The affine BMW algebra

The affine BMW algebra $W_{k}$

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

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}^{(ℓ)} \in C for ℓ \in ℤ

with

q

and

z

invertible. Let

Y_{i} = z X^{ε_{i}}

so that

Y_{1} = z X^{ε_{1}}, Y_{i} = T_{i - 1} Y_{i - 1} T_{i - 1}, and Y_{i} Y_{j} = Y_{j} Y_{i}, for 1 \leq i, j \leq k

(Ydf)

In the affine braid group

T_{i} Y_{i} Y_{i + 1} = Y_{i} Y_{i + 1} T_{i}

(YTc)

Assume $(q - q^{- 1})$ 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} - (q - q^{- 1}) Y_{i + 1} (1 - E_{i})

(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}^{\pm 1} = T_{i}^{\pm 1} E_{i} = z^{\mp 1} E_{i}, E_{i} T_{i - 1}^{\pm 1} E_{i} = E_{i} T_{i + 1}^{\pm 1} E_{i} = z^{\pm 1} E_{i}

(BW1)

E_{1} Y_{1}^{ℓ} E_{1} = Z_{0}^{(ℓ)} E_{1}, E_{i} Y_{i} Y_{i + 1} = E_{i} = Y_{i} Y_{i + 1} E_{i}

(BW2)

Since $Y_{i + 1}^{- 1} (T_{i} Y_{i}) Y_{i + 1} = Y_{i + 1}^{- 1} Y_{i} Y_{i + 1} T_{i} = Y_{i} T_{i}$ , conjugating (Edb) by $Y_{i + 1}^{- 1}$ gives

Y_{i} T_{i} = T_{i} Y_{i + 1} - (q - q^{- 1}) (1 - E_{i}) 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} = (q - q^{- 1}) (1 - E_{i})

, so that

E_{i} = 1 - \frac{T_{i} - T_{i}^{- 1}}{q - q^{- 1}} and 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

E_{i} E_{i + 1} E_{i} = E_{i} T_{i} T_{i + 1} E_{i} T_{i + 1}^{- 1} T_{i}^{- 1} E_{i} = E_{i} T_{i + 1} E_{i} T_{i + 1}^{- 1} E_{i} = z E_{i} T_{i + 1}^{- 1} E_{i} = E_{i},

so that

E_{i} E_{i \pm 1} E_{i} = E_{i} . Note that E_{i}^{2} = (1 + \frac{z - z^{- 1}}{q - q^{- 1}}) 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}^{(0)} = 1 + \frac{z - z^{- 1}}{q - q^{- 1}} and (T_{i} - z^{- 1}) (T_{i} + q^{- 1}) (T_{i} - q) = 0

(BWF)

since

(T_{i} - z^{- 1}) (T_{i} + q^{- 1}) (T_{i} - q) T_{i}^{- 1} = (T_{i} - z^{- 1}) (T_{i}^{2} - (q - q^{- 1}) T_{i} - 1) T_{i}^{- 1} = (T_{i} - z^{- 1}) (T_{i} - T_{i}^{- 1} - (q - q^{- 1})) = (T_{i} - z^{- 1}) (q - q^{- 1}) (- E_{i}) = - (z^{- 1} - z^{- 1}) (q - q^{- 1}) = 0

. The relations

$E_{i + 1} E_{i} = E_{i + 1} T_{i} T_{i + 1}, E_{i} E_{i + 1} = T_{i + 1}^{- 1} T_{i}^{- 1} E_{i + 1}$ ,	(BW6)
$T_{i} E_{i + 1} E_{i} = T_{i + 1}^{- 1} E_{i}, and E_{i + 1} E_{i} T_{i + 1} = E_{i + 1} T_{i}^{- 1}$ ,	(BW7)

follow from the computations

E_{i + 1} T_{i} T_{i + 1} = z (E_{i + 1} T_{i}^{- 1} E_{i + 1}) T_{i} T_{i + 1} = z (z^{- 1} E_{i + 1} T_{i + 1}^{- 1}) T_{i}^{- 1} E_{i + 1} T_{i} T_{i + 1} = E_{i + 1} E_{i},

T_{i + 1}^{- 1} T_{i}^{- 1} E_{i + 1} = T_{i + 1}^{- 1} T_{i}^{- 1} (z^{- 1} E_{i + 1} T_{i} E_{i + 1}) = T_{i + 1}^{- 1} T_{i}^{- 1} z^{- 1} E_{i + 1} T_{i} z T_{i + 1} E_{i + 1} = E_{i} E_{i + 1},

T_{i} E_{i + 1} E_{i} = T_{i} E_{i + 1} (T_{i}^{- 1} E_{i} z^{- 1}) = z^{- 1} T_{i + 1}^{- 1} E_{i} T_{i + 1} E_{i} z^{- 1} = T_{i + 1}^{- 1} z E_{i} z^{- 1} = T_{i + 1}^{- 1} E_{i}, and

E_{i + 1} E_{i} T_{i + 1} = E_{i + 1} T_{i + 1}^{- 1} z E_{i} T_{i + 1} = z E_{i + 1} T_{i} E_{i + 1} T_{i}^{- 1} = z z^{- 1} E_{i + 1} T_{i}^{- 1} = E_{i + 1} T_{i}^{- 1} .

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

(Z_{0}^{-} - \frac{z}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) (Z_{0}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) E_{1} = \frac{- (u^{2} - q^{2}) (u^{2} - q^{- 2})}{(u^{2} - 1) {(q - q^{- 1})}^{2}} E_{1},

where

Z_{0}^{+}

and

Z_{0}^{-}

are the generating functions

Z_{0}^{+} = \sum_{ℓ \in ℤ_{\geq 0}} Z_{0}^{(ℓ)} u^{- ℓ} and Z_{0}^{+} = \sum_{ℓ \in ℤ_{\leq 0}} Z_{0}^{(ℓ)} u^{- ℓ} .

This means that, unless the parameters

Z_{0}^{(ℓ)} are chosen carefully, it is likely that E_{1} = 0 in 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 $C = {z \in {ℂ [L_{1}^{\pm 1}, \dots, L_{r}^{\pm 1}]}^{S_{r}} | z (L_{1}, \dots, L_{r}) = z (L_{1}^{- 1}, L_{2}, \dots, L_{r})},$ where the symmetric group $S_{r}$ acts by permuting the variables $L_{1}, \dots, L_{r}$ . Here the constants $Z_{0}^{(ℓ)} \in C$ are given, explicitly, by setting the generating functions $Z_{0}^{+}$ and $Z_{0}^{-}$ equal, up to a normalization, to

\prod_{i = 1}^{r} \frac{(u - q L_{i})}{(u - q^{- 1} L_{i})} \cdot \frac{(u - q L_{i}^{- 1})}{(u - q^{- 1} L_{i}^{- 1})}

and

\prod_{i = 1}^{r} \frac{(u - q^{- 1} L_{i})}{(u - q L_{i})} \cdot \frac{(u - q^{- 1} L_{i}^{- 1})}{(u - q L_{i}^{- 1})}

This choice of

C

and the

Z_{0}^{(ℓ)}

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

E_{i} = 0, for i = 1, \dots, k - 1

(Ah)

Fix

b_{1}, \dots, b_{r} \in ℂ

. The cyclotomic BMW algebra

W_{r, k} (b_{1} \dots b_{r})

is the affine BMW algebra

W_{k}

with the additional relation

(Y_{1} - b_{1}) \dots (Y_{1} - b_{r}) = 0 .

(Cyc)

The cyclotomic Hecke algebra

H_{r, k} (b_{1} \dots b_{r})

is the affine Hecke algebra

H_{k}

with additional relation (Cyc).

A consequence of the relation (Cyc) in $W_{r, k} (b_{1} \dots b_{r})$ is

(Z_{0}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) E_{1} = (\frac{z}{q - q^{- 1}} - \frac{u z}{u^{2} - 1}) (\prod_{j = 1}^{r} \frac{u - b_{j}^{- 1}}{u - b_{j}}) E_{1}

(2.52)

This equation makes the data of the values

b_{i}

almost equivalent to the data of the

Z_{0}^{(ℓ)}

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)

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