The degenerate affine braid algebra 𝔹k

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 7 April 2011

The degenerate affine braid algebra 𝔹k

Let C be a commutative ring. The group algebra of the symmetric group Sk is

CSk =C-span tw | wSk , with   tu tv = tuv , (CSk)

for u,v Sk. Let s1, , sk-1 be the simple reflections in Sk so that si = i i+1 .

The degenerate affine braid algebra is the algebra 𝔹k over C generated by

tu,with uSk, κ0,κ1, and y1,, yk, (gbgA)
with relations
tu tv = tuv , yiyj = yjyi, κ0κ1 = κ1κ0, κ0yi = yiκ0, κ1yi = yiκ1, (gba1)
κ0tsi = tsiκ0, κ1ts1 κ1ts1 = ts1κ1 ts1κ1 and κ1tsj = tsjκ1 ,for j1, (gba2)
tsi yi + yi+1 = yi + yi+1 tsi , and yj tsi = tsi yj , and ji,i+1, (gba3)
and
tsi tsi+1 ti,i+1 ti+1 ti = ti+1,i+2 , where ti,i+1 = yi+1 - tsi yi tsi , (gba4)
for i=1,,k-1 .

In the degenerate affine braid group 𝔹k let c0=κ0 and

cj =κ+2 (y1 ++ yj ) so that yj = 12 cj - cj-1 , for j=1, ,k, (ctoy)
Then the relations (gba3) are equivalent to
tsi cj = cj tsi, forji. (gbp3)

The degenerate affine braid algebra 𝔹k has another presentation by generators CSk

tu, with uSk, κ0,, κk, and ti,j, for 0i,jk with ij, (gbgB)
and relations
tu tv = tuv , tw κi tw-1 = κw(i), tw ti,j tw-1 = tw(i), w(j), (gba5)
κi κj = κj κi , κi t,m = t,m κi , (gba6)
ti,j = tj,i, tp,r t,m = t,m tp,r , ti,j ( ti,r + tj,r )=( ti,r + tj,r ) ti,j , (gba7)
for p and pm and r and rm and ij, ir and jr.

The commutation relations between the κi and the ti,j can be written in the form

[ κr , t,m ]=0, [ ti,j , t,m ], and [ ti,j , ti,m ] = [ ti,m , tj,m ] , (gba8)
for all r and all i and im and j and jm.

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

κ0 = κ0 , κ1 = κ1 , tw = tw , (AtoB1)
t0,1 = y1 - 12 κ1, and tj,j+1 = yj+1 - tsj yj tsj, for j=1,,k-1, (AtoB2)
and
κm = tu κ1 tu-1 , t0,m = tu t0,1 tu-1 and ti,j = tv t1,2 tv-1 , (AtoB3)
for u,vSk such that u(1) =m, v(1) =i and v(2) =j.

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

κ0 = κ0 , κ1 = κ1 , tw = tw , and yj = 12 κj + 0<j t,j. (BtoA)
By the first formula in (ctoy) and the last formula in (BtoA)
cj = i=0 j κi +2 0𝓁<mj t𝓁,m. (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).

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.

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