Double affine braid groups and Hecke algebras of classical type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 12 March 2012


I thank M. Noumi for providing me with a copy of his article [N95].

Type (Cn,Cn), for n2

Use a graphical notation for relations so that gi gj means gigj = gjgi, gi gj means gigjgi = gjgigj,   and gi gj means gigjgigj = gjgigjgi.

The double affine braid group

The double affine braid group is the group 𝔅˜n is generated by q 1 2 T1...Tn and T0,T0,T0 with relations q 1 2 Z(𝔅˜n),  T0T1-1T0T1 = T1-1T0T1T0,  T0-1T1(T0)-1T1-1 = T1(T0)-1T1-1T0-1, (Daff 2.1) T0 T1 T2 Tn-2 Tn-1 Tn T0 T1 T2 Tn-2 Tn-1 Tn T0 T1 T2 Tn-2 Tn-1 Tn (Daff 2.2) There are redundant relations in this presentation of 𝔅˜n. In particular, the second relation in (Daff 2.1) implies T1T0T1-1T0 = T0T1T0-1T1, giving that T0 commutes with T1T0T1-1 and thus that T0-1 commutes with T1(T0)-1T1-1. But this is the last relation in (Daff 2.1).

The elements of the braid group on n+3 strands given by T0 = and Tn = T0 = and Ti = i i+1 , satisfy the relations in (Daff 2.1) and (Daff 2.2).

For j=1...n define Yεj and Xεj in 𝔅˜n by Yε1 = T0T1TnT1 and Yεj+1 = Tj-1 Yεj Tj-1, Xε1 = (T0)-1T1-1Tn-1T1-1 and Xεj+1 = TjXεjTj, (Daff 2.3) for j=1...n-1. Pictorially, Yεj = j and Xεj = j . The pictorial viewpoint makes it straightforward to verify that the subgroups X= Xε1,...,Xεn and Y= Yε1,...,Yεn are abelian.

In 𝔅˜n let Tsε1 = T1TnT1 and T0 = q-12 Xε1 T0-1 = q-12 (T0)-1 T1-1 Tn-1 T1-1 T0-1 = q-12 (T0)-1 Y-ε1 (Daff 2.4) DRAW A PICTURE OF THIS ELEMENT? Then, with Tsε1 = T1TnT1, T0 T1 T2 Tn-2 Tn-1 Tn and T0T0T0Tsε1 = q-12, since


The affine braid groups 𝔅n,𝔅n and 𝔅n are the subgroups of 𝔅˜n given by 𝔅n = T0,T1,...,Tn, 𝔅n = T0,T1,...,Tn and 𝔅n = T0,T1,...,Tn.

  1. The double affine braid group 𝔅˜n is generated by q12, the affine braid group 𝔅n, and the abelian group Y with additional relations q12 Z(𝔅˜n), Yεi+1 = Ti-1 Yεi Ti-1, for   i=1,...,n-1, and Yε1 Yε2 Yεj-1 Tj Yεj+2 Yεn-1 Yεn   for   j=0,...,n. (Daff 2.5)
  2. The double affine braid group 𝔅˜n is generated by q12, the affine braid group 𝔅n, and the abelian group X with additional relations q12 Z(𝔅˜n), and Xε1 Xε2 Xεj-1 Tj Xεj+2 Xεn-1 Xεn   for   j=0,...,n. (Daff 2.6)

(Daff 2.1) (a): The relations (Daff 2.5) for j0 follow easily from the pictorial expression for Yεj and the additional relations (Daff 2.5) for j=0 follow from the definition of Yεj and the relation T0T1-1T0T1 = T1-1T0T1T0.

(a) (Daff 2.1): Since T0 = Yε1T1-1 Tn-1-1Tn-1Tn-1-1 T1-1, the generators in (Daff 2.1) can be written in terms of the generators in (a). We need to show that the relations (Daff 2.5) imply T0T1T0T1 = T1T0T1T0, T0T1-1T0T1 = T1-1T0T1T0, and T0Ti = TiT0, for   i=2,3,...,n. Pictorially, Tsφ = = T1TnT1, so that TiTsφ-1 = Tsφ-1Ti, for i=2,3,...,n. Thus, T0Ti = Yε1Tsφ-1Ti = Yε1TiTsφ-1 = TiYε1Tsφ-1 = TiT0, for   i=2,...,n. Using T0Yε2 = Yε2T0,  Yε2 = T1-1 Yε1 T1-1 and T0Ti = TiT0 for i=2,...,n, T0 T1-1 T0T1 = T0 T1-1 Yε1 T1-1 Tn-1 T1-1 T1 = T0 Yε2 T2-1 Tn-1 T2-1 = Yε2 T0 T2-1 Tn-1 T2-1 = T1-1 Yε1 T1-1 T2-1 Tn-1 T2-1 T0 = T1-1 T0T1 T0. Since Yε1 = T0T1 Tn T1 and Yε2 = T1-1 Yε1 T1-1 = T1-1 T0T1 Tn T2 we have Yε1+ε2 = T0T1 Tn T2T0T1 Tn T2 = T0T1T0T2 Tn T2T1 Tn T2. Then T1-1 Yε1+ε2 = T1-1 Yε1 Yε2 = T1-1 Yε1 T1-1 Yε1 T1-1 = Yε2 Yε1 T1-1 = Yε2+ε1 T1-1 gives T1 Yε1+ε2 = T1T0T1T0T2 Tn T2T1 Tn T2 = Yε1+ε2 T1 = T0T1T0T2 Tn T2Tsφ = T0T1T0TsφT2 Tn T2 = T0T1T0T1T2 Tn T2T1 Tn T2, since TsφTi = TiTsφ for i=2,...,n. thus T1T0T1T0 = T0T1T0T1.

(b) Since ι(Yεi) = Xεi, applying the automorphism ι to the presentation in (a) gives the presentation in (b).

Letting (see [M03, (1.4.3)]) αn = 2εn and αi = εi-εi+1, for   i=1,...,n-1, the relations (Daff 2.5) are equivalent to Y being abelian and Ti Yλ = Ysiλ Ti, if   λ,αi = 0, and Ti-1 Yλ Ti-1 = Ysiλ,   if   λ,αi = 1. The last relation is vacuous for i=n since λ,2εn 2 for all λ 𝔥.

The presentation of the double affine braid group given in (???) is implicit in [H06, ???] and quite explicit in [?, ???].

Automorphisms of the double affine braid group

(Duality) [M03, (3.5.1)] There is an involutive automorphism ι: 𝔅˜n 𝔅˜n given by ι(q12) = q-12, ι(T0) = (T0)-1, ι(T0) = T0-1, and ι(Ti) = Ti-1   for   i=1,...,n.

The involution ι fixes the first relation in (Daff 2.1), switches the second and third relations in (Daff 2.1), and switches the relations in (Daff 2.2).

Note that ι(Yεi) = Xεi, for   i=1,...,n.

The double affine Weyl group

Type (Cn,Cn)

Let 𝔥 = i=1n εi and 𝔥* = i=1n εi, with εi,εj = δij. The Weyl group W0 is generated by s1,...,sn with s1 s2 sn-2 sn-1 sn and si2=1, (Daff 3.1) for i=1,...,n. The group W0 acts on 𝔥 by siεi = εi+1,   siεi+1 = εi, for   i=1,...,n-1   and   ji,i+1, and snεn = -εn, snεj = εj, for   jn. (Daff 3.2) Then W0 acts on 𝔥* by setting wμ,λ = μ,w-1λ, for   wW0, μ𝔥*, λ𝔥. (Daff 3.3)

The double affine Weyl group is W˜ = {qk/2XμwYλ  |  k, wW0, μ𝔥*, λ𝔥} (Daff 3.4) with q12 Z(W˜), wXμ = Xwμw, wYλ = Ywλw, (Daff 3.5) XμXν = Xμ+ν , YλYσ = Yλ+σ , and   XμYλ = qμ,λYλXμ, (Daff 3.6) for μ,ν𝔥*,λ,σ𝔥,wW0.

Noting that q= Xε1 sε1 Y-ε1 Xε1 sε1 Y-ε1   where   sε1 = s1s2 sn-1snsn-1 s2s1, (Daff 3.7) define s0 = sε1 Y-ε1 , s0 = Xε1 sε1 , s0 = q-12 Xε1 sε1 Y-ε1 . (Daff 3.8) The following proposition shows that W˜ is a quotient of that double affine braid group 𝔅˜n.

The group W˜ is presented by generators q12 s0 and relations q12 Z(W˜),  (s0)2 = 1, and si2 = 1,  for   i=0,...,n, s0 s1 s2 sn-2 sn-1 sn s0 s1 s2 sn-2 sn-1 sn and s0 s1s0s1 = s1s0s1 s0. (Daff 3.9)

With definitions of s0,s0 and s0 as in (Daff 3.8) it is straightforward to check that the relations of Proposition 3.1 hold in W˜.

Since Xε1 = s0 s1sns1, and Xεj+1 = sj Xεj sj, and Yε1 = s0 s1sns1, and Yεj+1 = sj Yεj sj, (Daff 3.10) the generators of W˜ can be written in terms of q12 s0 . Theorem 2.1(a) shows that the group presented in Proposition 3.1 satisfies the relations q12 Z(𝔅˜n), Yεi Yεj = Yεj Yεi ,for   1i,jn, (Daff 3.11) sn Yεj = Yεj sn,   for   j=1,...n-1, s0 Yεj = Yεj s0,  for   j=2,...,n, (Daff 3.12) Yεi+1 = si Yεi,  and   si Yεj = Yεj si, for   i=1,...,n-1   and   ji,i+1. (Daff 3.13) Similarly Theorem 2.1(b) shows that the group presented in Proposition 3.1 satisfies the relations q12 Z(𝔅˜n), Xεi Xεj = Xεj Xεi, for   1i,jn, (Daff 3.14) sn Xεj = Xεj sn,  for   j=1,...,n-1, s0 Xεj = Xεj s0,  for   j=2,...,n. (Daff 3.15) Xεi+1 = si Xεi si,   and   si Xεj = Xεj si, for   i=1,...,n-1   and   ji,i+1. (Daff 3.16) The relations in (Daff 3.11) and (Daff 3.14) imply the first relation in (Daff 3.5) and the first two relations of (Daff 3.6).

The relation s02=1 gives sε1 Y-ε1 = Yε1 sε1 which implies sn Y-εn = Yεn sn. Similarly   sn Xεn = X-εn sn follows from (s0)2=1. In combination with the relations in (Daff 3.12), (Daff 3.13), (Daff 3.15), (Daff 3.16) these imply the last two relations in (Daff 3.5).

The relation s0 Yεj = Yεj s0 gives Xε1 sε1 Yεj = Yεj Xε1 sε1 so that, for   j=2,...,n,   Xε1 Yεj = Yεj Xε1; similarly   Yε1 Xεj = Xεj Yε1 follows from s0 Xεj = Xεj s0, for j=2,...,n. Finally, (s0)2=1 gives Xε1 Yε1 = q Yε1 Xε1, which establishes the third relation of (Daff 3.6).

The group algebra of W˜ contains two Laurent polynomial rings [X±ε1,...,X±εn] &nsbp; and   [Y±ε1,...,Y±εn]   with   Xεi Yεj = qδij Yεj Xεi. This is analogous to the Weyl algebra generated by the two polynomial rings [x1,...,xn]   and   [ x1 ,..., xn ]   with   [ xj ,xi] =δij. CHECK AGAIN?

Affine root systems

The notation for affine root systems is WS = {sa  |  aS} and S=(S1,S2), where S1 = {aS  |  12aS} and S2 = {aS  |  2aS}.

Define O1 = {±εi+r  |  1in,r} O2 = {±2εi+2r  |  1in,r} O3 = {±εi+12(2r+1)  |  1in,r} O4 = {±2εi+2r+1  |  1in,r} O5- = {±(εi-εj)+r  |  1i<jn,r} O5+ = {±(εi+εj)+r  |  1i<jn,r} and O5=O5-O5+ so that the affine root systems of classical type are missing Note that εnO1,  2εnO2,  ε1+12δO3,  2ε1+δO4, and εi-εi+1O5,   for   i=1,...,n-1.

(Cn,Cn) = C-BCnII O4 O3 (Cn,BCn) = C-BCnI (BCn,Cn) = C-BCnIV O2 O3 O2 O1 Cn = C-Bn (Bn,Bn) = B-BCn BCn = C-BCnIII Cn O3 O2 O1 O4 O4 Bn Bn = B-Cn O1 O2 Dn O5+ An-1 O5-
Where the left notation is the notation in [M03, §1.3] and the right notation is that of [BT]. The Weyl groups of these are WS = { WCn, for types   Cn Cn BCn (BCn,Cn) (Cn,BCn) (Cn,Cn) , WBn, for types   Bn Bn (Bn,Bn) , WDn, for type   Dn } where WCn = W0𝔥 WBn = W0𝔥_ WDn = W0_𝔥_ with W0=G2,1,n, W0_=G2,2,n and 𝔥 = i=1n εi, and 𝔥_ = {λ1ε1++λnεn  |  λ1++λn =0 mod 2}, SAY SOMETHING ABOUT WS orbits on S. HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When n=2 define O1 = {±εi+r  |  1i2,r} O2 = {±2εi+2r  |  1i2,r} O3 = {±εi+12(2r+1)  |  1i2,r} O4 = {±2εi+2r+1  |  1i2,r} O5- = {±(ε1-ε2)+r  |  r} O5+ = {±(ε1+ε2)+r  |  r} Then the "classical" affine root systems of rank 2 are

(C2,C2) O3 O4 O2 O1 (BC2,C2) (C2,BC2) O2 O4 O3 O4 O3 BC2 (C2,C2) C2 C2 O1 O2 O1 O1 O3 O2 O4 O4 A1×A1 O5+ A1 O5-
When n=1 define O1 = {±ε1+r  |  r} O2 = {±2ε1+2r  |  r} O3 = {±ε1+12(2r+1)  |  r} O4 = {±2ε1+2r+1  |  r} Then the "classical" affine root systems of rank 1 are
(C1,C1) O3 O4 (BC1,C1) (C1,BC1) O2 O2 BC1 O4 A1     A1 O1 O1O3

Notes and References

Where are these from?



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