## The degenerate affine BMW algebra ${𝒲}_{k}$ and its quotients

Let $C$ be a commutative ring and let ${𝔹}_{k}$ be the degenerate affine braid algebra over $C$ as defined in The degenerate affine braid algebra. Define ${e}_{i}$ in the degenerate affine braid algebra by

 ${t}_{{s}_{i}}{y}_{i}={y}_{i+1}{t}_{{s}_{i}}-\left(1-{e}_{i}\right),\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i=1,\dots ,k-1,$ (edb)
so that, with ${t}_{i,i+1}$ as in (gba4),
 ${t}_{i,i+1}{t}_{{s}_{i}}=1-{e}_{i}.$ (edb′)

Fix constants

 $ϵ=±1\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{z}_{0}^{\left(\ell \right)}\in C$,    for $\ell \in {ℤ}_{\ge 0}$.
The degenerate affine Birman-Wenzl-Murakami (BMW) algebra ${𝒲}_{k}$ (with parameters $ϵ$ and ${z}_{0}^{\left(\ell \right)}$) is the quotient of the degenerate affine braid algebra ${𝔹}_{k}$ by the relations
 ${e}_{i}{t}_{{s}_{i}}={t}_{{s}_{i}}{e}_{i}=ϵ{e}_{i},\phantom{\rule{2em}{0ex}}{e}_{i}{t}_{{s}_{i-1}}{e}_{i}={e}_{i}{t}_{{s}_{i+1}}{e}_{i}=ϵ{e}_{i},$ (dbw1)
 $e1 y1𝓁 e1 = z1𝓁 e1, ei yi + yi+1 =0= yi + yi+1 ei.$ (dbw2)

Conjugating (edb) by ${t}_{{s}_{i}}$ and using the first relation in (dbw1) gives

 ${y}_{i}{t}_{{s}_{i}}={t}_{{s}_{i}}{y}_{i+1}-\left(1-{e}_{i}\right).$ (dbw3)

Then, by (edb′) and (gba4),

 ${t}_{i,i+1}={t}_{{s}_{i}}-ϵ{e}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{e}_{i+1}={t}_{{s}_{i}}{t}_{{s}_{i+1}}{e}_{i}{t}_{{s}_{i+1}}{t}_{{s}_{i}}.$ (dbw4)
Multiply the second relation in (dbw4) on the left and the right by ${e}_{i}$ and then use the relations in (dbw1) to get $ei ei+1 ei = ei tsi tsi+1 ei tsi+1 tsi tsi ei = ei tsi+1 ei tsi+1 ei = ϵei tsi+1 ei = ei,$ so that
 ${e}_{i}{e}_{i±1}{e}_{i}={e}_{i}.\phantom{\rule{3em}{0ex}}\text{Note that}\phantom{\rule{1em}{0ex}}{e}_{i}^{2}={z}_{1}^{\left(0\right)}{e}_{i}$ (dbw5)
is a special case of the first identity in (dbw2). The relations
 ${e}_{i+1}{e}_{i}={e}_{i+1}{t}_{{s}_{i}}{t}_{{s}_{i+1}},\phantom{\rule{2em}{0ex}}{e}_{i}{e}_{i+1}={t}_{{s}_{i+1}}{t}_{{s}_{i}}{e}_{i+1},$ (dbw6)
 ${t}_{{s}_{i}}{e}_{i+1}{e}_{i}={t}_{{s}_{i+1}}{e}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{e}_{i+1}{e}_{i}{t}_{{s}_{i+1}}={e}_{i+1}{t}_{{s}_{i}}$ (dbw7)
result from $ei+1 tsi tsi+1 = ϵ ei+1 tsi ei+1 tsi tsi+1 = ei+1 tsi+1 tsi ei+1 tsi tsi+1 = ei+1 ei, tsi+1 tsi ei+1 = ϵ tsi+1 tsi ei+1 tsi ei+1 = tsi+1 tsi ei+1 tsi tsi+1 ei+1 = ei ei+1, tsi ei+1 ei = ϵ tsi ei+1 tsi ei = ϵ tsi+1 ei tsi+1 ei = tsi+1 ei, and ei+1 ei tsi+1 = ϵ ei+1 tsi+1 ei tsi+1 = ϵ ei+1 tsi ei+1 tsi = ei+1 tsi.$

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

 $\left({z}_{0}\left(-u\right)-\left(\frac{1}{2}+ϵu\right)\right)\left({z}_{0}\left(u\right)-\left(\frac{1}{2}-ϵu\right)\right){e}_{1}=\left(\frac{1}{2}-ϵu\right)\left(\frac{1}{2}+ϵu\right){e}_{1},$
where ${z}_{0}\left(u\right)$ is the generating function $z0(u) = ∑ℓ∈ℤ ≥0 {z}_{0}^{\left(\ell \right)} u-ℓ.$ 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 [AS] and [DRV]) the natural choice of base ring is the center of the enveloping algebra of the orthogonal or symplectic Lie algebra, which, by the Harish-Chandra isomorphism, is isomorphic to the subring of symmetric functions given by $C= { z∈ ℂ[h1, …hr] Sr | z( h1,…hr )= z( -h1, h2,…hr ) },$ where the symmetric group ${S}_{r}$ acts by permuting the variables ${h}_{1},\dots {h}_{r}$. Here the constants ${z}_{0}^{\left(\ell \right)}\in C$ are given explicitly, by setting the generating function

 ${z}_{0}\left(u\right)$ equal, up to a normalization, to $\prod _{i=1}^{r}\frac{\left(u+\frac{1}{2}+{h}_{i}\right)\left(u+\frac{1}{2}-{h}_{i}\right)}{\left(u-\frac{1}{2}-{h}_{i}\right)\left(u-\frac{1}{2}+{h}_{i}\right)}.$
This choice of $C$ and the ${z}_{0}^{\left(\ell \right)}$ are the universal admissible parameters for ${𝒲}_{k}$.

### Quotients of ${𝒲}_{k}$

The degenerate affine Hecke algebra ${ℋ}_{k}$ is the quotient of ${𝒲}_{k}$ by the relations

 ${e}_{i}=0,\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}i=1,\dots ,k-1.$ (dah)
Fix ${u}_{1},\dots ,{u}_{r}\in ℂ.$ The degenerate cyclotomic BMW algebra ${𝒲}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$ is the degenerate affine BMW algebra with the additional relation
 $\left({y}_{1}-{u}_{1}\right)\cdots \left({y}_{1}-{u}_{r}\right)=0$. (cyc)
The degenerate cyclotomic Hecke algebra ${ℋ}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$ is the graded Hecke algebra with the additional relation (cyc).

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

 $\left({z}_{0}\left(u\right)+u-\frac{1}{2}\right){e}_{1}=\left(u-\frac{1}{2}{\left(-1\right)}^{r}\right)\left(\prod _{i=1}^{r}\frac{u+{b}_{i}}{u-{b}_{i}}\right){e}_{1}$.
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]. The definition of the degenerate affine BMW algebra here differs from the original definition of Nazarov [Naz] (used also in the paper [AMR]):

• (a) Instead of fixing an infinite number of parameters ${z}_{1}^{\left(𝓁\right)}$ we choose parameters ${h}_{1},\dots ,{h}_{r}$ and define the ${z}_{1}^{\left(𝓁\right)}$ in terms of the ${h}_{i}$.
• (b) We add an extra parameter $ϵ$.
The motivation for (b), the new parameter $ϵ$ is that we want to handle the symplectic case in tandem with the orthogonal case (see Actions of Tantalizers). In [Naz], Nazarov considered only the orthogonal case. The motivation for (a) comes from the approach of [OR] where the degenerate affine Hecke algebra is acting on a tensor space, and the ${z}_{1}^{\left(𝓁\right)}$ are naturally elements of the center of the enveloping algebra $\mathrm{U𝔤}$ which is in Schur-Weyl duality with the affine BMW algebra and the center of the enveloping algebra is, by the Harish-Chandra isomorphism, isomorphic to the ring of Weyl group symmetric polynomials in ${h}_{1},\dots ,{h}_{r}$, a basis of the Cartan subalgebra of $𝔤$. This new definition of the degenerate affine BMW algebra, provides the same finite dimensional representation theory as the algebra originally defined by Nazarov [Naz].

The factor $ϵ$ in (dbw1) is slightly unusual in the context of the Brauer algebra diagrams. For the precise conversion to the usual Brauer algebra see Degenerate BMW bases.

## 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)