The degenerate affine braid algebra ${𝔹}_{k}$

Let $C$ be a commutative ring. The group algebra of the symmetric group ${S}_{k}$ is

 (CSk)

for $u,v\in {S}_{k}$. Let ${s}_{1},\dots ,{s}_{k-1}$ be the simple reflections in ${S}_{k}$ so that ${s}_{i}=\left(i,i+1\right)$.

The degenerate affine braid algebra is the algebra ${𝔹}_{k}$ over $C$ generated by

 ${t}_{u},\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}u\in {S}_{k},\phantom{\rule{2em}{0ex}}{\kappa }_{0},{\kappa }_{1},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{y}_{1},\dots ,{y}_{k},$ (gbgA)
with relations
 ${t}_{u}{t}_{v}={t}_{uv},\phantom{\rule{2em}{0ex}}{y}_{i}{y}_{j}={y}_{j}{y}_{i},\phantom{\rule{2em}{0ex}}{\kappa }_{0}{\kappa }_{1}={\kappa }_{1}{\kappa }_{0},\phantom{\rule{2em}{0ex}}{\kappa }_{0}{y}_{i}={y}_{i}{\kappa }_{0},\phantom{\rule{2em}{0ex}}{\kappa }_{1}{y}_{i}={y}_{i}{\kappa }_{1},$ (gba1)
 (gba2)
 ${t}_{{s}_{i}}\left({y}_{i}+{y}_{i+1}\right)=\left({y}_{i}+{y}_{i+1}\right){t}_{{s}_{i}},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{y}_{j}{t}_{{s}_{i}}={t}_{{s}_{i}}{y}_{j},\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}j\ne i,i+1,$ (gba3)
and
 ${t}_{{s}_{i}}{t}_{{s}_{i+1}}{t}_{i,i+1}{t}_{i+1}{t}_{i}={t}_{i+1,i+2},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{2em}{0ex}}{t}_{i,i+1}={y}_{i+1}-{t}_{{s}_{i}}{y}_{i}{t}_{{s}_{i}},$ (gba4)
for $i=1,\dots ,k-1$.

In the degenerate affine braid group ${𝔹}_{k}$ let ${c}_{0}={\kappa }_{0}$ and

 (ctoy)
Then the relations (gba3) are equivalent to
 ${t}_{{s}_{i}}{c}_{j}={c}_{j}{t}_{{s}_{i}},\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}j\ne i.$ (gbp3)

The degenerate affine braid algebra ${𝔹}_{k}$ has another presentation by generators $C{S}_{k}$

 ${t}_{u},\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}u\in {S}_{k},\phantom{\rule{2em}{0ex}}{\kappa }_{0},\dots ,{\kappa }_{k},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{t}_{i,j},\phantom{\rule{0.5em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}0\le i,j\le k\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}i\ne j,$ (gbgB)
and relations
 ${t}_{u}{t}_{v}={t}_{uv},\phantom{\rule{2em}{0ex}}{t}_{w}{\kappa }_{i}{t}_{{w}^{-1}}={\kappa }_{w\left(i\right)},\phantom{\rule{2em}{0ex}}{t}_{w}{t}_{i,j}{t}_{{w}^{-1}}={t}_{w\left(i\right),w\left(j\right)},$ (gba5)
 ${\kappa }_{i}{\kappa }_{j}={\kappa }_{j}{\kappa }_{i},\phantom{\rule{2em}{0ex}}{\kappa }_{i}{t}_{\ell ,m}={t}_{\ell ,m}{\kappa }_{i},$ (gba6)
 ${t}_{i,j}={t}_{j,i},\phantom{\rule{2em}{0ex}}{t}_{p,r}{t}_{\ell ,m}={t}_{\ell ,m}{t}_{p,r},\phantom{\rule{2em}{0ex}}{t}_{i,j}\left({t}_{i,r}+{t}_{j,r}\right)=\left({t}_{i,r}+{t}_{j,r}\right){t}_{i,j},$ (gba7)
for $p\ne \ell$ and $p\ne m$ and $r\ne \ell$ and $r\ne m$ and $i\ne j$, $i\ne r$ and $j\ne r$.

The commutation relations between the ${\kappa }_{i}$ and the ${t}_{i,j}$ can be written in the form

 $\left[{\kappa }_{r},{t}_{\ell ,m}\right]=0,\phantom{\rule{2em}{0ex}}\left[{t}_{i,j},{t}_{\ell ,m}\right],\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left[{t}_{i,j},{t}_{i,m}\right]=\left[{t}_{i,m},{t}_{j,m}\right],$ (gba8)
for all $r$ and all $i\ne \ell$ and $i\ne m$ and $j\ne \ell$ and $j\ne m$.

Proof: The generators in (gbgB) are written in terms of the generators in (gbgA) by the formulas

 ${\kappa }_{0}={\kappa }_{0},\phantom{\rule{2em}{0ex}}{\kappa }_{1}={\kappa }_{1},\phantom{\rule{2em}{0ex}}{t}_{w}={t}_{w},$ (AtoB1)
 ${t}_{0,1}={y}_{1}-\frac{1}{2}{\kappa }_{1},\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{t}_{j,j+1}={y}_{j+1}-{t}_{{s}_{j}}{y}_{j}{t}_{{s}_{j}},\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}j=1,\dots ,k-1,$ (AtoB2)
and
 ${\kappa }_{m}={t}_{u}{\kappa }_{1}{t}_{{u}^{-1}},\phantom{\rule{2em}{0ex}}{t}_{0,m}={t}_{u}{t}_{0,1}{t}_{{u}^{-1}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{t}_{i,j}={t}_{v}{t}_{1,2}{t}_{{v}^{-1}},$ (AtoB3)
for $u,v\in {S}_{k}$ such that $u\left(1\right)=m$, $v\left(1\right)=i$ and $v\left(2\right)=j$.

The generators in (gbgA) are written in terms of the generators in (gbgB) by the formulas

 ${\kappa }_{0}={\kappa }_{0},\phantom{\rule{2em}{0ex}}{\kappa }_{1}={\kappa }_{1},\phantom{\rule{2em}{0ex}}{t}_{w}={t}_{w},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{y}_{j}=\frac{1}{2}{\kappa }_{j}+\sum _{0\le \ell (BtoA)
By the first formula in (ctoy) and the last formula in (BtoA)
 $cj = ∑ i=0 j κi +2 ∑ 0≤𝓁 (cytokt)

Let us show that the relations in (gba1-gba4) follow from the relations in (gba5-gba7).

To complete the proof let us show that the relations in (gba5-gba7) follow from the relations in (gba1-gba4).
$\square$

Notes and References

To our knowledge, the definition of the degenerate affine braid algebra has not appeared previously in the literature. It appears in [Da] and the presentation above is a part of the forthcoming paper [DRV]. Both the affine Hecke algebra and the affine BMW algebra are quotients of the group algebra of the affine group. In a similar manner it seemed desirable to define a degenerate affine braid algebra which has both the degenerate affine Hecke algebra and the degenerate affine BMW algebras as quotients. This is also justified by the analogy between the corresponding Schur-Weyl duality statements, see Actions of Tantalizers.

Bibliography

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