Affine and degenerate affine BMW algebras: The center

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

Last update: 21 October 2013

Identities in affine and degenerate affine BMW algebras

In [Naz1996], Nazarov defined some naturally occurring central elements in the degenerate affine BMW algebra $𝒲_{k}$ and proved a remarkable recursion for them. This recursion was generalized to analogous central elements in the affine BMW algebra $W_{k}$ by Beliakova-Blanchet [BBl1866492]. In both cases, the recursion was accomplished with an involved computation. In this section, we provide a new proof of the Nazarov and Beliakov-Blanchet recursions by lifting them out of the center, to intertwiner-like identities in $𝒲_{k}$ and $W_{k}$ (Propositions 3.1 and 3.5). These intertwiner-like identities for the degenerate affine and affine BMW algebras are reminiscent of the intertwiner identities for the degenerate affine and affine Hecke algebras found, for example, in [KRa2002, Prop. 2.5(c)] and [Ram2003, Prop. 2.14(c)], respectively. The central element recursions of [Naz1996] and [BBl1866492] are then obtained by multiplying the intertwiner-like identities by the projectors $e_{k}$ and $E_{k},$ respectively. We have carefully arranged the proofs so that the degenerate affine and the affine cases are exactly in parallel.

The degenerate affine case

Let $u$ be a variable, $\begin{matrix} u_{i}^{+} = \frac{1}{u - y_{i}}, and note that u_{i}^{+} u_{i + 1}^{+} = \frac{1}{2 u - (y_{i} + y_{i + 1})} (u i + u_{i + 1}^{+}) . & (3.1) \end{matrix}$ By (2.25) and the definition of $e_{i}$ in (2.21), $(u - y_{i + 1}) t_{s_{i}} = t_{s_{i}} (u - y_{i}) - (1 - e_{i}) and (u - y_{i}) t_{s_{i}} = t_{s_{i}} (u - y_{i + 1}) + (1 - e_{i}),$ which give $\begin{matrix} t_{s_{i}} u_{i}^{+} = u_{i + 1}^{+} t_{s_{i}} + u_{i + 1}^{+} e_{i} u_{i}^{+} - u_{i + 1}^{+} u_{i}^{+}, and t_{s_{i}} u_{i + 1}^{+} = u_{i}^{+} t_{s_{i}} - u_{i}^{+} e_{i} u_{i + 1}^{+} + u_{i + 1}^{+} u_{i}^{+}, & (3.2) \end{matrix}$ respectively.

In the degenerate affine BMW algebra $𝒲_{i + 1},$ $\begin{matrix} \begin{matrix} (e_{i} \frac{1}{1 - y_{i + 1}} - t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}) (e_{i} \frac{1}{1 - y_{i}} + t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}) \\ = \frac{- (2 u - (y_{i} + y_{i + 1}) + 1) (2 u - (y_{i} + y_{i + 1}) - 1)}{{(2 u - (y_{i} + y_{i + 1}))}^{2}}, \end{matrix} & (3.3) \end{matrix}$ and $\begin{matrix} \begin{matrix} (u_{i + 1}^{+} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) - u_{i}^{+} (u_{i + 1}^{+} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) u_{i}^{+} \\ = (t_{s_{i}} u_{i}^{+} t_{s_{i}} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) - u_{i + 1}^{+} (e_{i} u_{i}^{+} e_{i} + ε e_{i} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) u_{i + 1}^{+} . \end{matrix} & (3.4) \end{matrix}$

Proof.

Putting (3.1) into the first identity in (3.2) says that if $A = t_{s_{i}} + \frac{1}{2 u - (y_{i} + y_{i + 1})} and B = e_{i} u_{i}^{+} + t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}$ then $A u_{i}^{+} = u_{i + 1}^{+} B, and A e_{i} = e_{i} A$ follows from (2.23) and (2.24). So $\begin{matrix} (e_{i} u_{i + 1}^{+} - t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}) (e_{i} u_{i}^{+} + t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}) \\ = e_{i} u_{i + 1}^{+} B - A B = e_{i} A u_{i}^{+} - A B = A e_{i} u_{i}^{+} - A B = A (e_{i} u_{i}^{+} - B) \\ = - (t_{s_{i}} + \frac{1}{2 u - (y_{i} + y_{i + 1})}) (t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}), \end{matrix}$ and multiplying out the right hand side gives (3.3).

Multiplying the second relation in (3.2) by $t_{s_{i}}$ gives $u_{i + 1}^{+} - t_{s_{i}} u_{i + 1}^{+} u_{i}^{+} = t_{s_{i}} u_{i}^{+} t_{s_{i}} - t_{s_{i}} u_{i}^{+} e_{i} u_{i}^{-}$ and again using the relations in (3.2) gives $u_{i + 1}^{+} - u_{i}^{+} (t_{s_{i}} - e_{i} u_{i + 1}^{+} + u_{i + 1}^{+}) u_{i}^{+} = t_{s_{i}} u_{i}^{+} t_{s_{i}} - u_{i + 1}^{+} (t_{s_{i}} + e_{i} u_{i}^{+} - u_{i}^{+}) e_{i} u_{i + 1}^{+} .$ Using (3.1) and $adding t_{s_{i}} - e_{i} (\frac{1}{2 u - (y_{i} + y_{i + 1})}) - \frac{1}{2 u - (y_{i} + y_{i + 1})} u_{i}^{+} e_{i} u_{i + 1}^{+} to each side$ gives $\begin{matrix} (u_{i + 1}^{+} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) + u_{i}^{+} (u_{i + 1}^{+} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) u_{i}^{+} \\ = t_{s_{i}} u_{i}^{+} t_{s_{i}} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})} - u_{i + 1}^{+} (e_{i} u_{i}^{+} + t_{s_{i}} - \frac{1}{2 u - (y_{i} + y_{i + 1})}) e_{i} u_{i + 1}^{+} \\ = (t_{s_{i}} u_{i}^{+} t_{s_{i}} + t_{s_{i}} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) - u_{i + 1}^{+} (e_{i} u_{i}^{+} e_{i} + ε e_{i} - e_{i} \frac{1}{2 u - (y_{i} + y_{i + 1})}) u_{i + 1}^{+}, \end{matrix}$ completing the proof of (3.4).

$□$

Introduce notation $z_{i - 1}^{(ℓ)} e_{i}$ and the generating function $z_{i - 1} (u) e_{i}$ by $\begin{matrix} z_{i - 1} (u) e_{i} = \sum_{ℓ \in ℤ_{\geq 0}} z_{i - 1}^{(ℓ)} e_{i} u^{- ℓ} = e_{i} (\sum_{ℓ \in ℤ_{\geq 0}} y_{i}^{ℓ} u^{- ℓ}) e_{i} = e_{i} \frac{1}{1 - y_{i} u^{- 1}} e_{i}, & (3.5) \end{matrix}$ By [AMR0506467, Lemma 4.15], or the identity (3.9) below, $z_{i - 1}^{(ℓ)} \in 𝒲_{i - 1}$ for $ℓ \in ℤ_{\geq 0} .$ If $\begin{matrix} u_{i}^{-} = \frac{1}{u + y_{i}} then e_{i} u_{i + 1}^{+} = e_{i} u_{i}^{-}, u_{i + 1}^{+} e_{i} = u_{i}^{-} e_{i}, e_{i} u_{i}^{\pm} e_{i} = \frac{z_{i - 1} (\pm u)}{u} e_{i}, & (3.6) \end{matrix}$ where, for $i = 1,$ the last identity is a restatement of the first identity in (2.24). The identities (3.7), (3.8), and (3.9) of the following theorem are [Naz1996, Lemma 2.5], [Naz1996, Prop. 4.2] and [Naz1996, Lemma 3.8], respectively.

Let $z_{i - 1}^{(ℓ)}$ and $z_{i - 1} (u)$ be as defined in (3.5). Then $z_{i - 1}^{(ℓ)} \in Z (𝒲_{i - 1}),$ $\begin{matrix} (z_{i - 1} (- u) - (\frac{1}{2} + ε u)) (z_{i - 1} (u) - (\frac{1}{2} - ε u)) e_{i} = (\frac{1}{2} - ε u) (\frac{1}{2} + ε u) e_{i}, & (3.7) \\ (z_{i} (u) + ε u - \frac{1}{2}) e_{i + 1} = (z_{i - 1} (u) + ε u - \frac{1}{2}) (\frac{({(u + y_{i})}^{2} - 1) {(u - y_{i})}^{2}}{({(u - y_{i})}^{2} - 1) {(u + y_{i})}^{2}}) e_{i + 1}, and & (3.8) \\ (z_{k - 1} (u) + ε u - \frac{1}{2}) e_{i + 1} = (z_{0} (u) + ε u - \frac{1}{2}) \prod_{i = 1}^{k - 1} \frac{(u + y_{i} - 1) (u + y_{i} + 1) {(u - y_{i})}^{2}}{{(u + y_{i})}^{2} (u - y_{i} + 1) (u - y_{i} - 1)} e_{i + 1} . & (3.9) \end{matrix}$

Proof.

Since the generators $t_{s_{1}}, \dots, t_{s_{i - 2}}, e_{1} \dots, e_{i - 2}$ and $y_{1}, \dots, y_{i - 1}$ of $𝒲_{i - 1}$ all commute with $e_{i}$ and $y_{i}$ it follows that $z_{i - 1}^{(ℓ)} \in Z (𝒲_{i - 1}) .$

Multiply (3.3) on the right by $e_{i}$ to get (3.7), since $(\frac{1}{2} - u) (\frac{1}{2} + u) = (\frac{1}{2} - ε u) (\frac{1}{2} + ε u) .$

Multiplying (3.4) on the left and right by $e_{i + 1}$ and using the relations in (2.27), (2.28) and (2.29), $\begin{matrix} e_{i + 1} t_{s_{i}} u_{i}^{+} t_{s_{i}} e_{i + 1} = e_{i + 1} t_{s_{i}} t_{s_{i + 1}} u_{i}^{+} t_{s_{i + 1}} t_{s_{i}} e_{i + 1} = e_{i + 1} e_{i} u_{i}^{+} e_{i} e_{i + 1}, and \\ e_{i + 1} u_{i + 1}^{+} e_{i} u_{i + 1}^{+} e_{i + 1} = e_{i + 1} u_{i}^{-} e_{i} u_{i}^{-} e_{i + 1} = u_{i}^{-} e_{i + 1} e_{i} e_{i + 1} u_{i}^{-} = {(u_{i}^{-})}^{2} e_{i + 1}, \end{matrix}$ gives $(\frac{z_{i} (u)}{u} + ε - \frac{1}{2 u}) (1 - {(u_{i}^{+})}^{2}) e_{i + 1} = (\frac{z_{i - 1} (u)}{u} + ε - \frac{1}{2 u}) (1 - {(u_{i}^{-})}^{2}) e_{i + 1} .$ So (3.8) follows from $\frac{1 - {(u_{i}^{-})}^{2}}{1 - {(u_{i}^{+})}^{2}} = \frac{1 - {(\frac{1}{u + y_{i}})}^{2}}{1 - {(\frac{1}{u - y_{i}})}^{2}} = \frac{(u^{2} + 2 y_{i} u + y_{i}^{2} - 1) {(u - y_{i})}^{2}}{(u^{2} - 2 y_{i} u + y_{i}^{2} - 1) {(u + y_{i})}^{2}} = \frac{(u + y_{i} - 1) (u + y_{i} + 1) {(u - y_{i})}^{2}}{(u - y_{i} - 1) (u - y_{i} + 1) {(u + y_{i})}^{2}} .$ Finally, relation (3.9) follows, by induction, from (3.8).

$□$

Using the expansion $\frac{1}{u - a} = \frac{u^{- 1}}{1 - a u^{- 1}} = \sum_{ℓ \in ℤ_{\geq 1}} a^{ℓ - 1} u^{- ℓ},$ and taking the coefficient of $u^{- (ℓ + 1)}$ on each side of the relations in (3.2) gives $\begin{matrix} t_{s_{i}} y_{i}^{ℓ} & = & y_{i + 1}^{ℓ} t_{s_{i}} - (y_{i + 1}^{ℓ - 1} (1 - e_{i}) + y_{i + 1}^{ℓ - 2} (1 - e_{i}) y_{i} + \dots + (1 - e_{i}) y_{i}^{ℓ - 1}), and & (3.10) \\ t_{s_{i}} y_{i + 1}^{ℓ} & = & y_{i}^{ℓ} t_{s_{i}} + y_{i}^{ℓ - 1} (1 - e_{i}) + y_{i}^{ℓ - 2} (1 - e_{i}) y_{i + 1} + \dots + (1 - e_{i}) y_{i + 1}^{ℓ - 1}, & (3.11) \end{matrix}$ respectively.

Taking the coefficient of $u^{- s}$ on each side of (3.7) gives a trivial identity for even $s$ but, for odd $s = 2 ℓ + 1,$ gives $\begin{matrix} (2 z_{i - 1}^{(2 ℓ + 1)} + z_{i - 1}^{(2 ℓ)} - (z_{i - 1}^{(2 ℓ)} z_{i - 1}^{(0)} - z_{i - 1}^{(2 ℓ - 1)} z_{i - 1}^{(1)} + \dots + z_{i - 1}^{(0)} z_{i - 1}^{(2 ℓ)}) e_{i} = 0) & (3.12) \end{matrix}$ which is the admissibility relation inAMR0506467AMR, Remark 2.11] (see also [Naz1996, (4.6)].)

The affine case

Let $u$ be a variable, $\begin{matrix} U_{i}^{+} = \frac{Y_{i}}{u - Y_{i}}, and note that U_{i}^{+} U_{i + 1}^{+} = \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} (U_{i}^{+} + U_{i + 1}^{+} + 1) . & (3.13) \end{matrix}$ By the definition of $E_{i}$ in (2.41), $(u - Y_{i + 1}) T_{i} = T_{i} (u - Y_{i}) - (q - q^{- 1}) Y_{i + 1} (1 - E_{i}),$ and, by (2.44), $(u - Y_{i}) T_{i} = T_{i} (u - Y_{i + 1}) + (q - q^{- 1}) (1 - E_{i}) Y_{i + 1},$ so that $\begin{matrix} T_{i} \frac{1}{u - Y_{i}} & = & \frac{1}{u - Y_{i + 1}} T_{i} - (q - q^{- 1}) \frac{Y_{i + 1}}{u - Y_{i + 1}} (1 - E_{i}) \frac{1}{u - Y_{i}}, and & (3.14) \\ T_{i} \frac{1}{u - Y_{i + 1}} & = & \frac{1}{u - Y_{i}} T_{i} + (q - q^{- 1}) \frac{1}{u - Y_{i}} (1 - E_{i}) \frac{Y_{i + 1}}{u - Y_{i + 1}} . & (3.15) \end{matrix}$ The relations $\begin{matrix} T_{i} U_{i}^{+} & = & U_{i + 1}^{+} T_{i}^{- 1} - (q - q^{- 1}) U_{i + 1}^{+} (1 - E_{i}) U_{i}^{+} & (3.16) \\ = & U_{i + 1}^{+} (T_{i}^{- 1} - (q - q^{- 1}) (1 - E_{i}) U_{i}^{+}), and \end{matrix}$ $\begin{matrix} T_{i}^{- 1} U_{i + 1}^{+} & = & U_{i}^{+} T_{i} - (q - q^{- 1}) U_{i}^{+} E_{i} U_{i + 1}^{+} + (q - q^{- 1}) U_{i + 1}^{+} U_{i}^{+} & (3.17) \\ = & U_{i}^{+} (T_{i} + (q - q^{- 1}) (1 - E_{i}) U_{i + 1}^{+}) \end{matrix}$ are obtained by multiplying (3.14) and (3.15) on the right (resp. left) by $Y_{i}$ and using the relation $T_{i} Y_{i} = Y_{i + 1} T_{i}^{- 1} .$

Let $Q = q - q^{- 1} .$ Then, in the affine BMW algebra $W_{i + 1},$ $\begin{matrix} \begin{matrix} (E_{i} \frac{Y_{i + 1}}{u - Y_{i + 1}} - \frac{T_{i}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) (E_{i} \frac{Y_{i}}{u - Y_{i}} - \frac{T_{i}^{- 1}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) \\ = \frac{- (u^{2} - q^{2} Y_{i} Y_{i + 1}) (u^{2} - q^{- 2} Y_{i} Y_{i + 1})}{Q^{2} {(u^{2} - Y_{i} Y_{i + 1})}^{2}}, and \end{matrix} & (3.18) \\ \begin{matrix} (U_{i + 1}^{+} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) - Q^{2} (U_{i}^{+} + 1) (U_{i + 1}^{+} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) U_{i}^{+} \\ = (T_{i} U_{i}^{+} T_{i}^{- 1} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) - Q^{2} U_{i + 1}^{+} (E_{i} U_{i}^{+} E_{i} + z \frac{E_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) (U_{i + 1}^{+} + 1) . \end{matrix} & (3.19) \end{matrix}$

Proof.

Putting (3.13) into (3.16) says that if $A = \frac{T_{i}}{Q} + \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} and B = E_{i} U_{i}^{+} + \frac{T_{i}^{- 1}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}$ then $A U_{i}^{+} = U_{i + 1}^{+} B - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} . Next, A E_{i} = E_{i} A$ follows from (2.42) and (2.43). So $\begin{matrix} (E_{i} \frac{Y_{i + 1}}{u - Y_{i + 1}} - \frac{T_{i}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) (E_{i} \frac{Y_{i}}{u - Y_{i}} + \frac{T_{i}^{- 1}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) \\ = E_{i} (U_{i + 1}^{+} B) - A B = E_{i} (A U_{i}^{+} + \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) - A B = A (E_{i} U_{i}^{+} - B) + E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} \\ = - (\frac{T_{i}}{Q} + \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) (\frac{T_{i}^{- 1}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) + E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}, \end{matrix}$ and, by (2.45), multiplying out the right hand side gives (3.18).

Rewrite $T_{i}^{- 1} U_{i + 1}^{+} = U_{i}^{+} T_{i}^{- 1} + Q U_{i}^{+} (1 - E_{i}) (U_{i + 1}^{+} + 1)$ as $T_{i - 1} U_{i + 1}^{+} - Q (U_{i + 1}^{+} + 1) U_{i}^{+} = U_{i}^{+} T_{i}^{- 1} - Q U_{i}^{+} E_{i} (U_{i + 1}^{+} + 1),$ and multiply on the left by $T_{i}$ to get $\begin{matrix} U_{i + 1}^{+} - Q T_{i} (U_{i + 1}^{+} + 1) U_{i}^{+} = T_{i} U_{i}^{+} T_{i}^{- 1} - Q T_{i} U_{i}^{+} E_{i} (U_{i + 1}^{+} + 1) . & (3.20) \end{matrix}$ Then, since $T_{i} = T_{i}^{- 1} + Q (1 - E_{i}),$ equations (3.17) and (3.16) imply $T_{i} (U_{i + 1}^{+} + 1) = Q (U_{i}^{+} + 1) (\frac{T_{i}}{Q} + (1 - E_{i}) U_{i + 1}^{+}) and T_{i} U_{i}^{+} = Q U_{i + 1}^{+} (\frac{T_{i}^{- 1}}{Q} - (1 - E_{i}) U_{i}^{+}),$ and so (3.20) is $\begin{matrix} \begin{matrix} U_{i + 1}^{+} - Q^{2} (u_{i}^{+} + 1) (\frac{T_{i}}{Q} + (1 - E_{i}) U_{i + 1}^{+}) U_{i}^{+} \\ = T_{i} U_{i}^{+} T_{i}^{- 1} - Q^{2} U_{i + 1}^{+} (\frac{T_{i}^{- 1}}{Q} - (1 - E_{i}) U_{i}^{+}) E_{i} (U_{i + 1}^{+} + 1) . \end{matrix} & (3.21) \end{matrix}$ Using (3.13) and $adding \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} - Q^{2} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} (U_{i}^{+} + 1) E_{i} (U_{i + 1}^{+} + 1) to each side$ of (3.21) gives $\begin{matrix} U_{i + 1}^{+} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} - Q^{2} (U_{i}^{+} + 1) (U_{i + 1}^{+} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) U_{i}^{+} \\ = T_{i} U_{i}^{+} T_{i}^{- 1} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} - Q^{2} U_{i + 1}^{+} (E_{i} U_{i}^{+} + \frac{T_{i}^{- 1}}{Q} - \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) E_{i} (U_{i + 1}^{+} + 1) \\ = T_{i} U_{i}^{+} T_{i}^{- 1} + \frac{T_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}} - Q^{2} U_{i + 1}^{+} (E_{i} U_{i}^{+} E_{i} + z \frac{E_{i}}{Q} - E_{i} \frac{Y_{i} Y_{i + 1}}{u^{2} - Y_{i} Y_{i + 1}}) (U_{i + 1}^{+} + 1), \end{matrix}$ completing the proof of (3.19).

$□$

Introduce notation $Z_{i - 1}^{(ℓ)} e_{i}$ and generating functions $Z_{i - 1}^{+} E_{i}$ and $Z_{i - 1}^{-} E_{i}$ by $\begin{matrix} Z_{i - 1}^{+} E_{i} = \sum_{ℓ \in ℤ_{\geq 0}} Z_{i - 1}^{(ℓ)} E_{i} u^{- ℓ} = E_{i} (\sum_{ℓ \in ℤ_{\geq 0}} Y_{i}^{ℓ} u^{- ℓ}) E_{i} = E_{i} \frac{1}{1 - Y_{i} u^{- 1}} E_{i}, & (3.22) \\ Z_{i - 1}^{-} E_{i} = \sum_{ℓ \in ℤ_{\geq 0}} Z_{i - 1}^{(- ℓ)} E_{i} u^{- ℓ} = E_{i} (\sum_{ℓ \in ℤ_{\geq 0}} Y_{i}^{- ℓ} u^{- ℓ}) E_{i} = E_{i} \frac{1}{1 - Y_{i}^{- 1} u^{- 1}} E_{i} . & (3.23) \end{matrix}$ By [GHa0411155, Lemma 3.15(1)], or the identity (3.28) below, $Z_{i - 1}^{(ℓ)} \in W_{i - 1}$ for $ℓ \in ℤ .$ If $\begin{matrix} U_{i}^{-} = \frac{Y_{i}^{- 1}}{u - Y_{i}^{- 1}} then Z_{i - 1}^{(0)} = 1 + \frac{z - z^{- 1}}{q - q^{- 1}}, & (3.24) \end{matrix}$ by the second relation in (2.46), and $\begin{matrix} E_{i} U_{i + 1}^{+} = E_{i} U_{i}^{-}, U_{i + 1}^{+} E_{i} = U_{i}^{-} E_{i}, E_{i} U_{i}^{\pm} E_{i} = (Z_{i - 1}^{\pm} - Z_{i - 1}^{(0)}) E_{i}, & (3.25) \end{matrix}$ where, for $i = 1,$ the last identity is a restatement of the first identity in (2.43). In the following theorem, the identity (3.26) is equivalent to [GHa0411155, Lemma 2.8, parts (2)and (3)] or [GMo0612064, Lemma 2.6(4)] (see Remark 3.8) and the identity (3.27) is found in [BBl1866492, Lemma 7.4].

Let $Z_{i - 1}^{(ℓ)}$ and the generating functions $Z_{i - 1}^{+}$ and $Z_{i - 1}^{-}$ be as defined in (3.22) and (3.23). Then $Z_{i - 1}^{(ℓ)} \in Z (W_{i - 1}),$ $\begin{matrix} (Z_{i - 1}^{-} - \frac{z}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) (Z_{i - 1}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) E_{i} = \frac{- (u^{2} - q^{2}) (u^{2} - q^{- 2})}{{(u^{2} - 1)}^{2} {(q - q^{- 1})}^{2}} E_{i}, & (3.26) \\ \begin{matrix} (Z_{i}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) E_{i + 1} \\ = (Z_{i - 1}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) \frac{{(u - Y_{i})}^{2} (u - q^{- 2} Y_{i}^{- 1}) (u - q^{2} Y_{i}^{- 1})}{{(u - Y_{i}^{- 1})}^{2} (u - q^{2} Y_{i}) (u - q^{- 2} Y_{i})} E_{i + 1}, and \end{matrix} & (3.27) \\ \begin{matrix} (Z_{k - 1}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) E_{i + 1} \\ = (Z_{0}^{+} + \frac{z^{- 1}}{q - q^{- 1}} - \frac{u^{2}}{u^{2} - 1}) (\prod_{i = 1}^{k - 1} \frac{{(u - Y_{i})}^{2} (u - q^{- 2} Y_{i}^{- 1}) (u - q^{2} Y_{i}^{- 1})}{{(u - Y_{i}^{- 1})}^{2} (u - q^{2} Y_{i}) (u - q^{- 2} Y_{i})}) E_{i + 1} . \end{matrix} & (3.28) \end{matrix}$

Proof.

Since the generators $T_{1}, \dots, T_{i - 2},$ $E_{2}, \dots, E_{i - 2}$ and $Y_{1}, \dots, Y_{i - 1}$ of $W_{i - 1}$ all commute with $E_{i}$ and $Y_{i},$ it follows that $Z_{i - 1}^{(ℓ)} \in Z (W_{i - 1}) .$

Multiply (3.18) on the right by $W_{i}$ and use $Z_{i - 1}^{(0)} = 1 + (z - z^{- 1}) / (q - q^{- 1})$ to get (3.26).

Multiply (3.19) on the left and right by $E_{i + 1}$ and use the relations in (2.42), (2.43), (2.46), and $E_{i + 1} T_{i} U_{i}^{+} T_{i}^{- 1} E_{i + 1} = E_{i + 1} T_{i} T_{i + 1} U_{i}^{+} T_{i + 1}^{- 1} T_{i}^{- 1} E_{i + 1} = E_{i + 1} E_{i} U_{i}^{+} E_{i} E_{i + 1},$ to obtain $\begin{matrix} (Z_{i - 1}^{+} - Z_{i}^{(0)} + \frac{z}{q - q^{- 1}} - \frac{1}{u^{2} - 1}) (1 - {(q - q^{- 1})}^{2} U_{i}^{+} (U_{i}^{+} + 1)) E_{i + 1} \\ = (Z_{i - 1}^{+} - Z_{i - 1}^{(0)} + \frac{z}{q - q^{- 1}} - \frac{1}{u^{2} - 1}) (1 - {(q - q^{- 1})}^{2} U_{i}^{-} (U_{i}^{-} + 1)) E_{i + 1} . \end{matrix}$ Then (3.27) follows from $\begin{matrix} \frac{1 - {(q - q^{- 1})}^{2} U_{i}^{-} (U_{i}^{-} + 1)}{1 - {(q - q^{- 1})}^{2} U_{i}^{+} (U_{i}^{+} + 1)} = \frac{1 - {(q - q^{- 1})}^{2} \frac{Y_{i}^{- 1}}{u - Y_{i}^{- 1}} (\frac{Y_{i}^{- 1}}{u - Y_{i}^{- 1}} + 1)}{1 - {(q - q^{- 1})}^{2} \frac{Y_{i}}{u - Y_{i}} (\frac{Y_{i}}{u - Y_{i}} + 1)} \\ = \frac{({(u - Y_{i}^{- 1})}^{2} - {(q - q^{- 1})}^{2} Y_{i}^{- 1} u) \frac{1}{{(u - Y_{i}^{- 1})}^{2}}}{({(u - Y_{i})}^{2} - {(q - q^{- 1})}^{2} Y_{i} u) \frac{1}{{(u - Y_{i})}^{2}}} = \frac{(u - q^{- 2} Y_{i}^{- 1}) (u - q^{2} Y_{i}^{- 1}) {(u - Y_{i})}^{2}}{(u - q^{- 2} Y_{i}) (u - q^{2} Y_{i}) {(u - Y_{i}^{- 1})}^{2}} \end{matrix}$ and $Z_{i}^{(0)} = Z_{i - 1}^{(0)} = 1 + (z - z^{- 1}) / (q - q^{- 1}) .$ Finally, relation (3.28) follows, by induction, from (3.27).

$□$

Taking the coefficient of $u^{- (ℓ + 1)}$ on each side of (3.14) and (3.15) gives $\begin{matrix} T_{i} Y_{i}^{ℓ} = Y_{i + 1}^{ℓ} T_{i} - (q - q^{- 1}) (Y_{i + 1}^{ℓ} (1 - E_{i}) + Y_{i + 1}^{ℓ - 1} (1 - E_{i}) Y_{i} + \dots + Y_{i + 1} (1 - E_{i}) Y_{i}^{ℓ - 1}), & (3.29) \\ T_{i} Y_{i + 1}^{ℓ} = Y_{i}^{ℓ} T_{i} + (q - q^{- 1}) (Y_{i}^{ℓ - 1} (1 - E_{i}) Y_{i + 1} + Y_{i}^{ℓ - 2} (1 - E_{i}) Y_{i + 1}^{2} + \dots + (1 - E_{i}) Y_{i + 1}^{ℓ}) & (3.30) \end{matrix}$ respectively, for $ℓ \in ℤ_{\geq 0} .$ Therefore, $\begin{matrix} T_{i} Y_{i}^{- ℓ} = Y_{i + 1}^{- ℓ} T_{i} + (q - q^{- 1}) (Y_{i + 1}^{- (ℓ - 1)} (1 - E_{i}) Y_{i}^{- 1} + \dots + (1 - E_{i}) Y_{i}^{- ℓ}), & (3.31) \\ T_{i} Y_{i + 1}^{- ℓ} = Y_{i}^{- ℓ} T_{i} - (q - q^{- 1}) (Y_{i}^{- ℓ} (1 - E_{i}) + \dots + Y_{i}^{- 1} (1 - E_{i}) Y_{i + 1}^{- (ℓ - 1)}) . & (3.32) \end{matrix}$

Combining (3.26) and (3.28) yields a formula for $Z_{k - 1}^{-}$ in terms of $Z_{0}^{+}$ and $Y_{1}, Y_{2}, \dots, Y_{k - 1} .$ Using $Z_{i - 1}^{(0)} = 1 + \frac{z - z^{- 1}}{q - q^{- 1}},$ rewrite (3.26) as $\begin{matrix} \begin{matrix} (z Z_{i - 1}^{-} - z^{- 1} Z_{i - 1}^{+} - (z - z^{- 1}) Z_{i - 1}^{(0)}) E_{i} \\ = (q - q^{- 1}) (\frac{1}{u^{2} - 1} (Z_{i - 1}^{+} + Z_{i - 1}^{-} - Z_{i - 1}^{(0)}) - (Z_{i - 1}^{-} - Z_{i - 1}^{(0)}) (Z_{i - 1}^{+} - Z_{i - 1}^{(0)})) E_{i}, \end{matrix} & (3.33) \end{matrix}$ and take the coefficient of $u^{- ℓ}$ in (3.26) to get $\begin{matrix} \begin{matrix} (z Z_{i - 1}^{(- ℓ)} - z^{- 1} Z_{i - 1}^{(ℓ)}) E_{i} \\ = (q - q^{- 1}) (\binom{Z_{i - 1}^{(ℓ - 2)} + Z_{i - 1}^{(ℓ - 4)} + \dots + Z_{i - 1}^{(- (ℓ - 2))}}{- (Z_{i - 1}^{(ℓ - 1)} Z_{i - 1}^{(- 1)} + Z_{i - 1}^{(ℓ - 2)} Z_{i - 1}^{(- 2)} + \dots + Z_{i - 1}^{(1)} Z_{i - 1}^{(- (ℓ - 1))})}) E_{i}, \end{matrix} & (3.34) \end{matrix}$ from [GMo0612064, Lemma 2.6(4)].

Notes and references

This is a typed exert of the paper Affine and degenerate affine BMW algebras: The center by Zajj Daugherty, Arun Ram and Rahbar Virk.

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