## A basis of the degenerate affine BMW algebra ${𝒲}_{k}$

A diagram on $k$ dots is a graph with $k$ dots in the top row, $k$ dots in the bottom row and $k$ edges pairing the dots. For example,
 $d=$ is a Brauer diagram on 7 dots. (4.1)
Number the vertices of the top row, left to right, with $1,2,\dots k$ and the vertices in the bottom row, left to right, with $1\prime ,2\prime ,\dots ,k\prime$ so that the diagram above can be written $d= (13) (21′) (45) (66′) (74′) (2′7′) (3′5′) .$ The Brauer algebra is the vector space
 , (4.2)
and product given by placing diagrams on top of each other and changing each closed loop to $x$. For example, if
 ${d}_{1}=$ and ${d}_{2}=$
then
 ${d}_{1}{d}_{2}=$ $=x$ (4.3)
The Brauer algebra is generated by
 ${s}_{i}=$ ${e}_{1}=$ $\dots$ $,$ $,$ $1\le i\le k-1,$ $i$ $i+1$ $\dots$ $\dots$ $i$ $i+1$ $\dots$ (4.4)
and is a subalgebra of the degenerate affine BMW algebra ${𝒲}_{k}$. The Brauer algebra is also the quotient of ${𝒲}_{k}$ by ${y}_{1}=0$ and, hence, can be viewed as the degnerate cyclotomic BMW algebra ${𝒲}_{1,k}\left(0\right)$.

Let ${𝒲}_{k}$ be the degenerate affine BMW algebra and let ${𝒲}_{r,k}\left({u}_{1},\dots ,{u}_{r}\right)$ be the degnerate cyclotomic BMW algebra as defined in (2.11), (2.12) and (2.19), respectively. For ${n}_{1},\dots ,{n}_{k}\in ℤ$ and a diagram d on k dots let $d n1 ,…, nk = yi1n1 ,…, yiℓnℓ d y iℓ+1 nℓ+1 ,…, yik nk ,$ where, in the lexicographic ordering of the edges $\left({i}_{1},{j}_{1}\right),\dots ,\left({i}_{k},{j}_{k}\right)$ of $d$, ${i}_{1},\dots ,{i}_{\ell }$ are in the top row of $d$ and ${i}_{\ell +1},\dots ,{i}_{k}$ are in the bottom row of $d$. Let ${D}_{k}$ be the set of diagrams on $k$ dots as in (???).

(a)   If ?????????? and ??????????????? then $\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{.2em}{0ex}}{n}_{1},\dots {n}_{k}\in ℤ\right\}$ is a $C$-basis of ${𝒲}_{k}$.
(b)   If ?????????????? holds and ???????????? then $\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{0.5em}{0ex}}0\le {n}_{1},\dots ,{n}_{k}\le r-1\right\}$ is a $C$-basis of ${𝒲}_{r,k}\left({b}_{1},\dots ,{b}_{r}\right)$.

Part (a) of the theorem is [Naz, Theorem 4.6] (see also [AMR, Theorem 2.12]) and part (b) is [AMR, Prop. 2.15 and Theorem 5.5]. We refer to these references for the proof, remarking only that one key point in showing that $\left\{{d}^{{n}_{1},\dots ,{n}_{k}}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}d\in {D}_{k},\phantom{\rule{.2em}{0ex}}{n}_{1},\dots ,{n}_{k}\in ℤ\right\}$ spans ${𝒲}_{k}$ is that if $\left(i,j\right)$ is a top-to-bottom edge in $d$, then

 ${y}_{i}d=d{y}_{j}+\left(\text{terms with fewer crossings}\right)$,
and if $\left(i,j\right)$ is a top-to-top edge in $d$ then
 ${y}_{i}d=-{y}_{j}d+\left(\text{terms with fewer crossings}\right)$.
This is illustrated in the affine case in (3.14).

## Notes and references

This page is the result of joint work with Zajj Daugherty and Rahbar Virk [DRV]. The basis theorem for the degenerate cyclotomic BMW algebra may not be quite correct in its statement above and may require some conditions relating the parameters ${u}_{1},\dots ,{u}_{r}$ and the parameters ${z}_{1}^{\left(\ell \right)}$. See [AMR] for specifics.

## References

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