Affine and degenerate affine BMW algebras: Actions on tensor space

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

Last update: 15 October 2013

Symplectic and orthogonal higher Casimir elements

Our final goal in this paper will be to connect the central elements appearing naturally as parameters of the affine and degenerate affine BMW algebras (see Theorems 2.2 and 2.5) to higher Casimir elements for orthogonal and symplectic Lie algebras and quantum groups. In the degenerate case, we explain how the generating function for zV() derived in Theorem 3.3 can be matched up with the generating functions for central elements given by Perelomov-Popov in [PPo0214703, PPo0205621]. Expositions of the Perelomov-Popov results are also in [Mol2355506, §7.1] and [Zeh1973, §127]. In the affine case we show how the formula for ZV() in Corollary 3.6 can be derived as a special case of a remarkable identity for central elements in quantum groups discovered by Baumann [Bau1620662, Thm. 1].

The central elements zV() as higher Casimir elements

Returning to the notation developed in the preliminaries of Section 2, let 𝔤=𝔤𝔩r with nondegenerate ad-invariant form , as in (2.1) and operator γ=γ𝔤𝔩 as in (2.2). Then (idtrV) (γ)= i1,i2,,i Ei1i2 Ei2i3 Eii1 are the central elements of U𝔤𝔩n found, for example, in Gelfand [Gel1950, (3)]. Perelomov-Popov [PPo0214703, PPo0205621] generalized this construction to 𝔤=𝔰𝔬2r+1, 𝔰𝔭2r, and 𝔰𝔬2r, by letting Fij be the natural spanning set for 𝔤 given in (2.7), viewing F=(Fij)i,jVˆ as a matrix with entries in 𝔤, and writing trFk= i1,i2,,ikVˆ Fi1i2 Fi2i3 Fiki1 fork>0, (4.1) as an element of the enveloping algebra U𝔤 (see [Mol2355506, Thm. 7.1.7]). These elements are central in U𝔤 and Perelomov-Popov gave the following generating function formula for their Harish-Chandra images (see [Zeh1973, §127]). The proof we give below shows that the result of Perelemov-Popov is equivalent to Theorem 3.3 (which we obtained from the degenerate affine BMW algebra and Schur-Weyl duality). A proof of Theorem 4.1 using the theory of twisted Yangians is given in [Mol2355506, §7.1].

(Perelomov-Popov) [Mol2355506, Cor. 7.1.8] Let 𝔤=𝔰𝔬2r+1 or 𝔰𝔭2r or 𝔰𝔬2r, use notations for 𝔥* as in Section 2 and let h1,,hr be the basis of 𝔥 dual to the orthonormal basis ε1,,εr of 𝔥*. Let ρr=12-12y, l0=0 in the case that 𝔤=𝔰𝔬2r+1 and let li=-l-i=hi +ρi,fori =1,2,,r,where ρi=12 (y-2i+1). Then π0 ( 1+ x+12 x+12-12ε ( 0 (-1)tr(Fk) (x+ρr)+1 ) ) =iVˆ x+li+1 x+li . (4.2)


By (2.11), γ=12 i,jVˆ FijFji. Let η:𝔤End(V) be the defining representation. Since F-j,-i=-θijFij, (idη)(γ) = 12i,jVˆ Fij ( Eji-θji E-i,-j ) =12i,jVˆ ( FijEji- θjiF-j,-i Eji ) = 12i,jVˆ ( Fij-θji F-j,-i ) Eji =12i,jVˆ (Fij+Fij) Eji = i,jVˆ FijEji= Ft=-θFθ, whereθ= ( εid0 0id ) . Thus (idtrV) (γk)= tr((Ft)k) =tr((-θFθ)k) =(-1)ktr (θ2Fk)= (-1)ktr (Fk), (4.3) which provides the connection of the elements z0() appearing in Theorems 2.2 and 3.3 to the elements in (4.1).

In order to transform the generating function for the elements (idtrV)(γ) into the generating function for the elements (idtrV)((12y+γ)), notice 0 (idtrV) ((12y+γ)) u-=(idtrV) (11-(12y+γ)u-1) =(idtrV) (11-12yu-1-γu-1) =(idtrV) ( (11-12yu-1) (11-γu-11-12yu-1) ) =(idtrV) ( 11-12yu-1 0 γu- (1-12yu-1) ) =u ( 0 (idtrV)(γ) (u-12y)+1 ) = ( 0 (idtrV)(γ) (u-12+ρr)+1 ) ,whereρr= 12-12y. Then Theorem 3.3 is equivalent to π0 ( 1+εuεu-12 ( 0 (idtrV)(γ) (u-12+ρr)+1 ) ) =π0 ( 1+εεu-12 0 (idtrV) ((12y+γ)) u- ) = (εu+12) (εu-12) (u+12y-r) (u-12y+r) σρ ( i=1r (u+hi+12) (u+hi-12) (u-hi+12) (u-hi-12) ) = (εu+12) (εu-12) (u+12y-r) (u-12y+r) ( i=1r (u+hi+ρi+12) (u+hi-ρi+12) (u-hi-ρi+12) (u-hi-ρi-12) ) = (u-12+12ε+12) (u-12-12ε+12) (u-12+12y-r+12) (u-12-12y+r+12) ( i=1r (u-12+hi+ρi+1) (u-12+hi+ρi) (u-12-hi-ρi+1) (u-12-hi-ρi) ) . Replacing x=u-12, π0 ( 1+ x+12 x+12-12ε ( 0 (idtrV)(γ) (x+ρr)+1 ) ) = (x+12ε+12) (x-12ε+12) (x+12y-r+12) (x-12y+r+12) ( i=1r (x+hi+ρi+1) (x+hi+ρi) (x-hi-ρi+1) (x-hi-ρi) ) Since (x+12ε+12) (x-12ε+12) (x+12y-r+12) (x-12y+r+12) = { x+l0+1x+l0, if𝔤=𝔰𝔬2r+1, 1, if𝔤=𝔰𝔭2r or𝔰𝔬2r, it follows that π0 ( 1+ x+12 x+12-12ε ( 0 (idtrV)(γ) (x+ρr)+1 ) ) =iVˆ x+li+1 x+li . In combination with (4.3), this demonstrates the equivalence of the Perelomov-Popov theorem and Theorem 3.3.

Since π0 and ρ=ρ1ε1++ρnεn are defined via a specific choice of positive roots, each side of (4.2) depends on that choice, though the identity does not. The preferred choice of positive roots in [Mol2355506, p. 139] differs from our preferred choice in (2.12) by the action of the Weyl group element w= (1,-r) (2,-(r-1)) (r-1,-2) (r,-1) , in cycle notation.

The central elements ZV() as quantum higher Casimir elements

In this section, we show how the formula for the central elements ZV() in Corollary 3.6 is related to an identity for central elements in quantum groups discovered by Baumann in [Bau1620662, Thm. 1]. To do this we rewrite the Baumann identity for 𝔤=𝔰𝔭2r, 𝔰𝔬2r+1 and 𝔰𝔬2r and λ=ε1 in terms of Weyl characters indexed by partitions. Then a theorem of Turaev and Wenzl computing (idqtrL(ν))(21) provides a conversion between the expansion in Corollary 3.6 and the expansion obtained from Baumann’s identity.

For λ𝔥* define the Weyl character sλ=aλ+ρaρ ,whereaμ= wW0det(w) ewμ. The expressions sλ and aμ are elements of the group algebra of 𝔥*, [𝔥*]=-span{eν|ν𝔥*} with eμeν=eμ+ν. If wW0 then awμ=det(w) aμand swμ=det (w)sμ, (4.4) where the dot action of W0 on 𝔥* is given by wμ=w(μ+ρ) -ρ,forw W0,μ𝔥*. (4.5) The W0-invariants in [𝔥*] are [𝔥*]W0=-span{sλ|λdominant integral}. For 0 let Ψ: [𝔥*]W0 Z(Uh𝔤) sν (idqtrL(ν))((21)) . (4.6) By [Dri1990, Prop. 1.2], the map Ψ is a vector space isomorphism.

[Bau1620662, Thm. 1] For 0 define mλ()= wW0 q2wλ,ρ swλ.Then Ψ (mλ())= Ψ1 (mλ(1/)) .

In the same setting as in Theorem 3.5, let y= { 2r, if𝔤= 𝔰𝔬2r+1, 2r+1, if𝔤=𝔰𝔭2r , 2r-1, ifg=𝔰𝔬2r, ε= { 1, if𝔤= 𝔰𝔬2r+1, -1, if𝔤=𝔰𝔭2r , 1, ifg=𝔰𝔬2r, V=L(ε1), z=εqy, and let ZV() be the central elements in the Drinfeld-Jimbo quantum group Uh𝔤 which are given by ZV()=ε(idqtrL(ε1))((z21)). Then, for 1, ZV()=εΨ1 ( c+zm=max(-r+1,1) (-1)-m q-(-2m+1) s(m,1=m+1-1) +zm=max(y-+1,1)-y+r q-(-2m+1) (-1)m-+y s(m,1m-+y-1) ) , where c is given by c= { 1 if is even and<y, ory and𝔤= 𝔰𝔬2r+1, 0 otherwise.


The Weyl group W0 for 𝔤=𝔰𝔬2r+1 or 𝔤=𝔰𝔭2r is the group of signed permutations. With positive roots as in (2.12), the simple reflections are si=(i,i+1) (the transposition switching εi and εi+1, for i=1,2,,r-1) and sr=(r,-r) (the transposition switching the sign of εr). For 𝔤=𝔰𝔬2r, the Weyl group W0 consists of signed permutations with an even number of signs, with simple reflections si=(i,i+1) for i=1,2,,r-1, and sr=(r-1,-r)(r,-(r-1)).

To prove the desired identity, we will use the second identity in (4.4) to relabel the Weyl characters swλ appearing in mε1() =wW0 q2wε1,ρ swε1and mε1(1/) =wW0 q2wε1,ρ swε1 by dominant integral weights. By (2.13), ρi+1=ρi-1 and ρr=12(y-2r+1). So if μ=μ1ε1++μrεr then siμ=μ1++ μi-1εi-1+ (μi+1-1)εi +(μi+1)εi+1 +μi+2εi+2 ++μrεr for i=1,2,,r-1, and srμ=μ1ε1 +μr-1 εr-1+ (-μr-(y-2r+1)) εr,if𝔤= 𝔰𝔬2r+1or 𝔰𝔭2r, and srμ=μ1ε1+ μr-1εr-2+ (-μr-1)εr-1 +(-μr-1-1) εr,if𝔤= 𝔰𝔬2r. In particular, sεi=0 if 0<<i, and sεi= ss1s2si-1εi =(-1)i-1 s(-i+1)ε1+ε2++εi =(-1)i-1 s(-i+1,1i-1) if0<i. (4.7) Furthermore, if 𝔤=𝔰𝔬2r+1 or 𝔰𝔭2r, then sisr-1 srsr-1 si(-εi) =sisr-1 sr ( -(εi++εr-1) -(-(r-i))εr ) = sisr-1 ( -(εi++εr-1) +(-(r-i)-(y-2r+1)) εr ) = ( -(r-i)- (y-2r+1)- (r-i) ) εi= (+2i-y-1)εi. (4.8) Similarly, if 𝔤=𝔰𝔬2r, then sisr-2 srsr-1 si(-εi) =sisr-2 sr ( -(εi++εr-1) -(-(r-i))εr ) = sisr-2 ( -(εi++εr-2) +(-(r-i)-1) εr-1 ) = ( -(r-i)- 1- (r-i-1) ) εi= (+2i-y-1)εi. (4.9) So, letting Wε1 be the stabilizer of ε1, |Wε1|=2r-1(r-1)!, and combining (4.7) and (4.8) gives 1|Wε1| mε1() = i=1r q2εi,ρ sεi+ q2-εi,ρ s-εi= { q2ε1,ρ sε1+q-2eλεr,ρ sε-r if𝔤=𝔰𝔬2r+1, q2ε1,ρ sε1 if𝔤=𝔰𝔭2r or𝔤=𝔰𝔬2r. = q(y-1) sε1-awhere a= { q-s0 if𝔤=𝔰𝔬2r+1 0 otherwise, (4.10) since 1+2i-y-1=0 has a solution exactly if i=r and 𝔤=𝔰𝔬2r+1. If >0 then using det ( sisr-1sr sr-1si ) =-1=-det ( sisr-2 srsr-1si ) , +2i-y-1-(i-1)=-y+i, and (4.7), equations (4.8) and (4.9) give q2-εi,ρ s-εi= { -q-s0, if=y+1-2i and𝔤= 𝔰𝔬2r+1 or𝔰𝔭2r, q-s0, if=y+1-2i and𝔤=𝔰𝔬2r, (-1)i q-(y-2i+1) s(-y+i,1i-1), if-y+i1, 0, otherwise. Since -y+i1 when iy-+1, 1|Wε1| mε1(1/) = i=1r q2εi,ρ sεi+ q2-εi,ρ s-εi = ( i=1min(,r) qy-2i+1 (-1)i-1 s(-i+1,1i-1) ) -b+ ( i=max(y-+1,1)r q-(y-2i+1) (-1)i s(-y+u,1i-1) ) whereb=0if y, and if<y thenbis given by is odd is even 𝔤=𝔰𝔬2r+1 q-s0 0 𝔤=𝔰𝔭2r 0 q-s0 𝔤=𝔰𝔬2r 0 -q-s0 (so that b is nonzero exactly when =y+1-2i has a solution with 1ir). Then reindexing (with m=-i+1 in the first sum and m=-y+i in the second sum) gives 1|Wε1| mε1(1/) = -b+ m=max(-r+1,1) (-1)-m qy-2(-m)-1 s(m,1-m+1-1) + m=max(y-+1,1)-y+r qy-2(-m)-1 (-1)m-+y s(m,1m-+y-1). Notice that the last sum appears only if >y-r.

Theorem 4.3 applied in the case that λ=ε1 gives ZV() = ε(idqtrV) ((z21)) =εzΨ (sε1)=ε (εqy) Ψ ( q-(y-1) ( 1|Wε1| mε1()+a ) ) = ε+1a ( 1|Wε1| Ψ (mε1())+a ) =ε+1q ( 1|Wε1| Ψ1 (mε1(1/)) +a ) = εΨ1 ( εq ( mε1(1/) |Wε1| +a ) ) = εΨ1 ( εq(a-b)+z m=max(-r+1,1) (-1)-m q-(-2m+1) s(m,1-m+1-1) +zm=max(-r+1,1)-y+r q-(-2m+1) (-1)m-+y s(m,1m-+y-1) ) , since z=εqy. The result follows since c=εq(a-b).

We would like to connect Corollary 4.4 to the Harish-Chandra images of the parameters ZV() computed in Corollary 3.6. In order to do so, we will use the following result from [TWe1217386, Lemma 3.5.1] (also see [Dri1990, Prop. 5.3]).

Let 𝔤 be a finite-dimensional complex Lie algebra with a symmetric nondegenerate ad-invariant bilinear form and let U=Uh𝔤 be the corresponding Drinfeld-Jimbo quantum group with R-matrix . Let ν be a dominant integral weight so that the irreducible module L(ν) of highest weight ν is finite-dimensional and let sν be the Weyl character of L(ν). Then (idqtrL(ν)) (21)acts on L(μ)by ev2(μ+ρ) (sν)idL(μ), where evγ:[𝔥*] are the algebra homomorphisms given by evγ(eτ)=qγ,τ for γ,τ𝔥*.

For 𝔤=𝔰𝔬2r+1, 𝔰𝔭2r or 𝔰𝔬2r the Turaev-Wenzl identity almost provides an inverse to the Harish-Chandra homomorphism. With ε1,,εr as in (2.9), converting variable alphabets from Y=iVˆ eεito X=iVˆ Li2,then ev2λ(sμ(Y)) =evλ(sμ(X)). Thus, Theorem 4.5 in combination with the Harish-Chandra isomorphism in Theorem 3.1 says that evμ(π0(Ψ1(sν)))= ev2(μ+ρ)sν(Y)= evμ+ρsν(X)= evμ(σρ(sν(X))). Hence π0(Ψ1(sν))= σρ(sν(X)). (4.11) The modification rules of [KTe1987, §2.4] are used to convert the universal Weyl characters appearing in Corollary 3.6 to actual Weyl characters sλ. In general, either spλ(X)=0 or there is a unique dominant weight μ and a uniquely determined sign such that spλ(X)=±sμ, and similarly for the orthogonal cases. In particular, if (λ)<r then spλ(X)=sλ in the symplectic case and soλ(X)=sλ in the orthogonal case [KTe1987, Prop. 2.2.1]. In view of (4.11), the conversion from universal Weyl characters to actual Weyl characters provides the equivalence between Corollary 4.4 and Corollary 3.6.

Notes and references

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

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