## The Casimir element

A Lie algebra is a vector space $𝔤$ with a bilinear map $\left[,\right]:𝔤×𝔤\to 𝔤$ such that

(a)   if $x,y\in 𝔤$ then $\left[x,y\right]=-\left[y,x\right]$,
(b)   (Jacobi identity) if $x,y,z\in 𝔤$ then $\left[x,\left[y,z\right]\right]+\left[z,\left[x,y\right]\right]+\left[y,\left[z,x\right]\right]=0$.

A bilinear form $⟨,⟩:𝔤×𝔤\to ℂ$ is ad-invariant if, for all $x,y,z\in 𝔤$, $⟨ adx(y), z⟩ = -⟨ y, adx(z) ⟩, where adx(y) =[x,y] ,$ for $x,y\in 𝔤$. The Killing form is the particular inner product on $𝔤$ given by $⟨x,y⟩=\mathrm{Tr}\left({\mathrm{ad}}_{x}{\mathrm{ad}}_{y}\right)$. If the Killing form is nondegenerate then the Jacobi identity is equivalent to the fact that the Killing form is ad-invariant.

Let $𝔤$ be a finite dimensional Lie algebra with a nondegenerate ad-invariant bilinear form. The nondegeneracy of the form means that if $\left\{{b}_{i}\right\}$ is a basis of $𝔤$ then the dual basis $\left\{{b}_{i}^{*}\right\}$ of $𝔤$ with respect to $⟨,⟩$ exists. The enveloping algebra $U𝔤$ is the algebra generated by $𝔤$ with the relation $xy-yx=\left[x,y\right]$ for $x,y\in 𝔤$. The Casimir element of $𝔤$ is $κ= ∑i bi bi* , in U𝔤.$ The element $\kappa$ is central in $U𝔤$ since, for $y\in 𝔤$, $yκ = ∑i ybi bi* = ∑i ([y,bi] + biy) bi* = ∑ i,j ⟨[y,bi] , bj* ⟩ bj bi* + ∑i biy bi* = ∑ i,j -bj ⟨ bi, [y, bj* ] ⟩ bi* + ∑j bjy bj* = ∑ j -bj [y, bj* ] + bjy bj* = ∑ j bj bj* y = κy.$ If $c$ is a constant and $\left(,\right)=c⟨,⟩$ then $\left({b}_{i},{b}_{j}^{*}\right)=c⟨{b}_{i},{b}_{j}^{*}⟩={\delta }_{ij}c$ and the Casimir with respect to the form $\left(,\right)$ is $κ∼ = ∑i bi 1c bi* = 1c ∑i bi bi* = 1c κ.$

Let $𝔤$ be a finite dimensional Lie algebra with a nondegenerate ad-invariant bilinear form i.e., $⟨,⟩: 𝔤⊗𝔤 →ℂ, with ⟨[ x1, x2], x3⟩ + ⟨ x2, [ x1, x3] ⟩ =0$ and $⟨{x}_{1},{x}_{2}⟩=⟨{x}_{2},{x}_{1}⟩$, for ${x}_{1},{x}_{2},{x}_{3}\in 𝔤$. Let $\kappa$ be the Casimir with respect to $⟨,⟩$. The Casimir is $κ= b1 b1* + ⋯ + bn bn* = ∑b∈B b b* , and κ∈Z(U𝔤),$ the center of the enveloping algebra $U𝔤$. Since $\Delta \left(x\right)=x\otimes 1+1\otimes x$ for $x\in 𝔤$,

 $Δ(κ)= κ⊗1+ 1⊗κ +2t, where t= ∑ i=1 n bi ⊗ bi* = ∑ b∈B b⊗b*.$ (tdf)
For a $U𝔤$-module $M$ let $κM: M → M m ↦ κm$

Fix a triangular decomposition $𝔤={𝔫}^{-}\oplus 𝔥\oplus {𝔫}^{+}$ with Cartan subalgebra $𝔥$ and let ${R}^{+}$ be the set of positive roots. If $M$ is a $U𝔤$-module generated by a highest weight vector ${v}_{\lambda }^{+}$ of weight $\lambda$ then (see [Bou, VIII §2 no. 3 Prop. 6 and VIII §6 no. 4 Prop. 7) $κM = ⟨λ,λ+ρ⟩ idM where ρ= 1 2 ∑ α∈R+ α,$ and $⟨,⟩$ is the form on ${𝔥}^{*}$ obtained by restricting the form $⟨,⟩$ to $𝔥$ and identifying $𝔥$ with ${𝔥}^{*}$. More specifically, $⟨λ,μ⟩ = ∑ i=1 r λ(hi) μ( hi* ), for λ,μ∈ 𝔥* ,$ if ${h}_{1},\dots ,{h}_{r}$ is a basis of $𝔥$ and ${h}_{1}^{*},\dots ,{h}_{r}^{*}$ is the dual basis with respect to $⟨,⟩$. By equation (tdf), if $M=L\left(\mu \right)$, $N=L\left(\nu \right)$ are finite dimensional irreducible $U𝔤$-modules of highest weights $\mu$ and $\nu$ respectively, then $t$ acts on the $L\left(\lambda \right)$-isotypic component of the decomposition $L\left(\mu \right)\otimes L\left(\nu \right)\cong \underset{\lambda }{⨁}{L\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ by the constant

 $1 2 ( ⟨λ,λ+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨ν,ν+2ρ⟩ )$ (tval)

Let $U={U}_{h}𝔤$ be the Drinfel'd-Jimbo quantum group corresponding to $𝔤$. Let ${\rho }^{\vee }\in 𝔥$ be such that ${\alpha }_{i}\left({\rho }^{\vee }\right)=1$ for all simple roots ${\alpha }_{i}$. Then $v= e -hρ∨ u is a ribbon element in Uh𝔤.$ For a ${U}_{h}𝔤$-module $M$ let

 $CM: M → M m ↦ vm so that C M⊗N = ( Rˇ MN Rˇ NM ) -1 (CM ⊗ CN ).$ (casR)
If $M$ is a ${U}_{h}𝔤$-module generated by a highest weight vector ${v}^{+}$ of weight $\lambda$ then
 $CM = q -⟨λ, λ+2ρ⟩ idM,$ (qcas)
see [
LR, Prop. 2.14] or [Dr, Prop. 3.2]). From (qcas) and the relation (casR) it follows that if $M=L\left(\mu \right)$, $N=L\left(\nu \right)$ are finite dimensional irreducible ${U}_{h}𝔤$-modules of highest weights $\mu$ and $\nu$ respectively, then ${\stackrel{ˇ}{R}}_{MN}{\stackrel{ˇ}{R}}_{NM}$ acts on the $L\left(\lambda \right)$-isotypic component ${L\left(\lambda \right)}^{\oplus {c}_{\mu \nu }^{\lambda }}$ of the decomposition
 $L(μ) ⊗ L(ν) ≅ ⨁λ L(λ) ⊕ cμνλ by the scalar q ⟨λ,λ+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨ν,ν+2ρ⟩ .$ qtval

[Bou, VIII §2 no. 3 Prop. 6 and VIII §6 no. 4 Cor. to Prop. 7] Let $\kappa \in U𝔤$ be the Casimir and let ${e}^{-h\rho }u\in {U}_{h}𝔤$ be the quantum Casimir as defined in (??) and (???), respectively. Then $π0(κ) = σρ ( -⟨ρ,ρ⟩ + ∑ i=1 r hi hi* ) and evν (π0(κ)) = ⟨ν,ν+ρ ⟩,$ ,

Proof.

Let ${h}_{1},\dots ,{h}_{r}$ be a basis of $𝔥$ and let ${h}_{1}^{*},\dots ,{h}_{r}^{*}$ be the dual basis of $𝔥$ with respect to the restriction of $⟨,⟩$ to $𝔥$. The nondegenerate bilinear form $⟨,⟩:𝔥×𝔥\to ℂ$ is equivalent to a vector space isomorphism $ν:𝔥 ⟶∼ 𝔥* given by ν(h)= ⟨h,⋅⟩ and ν-1( λ)= ∑i=1r λ(hi) hi* ,$ for $h\in 𝔥$ and $\lambda \in {𝔥}^{*}$. The isomorphism $\nu :𝔥\stackrel{\sim }{⟶}{𝔥}^{*}$ and the form $⟨,⟩:𝔥×𝔥\to ℂ$ provide a form $⟨,⟩:{𝔥}^{*}×{𝔥}^{*}\to ℂ$ given by $⟨λ,μ⟩ = ⟨ν-1 (λ), ν-1(μ)⟩ = ⟨ ∑i=1r λ(hi) hi* , ∑j=1r μ(hj*) hj ⟩ = ∑i=1r λ(hi) μ(hi*) .$

Let ${x}_{\alpha }\in {𝔤}_{\alpha }$. There is a unique ${h}_{\alpha }\in 𝔥$ such that $ifh∈𝔥 then ⟨h,hα⟩ = α(h) . Thus ν(hα) =α.$ A different element ${h}_{{\alpha }^{\vee }}$ is obtained by choosing ${y}_{\alpha }\in {𝔤}_{-\alpha }$ and ${h}_{{\alpha }^{\vee }}\in 𝔥$ such that ${x}_{\alpha },{y}_{\alpha },{h}_{{\alpha }^{\vee }}$ span an ${𝔰𝔩}_{2}$-subalgebra of $𝔤$, $[xα, yα] = hα∨, [ hα∨ , xα ] = 2 xα , [ hα∨ , yα ] = -2 yα .$ Then $2=\alpha \left({h}_{{\alpha }^{\vee }}\right)$ and $2= α( hα∨ ) = ⟨ hα, hα∨ ⟩ = ⟨ hα, [xα, yα] ⟩ = - ⟨ [ xα,hα ] , yα ⟩ = α(hα) ⟨ xα,yα ⟩ = ⟨ hα,hα ⟩ ⟨ xα,yα ⟩$ gives $⟨ xα,yα ⟩ = 2 ⟨ hα,hα ⟩ and hα∨ = 2⟨hα, hα⟩ hα.$ Thus with respect to $⟨,⟩$.

Then $κ = ∑ i=1 r hi hi* + ∑ α∈R+ xα ⟨ hα,hα ⟩ 2 yα + ∑ α∈R+ ⟨ hα,hα ⟩ 2 yαxα = ∑ i=1 r hi hi* + ∑ α∈R+ ⟨ hα,hα ⟩ 2 [ xα, yα] +2 ∑ α∈R+ ⟨ hα,hα ⟩ 2 yαxα = ∑ i=1 r hi hi* + ∑ α∈R+ ⟨ hα,hα ⟩ 2 hα∨ +2 ∑ α∈R+ ⟨ hα,hα ⟩ 2 yαxα = ∑ i=1 r hi hi* + ∑ α∈R+ hα +2 ∑ α∈R+ ⟨ hα,hα ⟩ 2 yαxα$ and so $if ρ= ν( 12 ∑ α∈R+ hα ) = 12 ∑ α∈R+ α then 12 ∑ α∈R+ hα = ∑ i=1 r ρhi hi* .$ Then $π0(κ) = ∑ i=1 r hi hi* + ∑ α∈R+ hα = ∑ i=1 r hi hi* +ρhi hi* + hi ρ hi* = - ⟨ ρ,ρ ⟩ + ∑ i=1 r hi + ρhi hi* +ρ hi* = σρ ( - ⟨ ρ,ρ ⟩ + ∑ i=1 r hi hi* )$ since $⟨ρ,ρ⟩ = ∑ i=1 r ρ(hi) ρ(hi*) .$

WE NEED TO SHOW THAT $⟨ρ, αi∨ ⟩=1, for i=1,2,…, r.$ I THINK THIS IS THE IDENTITY WE WANT.

In the case where $𝔤=𝔤{𝔩}_{n}$, $κ = ∑ 1≤i,j≤n EijEji = ∑ i=1 n EiiEii + ∑ 1≤i Since ${E}_{ii}{v}_{\lambda }^{+}={\lambda }_{i}{v}_{\lambda }^{+}$ and ${E}_{ij}{v}_{\lambda }^{+}=0$ for $i, $\kappa {v}_{\lambda }^{+}={c}_{\lambda }{v}_{\lambda }^{+}$ where $cλ = ∑ i=1 n λi2 +0 + ∑ 1≤i

## The Drinfeld-Jimbo quantum group is a ribbon Hopf algebra

In the definition of the quantum group: If ${K}_{i}={K}^{{\alpha }_{i}}$ then

 ${K}_{i}{E}_{j}{K}_{i}^{-1}={K}^{{\alpha }_{i}}{E}_{j}{K}^{-{\alpha }_{i}}={K}^{{d}_{i}{\alpha }_{i}^{\vee }}{E}_{j}{K}^{-{d}_{i}{\alpha }_{i}^{\vee }}={q}^{⟨{d}_{i}{\alpha }_{i}^{\vee },{\alpha }_{j}⟩}{E}_{j}={q}^{{d}_{i}{a}_{ij}}{E}_{j}={{q}_{i}}^{{a}_{ij}}{E}_{j}$
and
 ${E}_{i}{F}_{j}-{F}_{j}{E}_{i}={\delta }_{ij}\frac{{K}^{{\alpha }_{i}}-{K}^{-{\alpha }_{i}}}{{q}^{{d}_{i}}-{q}^{-{d}_{i}}}={\delta }_{ij}\frac{{\left({K}^{{\alpha }_{i}^{\vee }}\right)}^{{d}_{i}}-{\left({K}^{{\alpha }_{i}^{\vee }}\right)}^{-{d}_{i}}}{{q}^{{d}_{i}}-{q}^{-{d}_{i}}}.$
The coproduct is given by
 $\Delta \left({E}_{i}\right)={E}_{i}\otimes 1+{K}_{i}\otimes {E}_{i},\phantom{\rule{2em}{0ex}}\Delta \left({K}_{i}\right)={K}_{i}\otimes {K}_{i},\phantom{\rule{2em}{0ex}}\Delta \left({F}_{i}\right)={F}_{i}\otimes {K}_{i}^{-1}+1\otimes {E}_{i},\phantom{\rule{2em}{0ex}}$
and
 $\epsilon \left({E}_{i}\right)=\epsilon \left({F}_{i}\right)=0,\phantom{\rule{2em}{0ex}}\epsilon \left({K}_{i}\right)=1,\phantom{\rule{2em}{0ex}}S\left({E}_{i}\right)=-{K}_{i}^{-1}{E}_{i},\phantom{\rule{2em}{0ex}}S\left({F}_{i}\right)=-{F}_{i}{K}_{i},\phantom{\rule{2em}{0ex}}S\left({K}_{i}\right)={K}_{i}^{-1}$.
Thus,
 $u{E}_{i}{u}^{-1}={S}^{2}\left({E}_{i}\right)=S\left(-{K}_{i}^{-1}{E}_{i}\right)={K}_{i}^{-1}{E}_{i}{K}_{i}={q}^{-⟨{d}_{i}{\alpha }_{i}^{\vee },{\alpha }_{i}⟩}{E}_{i}={q}^{-2{d}_{i}}{E}_{i}$.
Then
 ${K}^{\rho }{E}_{i}{K}^{-\rho }={q}^{⟨\rho ,{\alpha }_{i}⟩}{E}_{i}={q}^{⟨\rho ,{d}_{i}{\alpha }_{i}^{\vee }⟩}{E}_{i}={q}^{{d}_{i}}{E}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{K}^{2\rho }u\in Z\left(U\right)$.
This matches with [Bau, §2.1-2.3] and [Kac, Chapt 10] (note that [Kac] has the difference between $\rho$ and ${\rho }^{\vee }$ clear in [Kac, §10.8]; see also the clear use of ${\nu }^{-1}$ in the definition of the Casimir just before [Kac, (2.5.2)] ).

Let ${\stackrel{\sim }{H}}_{1},\dots ,{\stackrel{\sim }{H}}_{r}$ be an orthonormal basis of $𝔥$. The algebra ${U}_{h}𝔤$ is a quasitriangular Hopf algebra and the element $ℛ$ can be written in the form, see [Dr, Sect. 4],

 $ℛ= exp( h2t0 ) +∑ai+ ⊗bi-, where t0 = ∑i=1 r H∼i ⊗ H∼i ,$ (Rfm)
and the elements ${a}_{i}^{+}\in {{U}_{h}𝔤}^{\ge 0}$, and ${b}_{i}^{-}\in {{U}_{h}𝔤}^{\le 0}$, are homogeneous elements of degrees $\ge 1$ and $\le -1$, respectively.

[Dr, Prop. ???], [LR, Prop. (2.14)] Let ${U}_{h}𝔤$ be a Drinfeld-Jimbo quantum group and let $\rho$ be an element of $𝔥$ such that $⟨{\alpha }_{i},\rho ⟩=1$ for all simple roots ${\alpha }_{i}$. Let $u$ be as given in (???). Then

1. if $a\in {U}_{h}𝔤$ then ${e}^{h\rho }a{e}^{-h\rho }={S}^{2}\left(a\right)$,
2. ${e}^{-h\rho }u=ua{e}^{-h\rho }$ is a central element in ${U}_{h}𝔤$,
3. ${\left({e}^{-h\rho }\right)}^{2}-uS\left(u\right)=S\left(u\right)u$,
4. ${e}^{-h\rho }u$ acts in an irreducible representation $L\left(\lambda \right)$ of ${U}_{h}𝔤$ of highest weight $\lambda$ by the constant ${e}^{-\frac{h}{2}⟨\lambda ,\lambda +2\rho ⟩}={q}^{-⟨\lambda ,\lambda +2\rho ⟩}$,
5. $\Delta \left({e}^{-h\rho }u\right)={\left({ℛ}_{21}ℛ\right)}^{-1}\left({e}^{-h\rho }u\otimes {e}^{-h\rho }u\right)$,
6. $S\left({e}^{-h\rho }u\right)={e}^{-h\rho }u$, and
7. $\epsilon \left({e}^{-h\rho }u\right)=1$,
so that $\left({U}_{h}𝔤,ℛ,{e}^{-h\rho }u\right)$ is a ribbon Hopf algebra.

Proof.

1. Since both ${S}^{2}$ and conjugation by ${e}^{h\rho }$ are algebra homomorphisms it is sufficient to check this on generators. We shall show how this is done for the generator ${X}_{j}$. It follows from the fact that $\left[\rho ,{X}_{j}\right]=\rho {X}_{j}-{X}_{j}\rho =⟨{\alpha }_{j},\rho ⟩{X}_{j}$, that $ehρ Xj e-hρ = ehρ e-h (ρ- ⟨αj, ρ⟩) Xj = eh ⟨αj, ρ⟩) Xj = ehXj = q2Xj =S2 (Xj) .$
2. This follows from (1) and the relation $ua{u}^{-1}={S}^{2}\left(a\right)$, since ${e}^{-h\rho }ua{u}^{-1}{e}^{h\rho }={S}^{-2}\left({S}^{2}\left(a\right)\right)=a$.
3. Let ${\stackrel{\sim }{H}}_{1},\dots ,{\stackrel{\sim }{H}}_{r}$ be an orthonormal basis of $𝔥$. Let $L\left(\lambda \right)$ be an irreducible ${U}_{h}𝔤$-module of highest weight $\lambda$ and let ${v}_{\lambda }$ be a highest weight vector in $L\left(\lambda \right)$. Since element of ${{U}_{h}𝔤}^{\ge 0}$ which are of degree $\ge 1$ annihilate ${v}_{\lambda }$ it follows that $uvλ = exp( h2 ∑i=1 r S(H∼i) H∼i ) vλ = exp( -h2 ∑i=1 r H∼i H∼i ) vλ = exp( -h2 ∑i=1 r λ(H∼i) λ( H∼i ) ) vλ = exp( -h2 ⟨λ,λ⟩ ) vλ .$ The result follows since ${e}^{-h\rho }{v}_{\lambda }={e}^{-h⟨\lambda ,\rho ⟩}{v}_{\lambda }$.
4. This follows from $\Delta \left(u\right)={\left({ℛ}_{21}ℛ\right)}^{-1}\left(u\otimes u\right)$, since $Δ( e-hρu ) = Δ( ue-hρ ) = Δ(u) Δ( e-hρ ) = ( ℛ21 ℛ)-1 (u⊗u) ( e-hρ ⊗ e-hρ ) = ( ℛ21 ℛ)-1 ( e-hρu ⊗ e-hρu ) .$
5. and 6. and 7. follow from the equality ${e}^{h\rho }S\left(u\right)={e}^{-h\rho }u$ which is proved as follows. Since ${e}^{h\rho }S\left(u\right)=S\left({e}^{-h\rho }u\right)$ is a central element of ${U}_{h}𝔤$, it is sufficient to check that both ${e}^{h\rho }S\left(u\right)$ and ${e}^{-h\rho }u$ act by the same constant on an irreducible representation $L\left(\lambda \right)$ of ${U}_{h}𝔤$. The element ${e}^{h\rho }S\left(u\right)=S\left({e}^{-h\rho }u\right)$ acts on the module $L\left(\lambda \right)$ in the same way that $u{e}^{-h\rho }$ acts on the irreducible module ${L\left(\lambda \right)}^{*}$ which has highest weight $-{w}_{0}\lambda$ where ${w}_{0}$ is the longest element of the Weyl group. Thus, $u{e}^{-h\rho }$ acts on the irreducible module ${L\left(\lambda \right)}^{*}$ by the constant $e -h2 ⟨-w0λ , -w0λ ⟩ e -h ⟨-w0λ , ρ⟩ = e -h2 ⟨λ , λ+2ρ ⟩ = q - ⟨λ , λ+2ρ ⟩$ since ${w}_{0}\rho =-\rho$ and the inner product is invariant under the action of ${w}_{0}$.

## Notes and References

See [Bou, Ch. I §3 Prop. 11] for the fact that the Casimir element is in the center of $U𝔤$.

[Dr, §5 remark (1)] explains that the source of the element $\rho$ is coming from the rewriting of the Casimir in a preferred form so that higher degree terms act by 0 on a highest weight vector:

 $\sum _{\alpha \in R}{e}_{-\alpha }{e}_{\alpha }+\sum _{\mu }{I}_{\mu }^{2}=2\sum _{\alpha \in {R}^{+}}{e}_{-\alpha }{e}_{\alpha }+\sum _{\mu }{I}_{\mu }^{2}+2\rho$

## Bibliography

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

[Dr] V.G. Drinfeld, On almost cocommutative Hopf algebras, Leningrad Math. J. 1 (1990), 321-342. MR1025154 (91b:16046)

[LR] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: the Brauer, Birman-Wenzl and Type A Iwahori-Hecke algebras, Advances Math. 125 (1997), 1-94. MR1427801 (98c:20015)