The double affine braid group

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

Last update: 19 September 2012

The double affine braid group

Following Ion-Sahi [IS] the double affine braid group $\stackrel{\sim }{ℬ}$ is generated by $T_0,T_0^{\prime },T_0^{\vee },T_1,\dots ,T_n$ with

$$ℬ=⟨T_0,T_1,\dots ,T_n⟩,{ℬ}^{\prime }⟨T_0^{\prime },T_1,\dots ,T_n⟩,{ℬ}^{\vee }=⟨T_0^{\vee },T_1,\dots ,T_n⟩,$$

which are affine braid groups,

$$q\in Z\left(\stackrel{\sim }{ℬ}\right),\text{where}q=T_0{T}_0^{\prime }{T}_0^{\vee }{T}_{s_{\phi }},$$

and

$$T_0{T}_1^{-1}{T}_0^{\vee }{T}_1=T_1^{-1}{T}_0^{\vee }{T}_1{T}_0,\text{if nodes 0 and 1 are connected by a double edge.}$$

The conversion between presentations is given by

$$T_0=Y^{{\phi }^{\vee }}{T}_{s_{\phi }}^{-1},T_0^{\vee }=X^{\phi }{T}_{s_{\phi }},q=T_0{T}_0^{\prime }{T}_0^{\vee }{T}_{s_{\phi }}$$

We should extend this to include the effect of $\Omega$ by using the relations

$$g=Y^{{\omega }_g}{T}_{w_0{w}_g}^{-1}\text{and}{g}^{\vee }=X^{{\omega }_g}{T}_{w_0{w}_g},$$

or whatever the correct versions of these are.

The braid group ${𝒜}_3$ on 3 strands is generated by $a_1,a_2$ with relation $a_1{a}_2{a}_1=a_2{a}_1{a}_2\text{.}$ Using the automorphism of the Dynkin diagram PICTURE build

$${𝒜}_3⋊\mathbb{Z}/2\mathbb{Z}=\{be\mid b\in {𝒜}_3,e\in \mathbb{Z}/2\mathbb{Z}\},\text{with}e{a}_1=a_{2}e,e{a}_2=a_{1}e,e^2=1\text{.}$$

The isomorphism

$$\begin{array}{ccc}{𝒜}_3⋊\mathbb{Z}/2\mathbb{Z}& \cong & {GL}_2\left(\mathbb{Z}\right)\\ a_1& ⟼& \left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)\\ a_2& ⟼& \left(\begin{array}{cc}1& 0\\ -1& 1\end{array}\right)\\ e& ⟼& \left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)\end{array}$$

giving exact sequences

$$\left\{1\right\}⟶⟨{\left(a_1{a}_2{a}_1\right)}^4⟩⟶{𝒜}_3⟶S{L}_2\left(\mathbb{Z}\right)⟶\left\{1\right\}$$ $$\left\{1\right\}⟶Z\left({𝒜}_3\right)=⟨{\left(a_1{a}_2{a}_1\right)}^2⟩⟶{𝒜}_3⟶PS{L}_2\left(\mathbb{Z}\right)⟶\left\{1\right\}$$

The group $G{L}_2\left(\mathbb{Z}\right)$ acts on $\stackrel{\sim }{ℬ}$ by automorphisms via

$$\begin{array}{cccc}{a}_1:& \stackrel{\sim }{ℬ}& ⟶& \stackrel{\sim }{ℬ}\\ & T_0& ⟼& T_0^{\prime }\\ & T_0^{\prime }& ⟼& {\left(T_0^{\prime }\right)}^{-1}{T}_0{T}_0^{\prime }\\ & T_0^{\vee }& ⟼& T_0^{\vee }\\ & T_i& ⟼& T_i\end{array}\begin{array}{cccc}{a}_2:& \stackrel{\sim }{ℬ}& ⟶& \stackrel{\sim }{ℬ}\\ & T_0& ⟼& T_0\\ & T_0^{\prime }& ⟼& T_0^{\vee }\\ & T_0^{\vee }& ⟼& {\left(T_0^{\prime }\right)}^{-1}{T}_0^{\prime }{T}_0^{\vee }\\ & T_i& ⟼& T_i\end{array}$$ $$\begin{array}{cccc}e& \stackrel{\sim }{ℬ}& ⟶& \stackrel{\sim }{ℬ}\\ & T_0& ⟼& {\left(T_0^{\vee }\right)}^{-1}\\ & T_0^{\prime }& ⟼& {\left(T_0^{\prime }\right)}^{-1}\\ & T_0^{\vee }& ⟼& {\left(T_0\right)}^{-1}\\ & T_i& ⟼& T_i^{-1}\\ & X^{\mu }& ⟼& Y^{\mu }\end{array}$$

as the automorphism of $\stackrel{\sim }{cB}\text{.}$ The existence of the automorphism $e$ is sometimes called duality.

The affine braid group ${ℬ}^{\vee }$ is given by $T_0^{\vee },T_0^{\vee },\dots ,T_n^{\vee }$ and ${\Omega }^{\vee }$ with relations

$$\begin{array}{cc}\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_i^{\vee }{T}_j^{\vee }\dots }=\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_j^{\vee }{T}_i^{\vee }\dots },\text{and}{g}^{\vee }{T}_i^{\vee }{\left(g^{\vee }\right)}^{-1}=T_{g\left(i\right)}^{\vee },\text{for}{g}^{\vee }\in {\Omega }^{\vee }\text{.}& \text{(1)}\end{array}$$

The affine Weyl group

$$\begin{array}{cc}{W}^{\vee }=\{X^{\mu }w\mid \mu \in {𝔥}_{\mathbb{Z}}^{*},w\in W_0\}\text{acts on}\stackrel{\sim }{Y}=\{q^{k/e}{Y}^{{\lambda }^{\vee }}\mid k\in \mathbb{Z},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\},& \text{(2)}\end{array}$$

by conjugation. Write

$$\begin{array}{cc}{Y}^{v{\lambda }^{\vee }}=v{Y}^{{\lambda }^{\vee }}{v}^{-1},\text{for}v\in W^{\vee },{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\text{.}& \text{(3)}\end{array}$$

The double affine braid group $\stackrel{\sim }{B}$ is the group generated by $B^{\vee }$ and $\stackrel{\sim }{Y}$ with relations

$$\begin{array}{cc}\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_i^{\vee }{T}_j^{\vee }\dots }=\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_j^{\vee }{T}_i^{\vee }\dots },g^{\vee }{T}_i^{\vee }{\left(g^{\vee }\right)}^{-1}=T_{g\left(i\right)}^{\vee },g^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{g^{\vee }{\lambda }^{\vee }}{g}^{\vee },& \text{(4)}\end{array}$$ $$\begin{array}{cc}{\left(T_i^{\vee }\right)}^{-1}{Y}^{{\lambda }^{\vee }}=\{\begin{array}{ccc}{Y}^{s_i^{\vee }{\lambda }^{\vee }}{\left(T_i^{\vee }\right)}^{-1},& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=0,\\ Y^{s_i^{\vee }{\lambda }^{\vee }}{T}_i^{\vee },& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=1,\end{array}& \text{(5)}\end{array}$$

for $g^{\vee }\in {\Omega }^{\vee },{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}$ and $i=0,1,\dots ,n\text{.}$

For $w\in W^{\vee },$ view a reduced word $w=g{s}_{i_1}\dots s_{i_{\ell }}$ as a minimal length path $p$ from the fundamental alcove to $w$ in ${𝔥}_{ℝ}$ and define

with respect to the periodic orientation (see (??) and the pictures in the appendix). For $v\in W,$ view a reduced word $v=g^{\vee }{s}_{i_{\ell }}^{\vee }\dots s_{i_{\ell }}^{\vee }$ as a minimal length path $p^{\vee }$ from the fundamental alcove to $v$ in ${𝔥}_{ℝ}^{*}$ and define

Let $T_i^{\vee }=T_i,$ for $i=1,2,\dots ,n,$

$$\begin{array}{cc}{g}^{\vee }=X^{\omega g}{T}_{w_g{w}_0}^{\vee },{\left(T_0^{\vee }\right)}^{-1}=X^{\phi }{T}_{s_{\phi }}^{\vee },g=Y^{{\omega }_g^{\vee }}{T}_{w_0{w}_g}^{-1},T_0=Y^{{\phi }^{\vee }}{T}_{s_{\phi }}^{-1}\text{.}& \text{(8)}\end{array}$$

where $\phi$ and ${\phi }^{\vee }$ are as in (??) and, using the action in (2.7), ${\omega }_g=g^{\vee }·0$ and $w_g$ is the longest element of the stabilizer of ${\omega }_g$ in $W_0\text{.}$

The following theorem, discovered by Cherednik [1, Thm. 2.2], is proved in [15, 3.5-3.7], in [7], and in [6, 4.13-4.18].

(Duality) Let $Y^d=q^{-1}\text{.}$ The double affine braid group $\stackrel{\sim }{ℬ}$ is generated by $T_0^{\vee },T_1^{\vee },\dots ,T_n^{\vee },{\Omega }^{\vee }$ and $Y$ with relations

$$\begin{array}{cc}\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_i^{\vee }{T}_j^{\vee }\dots }=\underset{\underset{m_{ij}^{\vee }}{⏟}}{T_j^{\vee }{T}_i^{\vee }\dots },g^{\vee }{T}_i^{\vee }{\left(g^{\vee }\right)}^{-1}=T_{{\sigma }^{\vee }\left(i\right)}^{\vee },g^{\vee }{Y}^{{\lambda }^{\vee }}=Y^{g^{\vee }{\lambda }^{\vee }}{g}^{\vee },& \text{(2.28)}\end{array}$$

for $g^{\vee }\in {\Omega }^{\vee },$ and

$$\begin{array}{cc}\begin{array}{ccc}{T}_i^{\vee }=Y^{s_i^{\vee }{\lambda }^{\vee }}{T}_i^{\vee },& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=0,\\ {\left(T_i^{\vee }\right)}^{-1}{Y}^{{\lambda }^{\vee }}{\left(T_i^{\vee }\right)}^{-1}=Y^{s_i^{\vee }{\lambda }^{\vee }},& & \text{if}⟨{\lambda }^{\vee },{\alpha }_i⟩=1,\end{array}\text{for}i=0,1,\dots ,n,& \text{(2.29)}\end{array}$$

where the action of $W$ on ${𝔥}_{\mathbb{Z}}\oplus \mathbb{Z}d$ is as in (2.10).

Notes and References

This page is taken from a paper entitled Relating double affine Hecke algebras and Rational Cherednik algebras by Stephen Griffeth and Arun Ram, May 4, 2009. (2.7) is a reference to the section entitled The double affine Weyl group.

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