Identities in affine and degenerate affine BMW algebras

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

Last updates: 13 April 2011

The degenerate affine case

Let $u$ be a variable and define $z_{i-1}^{\left(\ell \right)}\in {𝒲}_{i-1}$ by the generating function

$z_{i-1}\left(u\right)e_i=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{z}_{i-1}^{\left(\ell \right)}{u}^{-\ell }{e}_i=e_i\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{y}_i^{\ell }{u}^{-\ell }\right)e_i=e_i{\displaystyle \frac{u}{u-y_i}}{e}_i$.

(zid)

Let

$u_i^{+}={\displaystyle \frac{1}{u-y_i}}\text{and}{u}_i^{-}={\displaystyle \frac{1}{u+y_i},\text{and note that}}$

(upm)

$u_i^{+}{u}_{i+1}^{+}={\displaystyle \frac{1}{2u-(y_i+y_{i+1})}\left(u_i^{+}+u_{i+1}^{+}\right)},$

(uup)

$e_i{u}_{i+1}^{+}=e_i{u}_i^{-},u_{i+1}^{+}=u_i^{-}{e}_i,e_i{u}_i^{\pm }{e}_i={\displaystyle \frac{z_{i-1}\left(\pm u\right)}{u}{e}_i,}$

(eup)

where, for $i=1$, the last identity is a restatement of the first identity in (dbw2). By (dbw3) and the definition of $e_i$ in (edb),

$(u-y_{i+1})t_{s_i}=t_{s_i}(u-y_i)-(1-e_i)\text{and}(u-y_i)t_{s_i}=t_{s_i}(u-y_{i+1})+(1-e_i)$,

which give

$$t_{s_i}{u}_i^{+}=u_{i+1}^{+}{t}_{s_i}+u_{i+1}^{+}{e}_i{u}_i^{+}-u_{i+1}^{+}{u}_i^{+},\text{and}{t}_{s_i}{u}_{i+1}^{+}=u_i^{+}{t}_{s_i}-u_i^{+}{e}_i{u}_{i+1}^{+}+u_{i+1}^{+}{u}_i^{+},$$

(tuc)

respectively.

In the degenerate affine BMW algebra ${𝒲}_{i+1}$,

$\begin{array}{l}(e_i{\displaystyle \frac{1}{1-y_{i+1}}}-t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})(e_i{\displaystyle \frac{1}{1-y_i}}+t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})\\ ={\displaystyle \frac{-(2u-(y_i+y_{i+1})+1)(2u-(y_i+y_{i+1})-1)}{{(2u-(y_i+y_{i+1}\left)\right)}^2},}\end{array}$

(itw1)

$\begin{array}{l}(u_{i+1}^{+}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})-u_i^{+}(u_{i+1}^{+}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})u_i^{+})\\ ={\displaystyle (t_{s_i}{u}_i^{+}{t}_{s_i}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})-u_{i+1}^{+}(e_i{u}_i^{+}{e}_i+ϵe_i-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})u_{i+1}^{+}.}\end{array}$

(itw2)

		Proof.
	Putting (uup) into the first identity in (tuc) says that if $A=t_{s_i}+{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}}\text{and}B=e_i{u}_i^{+}+t_{s_i}+{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}}$ then $A{u}_i^{+}=u_{i+1}^{+}B,\text{and}A{e}_i=e_{i}A$ follows from (dbw1) and (dbw2). So $\begin{array}{l}(e_i{u}_{i+1}^{+}-t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})(e_i{u}_i^{+}+t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})\\ =e_i{u}_{i+1}^{+}B-AB=e_{i}A{u}_i^{+}-AB=A{e}_i{u}_i^{+}-AB=A(e_i{u}_i^{+}-B)\\ =-(t_{s_i}+{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})(t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})\end{array}$ and mutiplying out the right hand side gives (itw1). Multiplying the second relation in (tuc) by $t_{s_i}$ $u_{i+1}^{+}-t_{s_i}{u}_{i+1}^{+}{u}_i^{+}=t_{s_i}{u}_i^{+}{t}_{s_i}-t_{s_i}{u}_i^{+}{e}_i{u}_i^{-}$ and again using the relations in (tuc) then $u_{i+1}^{+}-u_i^{+}(t_{s_i}-e_i{u}_{i+1}^{+}+u_{i+1}^{+})u_i^{+}=t_{s_i}{u}_i^{+}{t}_{s_i}-u_{i+1}^{+}(t_{s_i}+e_i{u}_i^{+}-u_i^{+})e_i{u}_{i+1}^{+}.$ Using (uup) and adding    $t_{s_i}-e_i\left({\displaystyle \frac{1}{2u-(y_i+y_{i+1})}}\right)-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}}{u}_i^{+}{e}_i{u}_{i+1}^{+}$     to each side gives $\begin{array}{l}(u_{i+1}^{+}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})-u_i^{+}(u_{i+1}^{+}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})u_i^{+}\\ =t_{s_i}{u}_i^{+}{t}_{s_i}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}}-u_{i+1}^{+}(e_i{u}_i^{+}+t_{s_i}-{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})e_i{u}_{i+1}^{+}\\ ={\displaystyle (t_{s_i}{u}_i^{+}{t}_{s_i}+t_{s_i}-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})-u_{i+1}^{+}(e_i{u}_i^{+}{e}_i+ϵe_i-e_i{\displaystyle \frac{1}{2u-(y_i+y_{i+1})}})u_{i+1}^{+},}\end{array}$ completing the proof of (itw2). $\square$

The identities (4.2), (4.3), and (4.4) of the following theorem are [Naz, Lemma 2.5], [Naz, Prop. 4.2] and [Naz, Lemma 3.8], respectively.

As in (zid), define $z_{i-1}^{\left(\ell \right)}\in {𝒲}_{i-1}$ by the generating function

$z_{i-1}\left(u\right)e_i=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{z}_{i-1}^{\left(\ell \right)}{u}^{-\ell }{e}_i=e_i\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{y}_i^{\ell }{u}^{-\ell }\right)e_i=e_i{\displaystyle \frac{u}{u-y_i}}{e}_i$.

(zid)

Then

$(z_{i-1}(-u)-(\frac{1}{2}+ϵu))\left(z_{i-1}\left(u\right)-\left(\frac{1}{2}-ϵu\right)\right)e_i=\left(\frac{1}{2}-ϵu\right)\left(\frac{1}{2}+ϵu\right)e_i,$

(zmz)

$\left(z_i\left(u\right)+ϵu-\frac{1}{2}\right)e_{i+1}=\left(z_{i-1}\left(u\right)+ϵu-\frac{1}{2}\right)\left({\displaystyle \frac{\left({\left(u+y_i\right)}^2-1\right){\left(u-y_i\right)}^2}{\left({\left(u-y_i\right)}^2-1\right){\left(u+y_i\right)}^2}}\right)e_{i+1},\text{<em>and</em>}$

(zrc)

$\left(z_{k-1}\left(u\right)+ϵu-\frac{1}{2}\right)e_{i+1}=\left(z_0\left(u\right)+ϵu-\frac{1}{2}\right){\displaystyle \prod _{i=1}^{k-1}}{\displaystyle \frac{\left(u+y_i-1\right)\left(u+y_i+1\right){\left(u-y_i\right)}^2}{{\left(u+y_i\right)}^2\left(u-y_i+1\right)\left(u-y_i-1\right)}}{e}_{i+1}.$

(zgn)

		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 $z_{i-1}^{\left(\ell \right)}\in Z\left({𝒲}_{i-1}\right)$. Multiply (itw1) on the right by $e_i$ to get (zmz), since $(\frac{1}{2}-u)(\frac{1}{2}+u)=(\frac{1}{2}-ϵu)(\frac{1}{2}+ϵu)$. Multiplying (itw2) on the right by $e_{i+1}$ and using the relations in (dbw5), (dbw6), and (dbw7), $e_{i+1}{t}_{s_i}{u}_i^{+}{t}_{s_i}{e}_{i+1}=e_{i+1}{t}_{s_i}{t}_{s_{i+1}}{u}_i^{+}{t}_{s_{i+1}}{t}_{s_i}{e}_{i+1}=e_{i+1}{e}_i{u}_i^{+}{e}_i{e}_{i+1}$,  and $e_{i+1}{u}_{i+1}^{+}{e}_i{u}_{i+1}^{+}{e}_{i+1}=e_{i+1}{u}_i^{-}{e}_i{u}_i^{-}{e}_{i+1}=u_i^{-}{e}_{i+1}{e}_i{e}_{i+1}{u}_i^{-}={\left(u_i^{-}\right)}^2{e}_{i+1}$, gives $$({\displaystyle \frac{z_i\left(u\right)}{u}}+ϵ-{\displaystyle \frac{1}{2u}})\left(1-{\left(u_i^{+}\right)}^2\right)e_{i+1}=(\frac{z_{i-1}\left(u\right)}{u}+ϵ-{\displaystyle \frac{1}{2u}})\left(1-{\left(u_i^{-}\right)}^2\right)e_{i+1}.$$ So (zrc) follows from $$\frac{1-{\left(u_i^{-}\right)}^2}{1-{\left(u_i^{+}\right)}^2}=\frac{1-{\left(\frac{1}{u+y_i}\right)}^2}{1-{\left(\frac{1}{u-y_i}\right)}^2}=\frac{\left(u^2+2y_{i}u+y_i^2-1\right){\left(u-y_i\right)}^2}{\left(u^2-2y_{i}u+y_i^2-1\right){\left(u+y_i\right)}^2}=\frac{\left(u+y_i-1\right)\left(u+y_i+1\right){\left(u-y_i\right)}^2}{\left(u-y_i-1\right)\left(u-y_i+1\right){\left(u+y_i\right)}^2}.$$ Finally, relation (zgn) follows, by induction, from (zrc). $\square$

Remark. Taking the coefficient of $u^{-s}$ on each side of (4.2) gives a trivial identity for even $s$, but for odd $s=2l+1$, gives

$$\left(2z_i^{(2l+1)}+z_i^{\left(2l\right)}-\left(z_i^{\left(2l\right)}{z}_i^{\left(0\right)}-z_i^{(2l-1)}{z}_i^{\left(1\right)}+\cdots +z_i^{\left(0\right)}{z}_i^{\left(2l\right)}\right)\right)e_i=0$$

4.10

which is the admissibltiy relation in [AMR, Remark 2.11] (see also [Naz, (4.6)]).

The affine case

Let $u$ be a variable and let

$U_i^{+}={\displaystyle \frac{Y_i}{u-Y_i}}\text{and}{U}_i^{-}={\displaystyle \frac{Y_i^{-1}}{u-Y_i^{-1}}}$,

(3.21)

and note that

$U_i^{+}{U}_{i+1}^{+}={\displaystyle \frac{1}{u^2-Y_i{Y}_{i+1}}}(U_i^{+}+U_{i+1}^{+}+1)$.

(3.22)

The second relation in (BW2) and the definition of $Z_{i-1}^{\pm }$ give

$E_i{U}_{i+1}^{+}=E_i{U}_i^{-},U_{i+1}^{+}{E}_i=U_i^{-}{E}_i,E_i{U}_i^{\pm }{E}_i=(Z_{i-1}^{\pm }-Z_{i-1}^{\left(0\right)})E_i,$

(3.23)

and the relations

$$\begin{array}{rcl}{T}_i{U}_i^{+}& =& U_{i+1}^{+}{T}_i^{-1}-(q-q^{-1})U_{i+1}^{+}(1-E_i)U_i^{+}\\ & =& U_{i+1}^{+}(T_i^{-1}-(q-q^{-1}\left)\right(1-E_i\left)U_i^{+}\right),\text{and}\end{array}$$

(3.24)

$\begin{array}{rcl}{T}_i^{-1}{U}_{i+1}^{+}& =& U_i^{+}{T}_i-(q-q^{-1})U_i^{+}{E}_i{U}_i^{-}+(q-q^{-1})U_{i+1}^{+}{U}_i^{+}\\ & =& U_i^{+}(T_i+(q-q^{-1}\left)\right(1-E_i\left)U_{i+1}^{+}\right)\end{array}$

(3.25)

are obtained by multiplying (3.15) (resp. (3.16)) on the right (resp. left) by $Y_i$ and using the relation $T_i{Y}_i=Y_{i+1}{T}_i^{-1}$.

Let $Q=q-q^{-1}$. In the affine BMW algebra $W_{i+1}$,

$\begin{array}{l}(E_i{\displaystyle \frac{Y_{i+1}}{u-Y_{i+1}}}-{\displaystyle \frac{T_i}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})(E_i{\displaystyle \frac{Y_i}{u-Y_i}}+{\displaystyle \frac{T_i^{-1}}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})\\ ={\displaystyle \frac{-(u^2-q^2{Y}_i{Y}_{i+1})(u^2-q^{-2}{Y}_i{Y}_{i+1})}{Q^2{(u^2-Y_i{Y}_{i+1})}^2}},\end{array}$

(Itw1)

and

$\begin{array}{l}(U_{i+1}^{+}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})-Q^2(U_i^{+}+1)(U_{i+1}^{+}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})U_i^{+}\\ ={\displaystyle (T_i{U}_i^{+}{T}_i^{-1}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})-Q^2{U}_{i+1}^{+}(E_i{U}_i^{+}{E}_i+z{\displaystyle \frac{E_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})(U_{i+1}^{+}+1).}\end{array}$

(Itw2)

Proof. Putting (UUp) into (TUc) says that if

$A={\displaystyle \frac{T_i}{Q}}+{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}\text{and}B=E_i{U}_i^{+}+{\displaystyle \frac{T_i^{-1}}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}$

then

$A{U}_i^{+}=U_{i+1}^{+}B-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}.\text{Next,}A{E}_i=E_{i}A$

follows from (BW1) and (BW2). So

$\begin{array}{l}(E_i{\displaystyle \frac{Y_{i+1}}{u-Y_{i+1}}}-{\displaystyle \frac{T_i}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})(E_i{\displaystyle \frac{Y_i}{u-Y_i}}+{\displaystyle \frac{T_i^{-1}}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})\\ =E_i\left(U_{i+1}^{+}B\right)-AB=E_i(A{U}_i^{+}+{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})-AB=A(E_i{U}_i^{+}-B)+E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}\\ =-({\displaystyle \frac{T_i}{Q}}+{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})({\displaystyle \frac{T_i^{-1}}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})+E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}\end{array}$

and, by (BW4), multiplying out the right hand side gives (Itw1).

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

$U_{i+1}^{+}-Q{T}_i(U_{i+1}^{+}+1)U_i^{+}=T_i{U}_i^{+}{T}_i^{-1}-Q{T}_i{U}_i^{+}{E}_i(U_{i+1}^{+}+1)$.

Then, since $T_i=T_i^{-1}+Q(1-E_i)$, equations (3.25) and (3.24) imply

$T_i(U_{i+1}^{+}+1)=Q(U_{i+1}^{+}+1)({\displaystyle \frac{T_i}{Q}}+(1-E_i)U_{i+1}^{+})\text{and}{T}_i{U}_i^{+}=Q{U}_{i+1}^{+}({\displaystyle \frac{T_i^{-1}}{Q}}-(1-E_i)U_i^{+})$.

and so (3.28) is

$U_{i+1}^{+}-Q^2(U_i^{+}+1)({\displaystyle \frac{T_i}{Q}}+(1-E_i)U_{i+1}^{+})U_i^{+}=T_i{U}_i^{+}{T}_i^{-1}-Q^2{U}_{i+1}^{+}({\displaystyle \frac{T_i^{-1}}{Q}}-(1-E_i)U_i^{+})E_i(U_{i+1}^{+}+1)$.

(3.29)

Using (3.22) and

adding    ${\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}-Q^2{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}(U_i^{+}+1)E_i(U_{i+1}^{+}+1)$    to each side

of (3.29) gives $$\begin{array}{l}{U}_{i+1}^{+}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}-Q^2(U_i^{+}+1)(U_{i+1}^{+}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})U_i^{+}\\ =T_i{U}_i^{+}{T}_i^{-1}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}-Q^2{U}_{i+1}^{+}(E_i{U}_i^{+}+{\displaystyle \frac{T_i^{-1}}{Q}}-{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})E_i(U_{i+1}^{+}+1)\\ =T_i{U}_i^{+}{T}_i^{-1}+{\displaystyle \frac{T_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}}-Q^2{U}_{i+1}^{+}(E_i{U}_i^{+}{E}_i+z{\displaystyle \frac{E_i}{Q}}-E_i{\displaystyle \frac{Y_i{Y}_{i+1}}{u^2-Y_i{Y}_{i+1}}})(U_{i+1}^{+}+1),\end{array}$$ completing the proof of (3.27). $\square$

The identities (4.13) and (4.14) of the following theorem are found in [GH1, Lemma 2.8(4)] and [BB, Lemma 7.4], respectively.

Define central elements $Z_{i-1}^{\left(\ell \right)}\in Z\left(W_{i-1}\right)$ by the generating functions $Z_{i-1}^{+}$ and $Z_{i-1}^{-}$ given by

$Z_{i-1}^{+}{E}_i=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_{i-1}^{\left(\ell \right)}{u}^{-\ell }{E}_i=E_i\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Y}_i^{\ell }{u}^{-\ell }\right)E_i=E_i{\displaystyle \frac{1}{1-Y_i{u}^{-1}}}{E}_i,$

(Zpd)

$Z_{i-1}^{-}{E}_i=\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Z}_{i-1}^{(-\ell )}{u}^{-\ell }{E}_i=E_i\left(\sum _{\ell \in {\mathbb{Z}}_{\ge 0}}{Y}_i^{-\ell }{u}^{-\ell }\right)E_i=E_i{\displaystyle \frac{1}{1-Y_i^{-1}{u}^{-1}}}{E}_i.$

(Zmd)

Then

$\begin{array}{l}(Z_{i-1}^{-}-{\displaystyle \frac{z}{q-q^{-1}}}-{\displaystyle \frac{u^2}{u^2-1}})(Z_{i-1}^{+}+{\displaystyle \frac{z^{-1}}{q-q^{-1}}}-{\displaystyle \frac{u^2}{u^2-1}})E_i\\ ={\displaystyle \frac{-(u^2-q^2)(u^2-q^{-2})}{{(u^2-1)}^2{(q-q^{-1})}^2}}{E}_i,\end{array}$

4.13

$$\begin{array}{l}\left(Z_{i+1}^{+}-Z_{i+1}^{\left(0\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1}\right)E_{i+1}\\ =\left(Z_i^{+}-Z_i^{\left(1\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1}\right)\frac{{\left(u-Y_i\right)}^2\left(u-q^{-2}{Y}_i^{-1}\right)\left(u-q^2{Y}_i^{-1}\right)}{{\left(u-Y_i^{-1}\right)}^2\left(u-q^2{Y}_i\right)\left(u-q^{-2}{Y}_i\right)}{E}_{i+1},\end{array}$$

4.14

$$\begin{array}{l}\left(Z_k^{+}-Z_k^{\left(0\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1}\right)E_{i+1}\\ =\left(Z_1^{+}-Z_1^{\left(0\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1}\right)\left(\prod _{i=1}^{k-1}\frac{{\left(u-Y_i\right)}^2\left(u-q^{-2}{Y}_i^{-1}\right)\left(u-q^2{Y}_i^{-1}\right)}{{\left(u-Y_i^{-1}\right)}^2\left(u-q^2{Y}_i\right)\left(u-q^{-2}{Y}_i\right)}\right)E_{i+1}\end{array}$$

4.15

		Proof.
	Since the generators $T_1,\dots ,T_{i-2}$, $E_1,\dots ,E_{i-2}$, and $Y_1,\dots ,Y_{i-1}$ of $W_{i-1}$ all commute with $E_i$ and $Y_i$ it follows that $Z_{i-1}^{\left(\ell \right)}\in Z\left(W_{i-1}\right)$. Multiply (3.26) on the right by $E_i$ and use $Z_{i-1}^{\left(0\right)}=1+(z-z^{-1})/(q-q^{-1})$ to get (3.32). Multiply (3.27) 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 $$\begin{array}{l}(Z_i^{+}-Z_i^{\left(0\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1})(1-{(q-q^{-1})}^2{U}_i^{+}(U_i^{+}+1))E_{i+1}\\ =(Z_{i-1}^{+}-Z_{i-1}^{\left(0\right)}+\frac{z}{q-q^{-1}}-\frac{1}{u^2-1})(1-{(q-q^{-1})}^2{U}_i^{-}(U_i^{-}+1))E_{i+1}.\end{array}$$ Then (3.33) follows from $$\begin{array}{l}\frac{1-{(q-q^{-1})}^2{U}_i^{-}(U_i^{-}+1)}{1-{(q-q^{-1})}^2{U}_i^{+}(U_i^{+}+1)}=\frac{1-{(q-q^{-1})}^2\frac{Y_i^{-1}}{u-Y_i^{-1}}\left(\frac{Y_i^{-1}}{u-Y_i^{-1}}+1\right)}{1-{\left(q-q^{-1}\right)}^2\frac{Y_i}{u-Y_i}\left(\frac{Y_i}{u-Y_i}+1\right)}\\ =\frac{\left({\left(u-Y_i^{-1}\right)}^2-{(q-q^{-1})}^2{Y}_i^{-1}u\right)\frac{1}{{\left(u-Y_i^{-1}\right)}^2}}{\left({\left(u-Y_i\right)}^2-{(q-q^{-1})}^2{Y}_{i}u\right)\frac{1}{{\left(u-Y_i\right)}^2}}=\frac{\left(u-q^{-2}{Y}_i^{-1}\right)\left(u-q^2{Y}_i^{-1}\right){\left(u-Y_i\right)}^2}{\left(u-q^{-2}{Y}_i\right)\left(u-q^2{Y}_i\right){\left(u-Y_i^{-1}\right)}^2}.\end{array}$$ and $Z_i^{\left(0\right)}=Z_{i-1}^{\left(0\right)}=1+(z-z^{-1})/(q-q^{-1})$. Finally, relation (3.34) follows, by induction, from (3.33). $\square$

Remark. Combining (4.13) and (4.15) yields a formula for $Z_{k-1}^{-}$ in terms of $Z_0^{+}$ and $Y_1,Y_2,\dots ,Y_{k-1}$. Using $Z_{i-1}^{\left(0\right)}=1+\frac{z-z^{-1}}{q-q^{-1}}$, rewrite (3.32) as

$\begin{array}{l}(z{Z}_{i-1}^{-}-z^{-1}{Z}_{i-1}^{+}-(z-z^{-1})Z_{i-1}^{\left(0\right)})E_i\\ =(q-q^{-1})({\displaystyle \frac{1}{u^2-1}}(Z_{i-1}^{+}+Z_{i-1}^{-}-Z_{i-1}^{(0)})-(Z_{i-1}^{-}-Z_{i-1}^{\left(0\right)})(Z_{i-1}^{+}-Z_{i-1}^{\left(0\right)}))E_i,\end{array}$

(3.35)

and take the coefficient of $u^{-\ell }$ in (3.32) to get

$\begin{array}{l}(z{Z}_{i-1}^{(-\ell )}-z^{-1}{Z}_{i-1}^{\left(\ell \right)})E_i\\ =(q-q^{-1})\left(\begin{array}{l}{Z}_{i-1}^{(\ell -2)}+Z_{i-1}^{(\ell -4)}+\cdots +Z_{i-1}^{(-(\ell -2\left)\right)}\\ -(Z_{i-1}^{(\ell -1)}{Z}_{i-1}^{(-1)}+Z_{i-1}^{(\ell -2)}{Z}_{i-1}^{(-2)}+\cdots +Z_{i-1}^{\left(1\right)}{Z}_{i-1}^{(-(\ell -1\left)\right)})\end{array}\right)E_i,\end{array}$

(3.36)

from [GH1, Lemma 2.8(4)].

Notes and References

This section is based on forthcoming joint work with Z. Daugherty and R. Virk [DRV]. The remarkable recursions for generating central elements which appear in Theorems ??? and ??? were given by Nazarov [Naz] in the degenerate case, and then extended to the affine BMW algebra by Beliakova-Blanchet [BB]. Another proof in the affine cyclotomic case appears in [RX2, Lemma 4.21] and, in the degenerate case, in [AMR, Lemma 4.15]. In all of these proofs, the recursion is obtained by a rather mysterious and tedious computation. We show that there is an "intertwiner like identity in the full algebra which, when "projected to the center" produces the Nazarov recursions. Our approach dramatically simplifies the proof and gives some insight into where these recursions are coming from. Moreover, the proof is exactly analogous in both the degenerate and the affine cases, and includes the parameter $ϵ$, so that both the orthogonal and symplectic cases are treated simultaneously.

References

[AMR] S. Ariki, A. Mathas, and H. Rui, Cyclotomic Nazarov-Wenzl algebras, Nagoya Math. J. 182 (2006), 47-134. MR2235339 (2007d:20005)

[BB] A. Beliakova and C. Blanchet, Skein construction of idempotents in Birman-Murakami-Wenzl algebras, Math. Ann. 321 (2001), 347-373. MR1866492 (2002h:57018)

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

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

[GH1] F. Goodman and H. Hauschild Mosley, Cyclotomic Birman-Wenzl-Murakami algebras. I. Freeness and realization as tangle algebras, J. Knot Theory Ramifications 18 (2009), 1089-1127. MR2554337 (2010j:57014)

[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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