}\left({b}_{1}\dots {b}_{r}\right)$ is the affine Hecke algebra ${H}_{k}$ with additional relation (Cyc).

A consequence of the relation (Cyc) in ${W}_{r,k}\left({b}_{1}\dots {b}_{r}\right)$ is

 $\left({Z}_{0}^{+}+\frac{{z}^{-1}}{q-{q}^{-1}}-\frac{{u}^{2}}{{u}^{2}-1}\right){E}_{1}=\left(\frac{z}{q-{q}^{-1}}-\frac{uz}{{u}^{2}-1}\right)\left(\prod _{j=1}^{r}\frac{u-{b}_{j}^{-1}}{u-{b}_{j}}\right){E}_{1}$. (2.52)
This equation makes the data of the values ${b}_{i}$ almost equivalent to the data of the ${Z}_{0}^{\left(\ell \right)}$.

## Notes and References

This section is based on forthcoming joint work with Z. Daugherty and R. Virk [DRV].

## Bibliography

[AMR] S. Ariki, A. Mathas and H. Rui, Cyclotomic Nazarov Wenzl algebras, Nagoya Math. J. 182, (2006), 47-134. MR2235339 (2007d:20005)

[DRV] Z. Daugherty, A. Ram, and R. Virk, Affine and graded BMW algebras, in preparation.

[Naz] M. Nazarov, Young's orthogonal form for Brauer's centralizer algebra, J. Algebra 182 (1996), no. 3, 664--693. MR1398116 (97m:20057)

[OR] R. Orellana and A. Ram, Affine braids, Markov traces and the category $𝒪$, Algebraic groups and homogeneous spaces, 423-473, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007. MR2348913 (2008m:17034)