Spectral sublagebras

Spectral subalgebras

Then since, if ${l}_{1},{l}_{2}\in {C}_{0}$ and $a\in A$ then $l 2 l 1 a = l 1 ⊗ l 2 Δ op a = l 1 ⊗ l 2 ℛΔ a ℛ -1 = l 1 ⊗ l 2 Δ a ℛ -1 ℛ= l 1 ⊗ l 2 Δ a= l 1 l 2 a ,$ where the third equality uses the definition of ${C}_{0}.$

If $\left(A,ℛ\right)$ is a quasitriangular Hopf algebra the $ℛ$ satisfies the quantum Yang-Baxter equation, (QYBE),$ℛ 12 ℛ 13 ℛ 23 = ℛ 12 Δ⊗id ℛ = Δ op ⊗id ℛ ℛ 12 = ℛ 23 ℛ 13 ℛ 12 .$

Since $ℛ= ε⊗id⊗id Δ⊗id ℛ = ε⊗id⊗id ℛ 13 ℛ 23 = ε⊗id ℛ .ℛ,$ and $ℛ= id⊗id⊗ε id ⊗ Δ ℛ = id⊗id⊗ε ℛ 13 ℛ 23 = id⊗ε ℛ .ℛ,$ and so $ε⊗id ℛ =1and id⊗ε ℛ =1.$

Then, since $ℛ S⊗id ℛ = m⊗id id⊗S⊗id ℛ 13 ℛ 23 = m⊗id id⊗S⊗id Δ⊗id ℛ = ε⊗id ℛ =1,$ it follows that $S⊗id ℛ = ℛ -1 .$ Applying this to the pair $\left({A}^{\mathrm{op}},{ℛ}^{21}\right)$ gives $\left({S}^{-1}\otimes \mathrm{id}\right)\left({ℛ}^{21}\right)={\left({ℛ}^{21}\right)}^{\mathrm{op}},$ and so $id⊗ S -1 ℛ = ℛ -1 .$ Then $S⊗S ℛ = id⊗S S⊗id ℛ = id⊗S ℛ -1 = id⊗S id⊗ S -1 ℛ =ℛ.$

The map $\phi :C\to Z\left(A\right)$ in the following proposition is ananalogue of the Harish-Chandra homomorphism.

Let $\left(A,ℛ\right)$ be a quasitrtiangular Hopf algebra. Then $C= λ∈A*| λ xy =λ y S 2 x is a commutative algebra$ and the map $φ: C → Z A l ↦ id⊗l ℛ 21 ℛ is a well defined algebra homomorphism.$

 Proof. If ${l}_{1},{l}_{2}\in A*$ and $a\in A$ then $l 1 l 2 a = l 1 ⊗ l 2 Δ op a = l 1 ⊗ l 2 ℛΔ a ℛ -1 = l 1 ⊗ l 2 Δ a ℛ -1 S 2 ⊗ S 2 ℛ ,by definiton of C, = l 1 ⊗ l 2 Δ a ℛ -1 ℛ = l 1 ⊗ l 2 Δ a = l 1 l 2 a ,$ and hence $C$ is a commutative algebra. Let $a\in A.$ First note that $a⊗1 = id⊗ε Δ a = id⊗m id⊗ S -1 ⊗id id⊗ Δop Δ a = ∑ a a 1 ⊗ S -1 a 3 a 2 = ∑ a 1⊗ S -1 a 2 a 11 ⊗ a 12 = ∑a 1⊗ S -1 a 2 Δ a ,$ since ${S}^{-1}$ is the antipode of ${A}^{\mathrm{op}}$, and $a⊗1 = id⊗ε Δ a = id⊗m id⊗id⊗S id⊗Δ Δ a = ∑ a a 1 ⊗ a 2 S a 3 = ∑ a a 11 ⊗ a 12 1⊗S a 2 = ∑a Δ a 1 1⊗S a 2 , .$ Then, since $ℛ 21 ℛΔ a = ℛ 21 Δ op a ℛ=Δ a ℛ 21 ℛ,$ and so $\phi \left(l\right)\in Z\left(A\right).$ Since and so $\phi$ is a homomorphism. $\square$

Central elements $\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left(a\right)$

The following proposition provides Drinfeld's "second central element construction" [Dr, Prop. 1.2 and Prop 3.3]. The model example is the case that and $a={ℛ}_{21}ℛ.$

Let $\left(U,ℛ\right)$ be a quasitriangular Hopf algebra with antipode $S.$ Let ${x}_{0}\in U$ and $a\in U\otimes U$ be invertible elements such that respectively. Let Let $Z\left(U\right)$ be the center of $U$ and let $\mathrm{Rep}$ be the Grothendieck group of finite dimensional representations of $U.$ Define for $u\in U.$ Then

1. is a commutative subalgebra of ${U}^{*};$
2. the maps ${x}_{0}^{*}$ and ${f}_{a}$ are well defined;
3. $\mathrm{tr}:\mathrm{Rep}\to {C}_{0}$ is an algebra homomorphism;
4. if $\Delta \left({x}_{0}\right)={x}_{0}\otimes {x}_{0}$ then ${x}_{0}^{*}:{C}_{0}\to {C}_{1}$ is an algebra homomorphism; and
5. if $a={ℛ}_{21}ℛ$ then ${f}_{a}:{C}_{1}\to Z\left(U\right)$ is an algebra homomorphism.

 Proof. Let ${\ell }_{1},{\ell }_{2}\in {C}_{k}.$ Then $(ℓ1ℓ2)(xy) = (ℓ1⊗ℓ2) (Δ(xy)) = (ℓ1⊗ℓ2) (Δ(x)Δ(y)) = (ℓ1⊗ℓ2) (Δ(y)(S2k⊗S2k)Δ(x)) = (ℓ1⊗ℓ2) (Δ(y)Δ(S2k(x))) = (ℓ1⊗ℓ2) (Δ(yS2k(x))) = (ℓ1ℓ2) (yS2k(x)),$ where the fourth equality follows from the fact that ${S}^{2}$ is a coalgebra automorphism. Thus ${C}_{k}$ is a subalgebra of ${U}^{*}.$ Since $\left(S\otimes S\right)\left(ℛ\right)=ℛ,$ $(ℓ1ℓ2)(x) = (ℓ1⊗ℓ2) Δ(x) = (ℓ2⊗ℓ1) Δop(x) = (ℓ2⊗ℓ1) (ℛΔ(x)ℛ-1) = (ℓ2⊗ℓ1) (Δ(x)ℛ-1(S2k⊗S2k)(ℛ)) = (ℓ2⊗ℓ1) (Δ(x)ℛ-1ℛ = (ℓ2⊗ℓ1) Δ(x) = ℓ2ℓ1(x).$ So ${C}_{k}$ is commutative. Let $\ell ={x}_{0}^{*}\left(l\right).$ Then $ℓ(xy) = l(xyx0) = l(yx0x) = l(yx0xx0-1x0) = l(yS2(x)x0) = ℓ(yS2(x)),$ and thus ${x}_{0}^{*}\left(l\right)\in {C}_{1}$ and ${x}_{0}^{*}$ is well defined. The identities $ε(x) = ∑xS-1(x(2))x(1) and ∑xx(1)S(x(2)) = ε(x), (Ss 2.4)$ from the definition of a Hopf algebra are the relations which provide the isomorphisms in [DRV] (A.8). For $x\in U,$ So $\left(\mathrm{id}\otimes \ell \right)\left(a\right)\in Z\left(U\right).$ Hence ${f}_{a}\left(\ell \right)=\left(\mathrm{id}\otimes \ell \right)\left(a\right)$ is an element of the center of $U,$ and ${f}_{a}$ is well defined. Let $M,N\in \mathrm{Rep}.$ Let $\left\{{m}_{i}\right\}$ be a basis of $M,$ $\left\{{m}^{i}\right\}$ a dual basis in ${M}^{*},$ $\left\{{n}_{j}\right\}$ a basis of $N$ and $\left\{{n}^{j}\right\}$ a dual basis in ${N}^{*}.$ Then $(trM⋅trN)(x) = ∑x trM(x(1)) trN(x(2)) = ∑x,i,j ⟨x(1)mi⊗ x(2)nj, mi⊗nj⟩ = ∑i,j ⟨x(mi⊗nj), mi⊗nj⟩ = trM⊗N(x).$ So $\mathrm{tr}:\mathrm{Rep}\to {C}_{0}$ is an algebra homomorphism. Assume $\Delta \left({x}_{0}\right)={x}_{0}\otimes {x}_{0}.$ Let ${l}_{1},{l}_{2}\in {C}_{0}.$ Then $x0* (l1l2)(x) = (l1l2) (xx0) = (l1⊗l2) (Δ(xx0)) = (l1⊗l2) (Δ(x)(x0⊗x0)) = (x0*(l1) ⊗ x0*(l2)) (Δ(x)) = (x0*(l1) x0*(l2)) (x).$ So ${x}_{0}^{*}:{C}_{0}\to {C}_{1}$ is an algebra homomorphism. Assume $a={ℛ}_{21}ℛ.$ Let ${\ell }_{1},{\ell }_{2}\in {C}_{1}.$ Then $(id⊗ℓ1ℓ2) (ℛ21ℛ) = (id⊗ℓ1⊗ℓ2) (id⊗Δ) (ℛ21ℛ) = (id⊗ℓ1⊗ℓ2) (ℛ21) (ℛ21ℛ13ℛ) = (id⊗ℓ1) (ℛ21⋅((id⊗ℓ2)(ℛ21ℛ)⊗1)⋅ℛ).$ Since $\left(\mathrm{id}\otimes {\ell }_{2}\right)\left({ℛ}_{21}ℛ\right)\in Z\left(U\right),$ $fa (ℓ1ℓ2) = (id⊗ℓ1ℓ2) (ℛ21ℛ) = (id⊗ℓ1) (ℛ21ℛ⋅((id⊗ℓ2)(ℛ21ℛ)⊗1)) = (id⊗ℓ1) (ℛ21ℛ) (id⊗ℓ2) (ℛ21ℛ) = fa(ℓ1) fa(ℓ2).$ So ${f}_{a}:{C}_{1}\to Z\left(U\right)$ is an algebra homomorphism. $\square$

Let $U=U𝔤$ and let $a=\gamma =\sum _{b}b\otimes {b}^{*}.$ Let ${\ell }_{1},{\ell }_{2}\in {C}_{1}.$ Then $fa (ℓ1ℓ2) = (id⊗ℓ1ℓ2) (γ) = (id⊗ℓ1⊗ℓ2) (id⊗Δ) (γ) = (id⊗ℓ1⊗ℓ2) ∑bb⊗b*⊗1 + b⊗1⊗b* = fa (ℓ1) ℓ2(1) + ℓ1(1) fa (ℓ2).$ Thus, since $\left({\ell }_{1}{\ell }_{2}\right)\left(1\right)=\left({\ell }_{1}\otimes {\ell }_{2}\right)\left(\Delta \left(1\right)\right)=\left({\ell }_{1}\otimes {\ell }_{2}\right)\left(1\otimes 1\right)={\ell }_{1}\left(1\right){\ell }_{2}\left(1\right),$ $fa(ℓ1ℓ2) (ℓ1ℓ2)(1) = f1(ℓ1) ℓ1(1) + fa(ℓ2) ℓ2(1) . (Ss 2.5)$

$\left(\mathrm{id}\otimes {\mathrm{qtr}}_{L\left(\nu \right)}\right)\left({ℛ}_{21}ℛ\right)$ when $U={U}_{h}𝔤$

In the case that $U={U}_{h}𝔤$ and ${x}_{0}={v}^{-1}u={e}^{h\rho },$ $M$ and $N$ are $U-$modules, and ${\ell }_{1}={\mathrm{qtr}}_{M}$ and ${\ell }_{2}={\mathrm{qtr}}_{N}$ then (d) and (e) above give that, as elements of ${U}^{*},$ $(qtrM⋅qtrN)(x) = qtrM⊗N(x),$ and $(id⊗qtrM⊗N) (ℛ21ℛ) = (id⊗qtrM) (ℛ21ℛ) (id⊗qtrN) (ℛ21ℛ)$ (as explained in [Dr, Prop. 3.3] and [Bau, Prop. 2]). In this case, the computation in the proof of (d) and (e), pictorially, is $= = = =$

Fix a Cartan subalgebra $𝔥$ in $𝔤.$ For $\gamma \in {𝔥}^{*}$ define the Weyl character $sγ = aγ+ρ aρ , where aμ = ∑w∈W0 det(w)Xwμ.$ The expressions ${s}_{\gamma }$ and ${a}_{\mu }$ are elements of the group algebra of ${𝔥}^{*},$ and, if $w\in {W}_{0}$ then $awμ = det(w)aμ, and sw∘μ = det(w)sμ,$ where the dot action of ${W}_{0}$ on ${𝔥}^{*}$ is given by The Weyl denominator formula says that $aρ = ∏α∈R+ (Xα/2-X-α/2) (Ss 3.2)$ and the Weyl character formula says that if $\gamma$ is a dominant integral weight, $L\left(\lambda \right)$ is the simple $𝔤-$module of highest weight $\gamma ,$ and ${v}_{1},...,{v}_{n}$ is a basis of $L\left(\lambda \right)$ consisting of weight vectors then $sλ = ∑i=1n Xwt(vi) = ∑ν∈P Kλν Xν, where Kλν = dim(L(λ)ν), (Ss 3.3)$ the dimension of the $\nu$ weight space of $L\left(\lambda \right).$ For $\mu \in {𝔥}^{*}$ define an algebra homomorphism $evμ: ℂ[𝔥*]→ ℂ[𝔥*] by evμ(Xν) = q⟨μ,ν⟩.$ Then $dimq(L(ν)) = ev2ρ(sν), (Ss 3.4)$ since ${\mathrm{dim}}_{q}\left(L\left(\nu \right)\right)={\mathrm{tr}}_{L\left(\nu \right)}\left({e}^{h\rho }\right)$ and $q={e}^{\frac{h}{2}}.$

[TW, Lemma 3.5.1] Let $𝔤$ be a finite dimensional complex semisimple Lie algebra and let ${U}_{h}𝔤$ be the corresponding Drinfeld-Jimbo quantum group. etc etc ???? Let $\nu$ be a dominant integral weight so that the irreducible module $L\left(\nu \right)$ of highest weight $\nu$ is finite dimensional. Then

 Proof. Let ${h}_{1},...,{h}_{r}$ be an orthonormal basis of $𝔥.$ By [Dr, §4] (see [LR, (2.13)]) there is an expression $ℛ= e 1 2 hγ0 + ∑bj+⊗bj-, where γ0 = ∑l=1r hl⊗hl,$ and ${b}_{j}^{+}\in {U}^{+}$ and ${b}_{j}^{-}\in {U}^{-}$ are homogeneous elements of degree greater than $0.$ Let ${v}_{1},...,{v}_{n}$ be a basis of weight vectors of $L\left(\nu \right)$ and let ${v}^{1},...,{v}^{n}$ be the dual basis in $L{\left(\nu \right)}^{*}.$ Let ${v}_{\mu }^{+}$ be a highest weight vector in $L\left(\mu \right).$ Since ${b}_{j}^{+}{v}_{\mu }^{+}=0$ and $q={e}^{\frac{h}{2}},$ ${ℛ}_{21}ℛ\otimes 1$ acts on ${v}_{\mu }^{+}\otimes {v}_{i}\otimes {v}^{i}\in L\left(\mu \right)\otimes L\left(\nu \right)\otimes L{\left(\nu \right)}^{*}$ by $(ℛ21ℛ⊗id) (vμ+⊗vi⊗vi) = qγ0 + ∑j (bj-⊗bj+) qγ0 + ∑j (bj+⊗bj-) (vμ+⊗vi) ⊗vi = qγ0 qγ0 + qγ0 ∑j (bj-⊗bj+) (vμ+⊗vi) ⊗vi = qγ0 qγ0 (vμ+⊗vi⊗vi) + qγ0 ∑j (bj-vμ+⊗bj+vi⊗vi) (vμ+⊗vi) ⊗vi$ Since $\left(\mathrm{id}\otimes {\mathrm{qtr}}_{L\left(\nu \right)}\right)\left({ℛ}_{21}ℛ\right)$ is central in $U,$ it acts on ${v}_{\mu }^{+}$ by a scalar. Therefore, since ${b}_{j}^{-}$ is a lowering operator, $(id⊗qtrL(ν)) (ℛ21ℛ) = (id⊗qtrL(ν)) (q2γ0),$ as proved in [Dr, Prop. 5.3]. Then $(id⊗qtrL(ν)) (ℛ21ℛ) = (id⊗qtrL(ν)) (q2γ0) = ∑i q2γ0 (1⊗ehρ) (vμ+⊗vi) |vμ+⊗vi = ∑ivi q 2 ∑l=1r μ(hl) wt(vi) (hl) q⟨ 2ρ,wt(vi)⟩ (vμ+⊗vi) |vμ+⊗vi = ∑i q2 ⟨μ+ρ,wt(vi)⟩ = ev2(μ+ρ) (sν) = ∑w∈W0 det(w) q2⟨μ+ρ,w(ν+ρ)⟩ ∑w∈W0 det(w) q2⟨μ+ρ,wρ⟩ .$ $\square$

The Turaev-Wenzl identity almost provides an inverse to the Harish-Chandra homomorphism. In the case of $U={U}_{h}{\mathrm{𝔰𝔩}}_{2},$ $(id⊗qtrL(ω1)) (ℛ21ℛ) = (id⊗qtrL(ω1)) e h( 1 2 (H⊗H)) = e h 2 (H+1) + e - h 2 (H+1) ,$ and this element acts on $L\left(\mu {\omega }_{1}\right)$ by the constant $ev2(μ+ρ) (Xω1+X-ω1) = qμ+1 + q-(μ+1) = evμ+ρ (eh2H + e-h2H) = evμ+ρ (K+K-1).$

$\left(\mathrm{id}\otimes {\mathrm{qtr}}_{L\left(\nu \right)}\left({\left({ℛ}_{21}ℛ\right)}^{l}\right)\right)$ when $U={U}_{h}𝔤$

Drinfeld [Dr, last par. of §4] explains the connection between the construction of central elements in (Ss 2.3) and the construction of central elements in [RTF, Theorem 14]. This is expanded by Baumann [Bau, 3rd par. of §3] as follows. Let $U={U}_{h}𝔤$ and let $\pi :U\to {M}_{n}\left(ℂ\right)$ be a representation of ${U}_{h}𝔤$ and let ${\pi }_{ij}:U\to ℂ$ be the matrix coefficients of $\pi .$ As in [RTF, Theorem 16(1) and Theorem 18] set $L+ = (lij+) = (π⊗id) (ℛ21) so that lij+ = (id⊗πij) (ℛ).$ Let $L- = (lij-) = (π⊗id) (id⊗S-1)(ℛ) = (π⊗id) ℛ-1) so that S(lij-) = (πij⊗id) (ℛ).$ Then ${e}^{h\rho }{\left({L}^{+}S\left({L}^{-}\right)\right)}^{k}$ is a matrix with entries in $U$ and

Examples. Since ${\gamma }^{0}={\left({ℛ}_{21}ℛ\right)}^{0}=1\otimes 1$ the definition of $\mathrm{dim}\left(L\left(\nu \right)\right)$ and ${\mathrm{dim}}_{q}\left(L\left(\nu \right)\right)$ give $(id⊗trL(ν)) (γ0) = dim(L(ν)) and (id⊗qtrL(ν)) ((ℛ21ℛ)0) = dimq(L(ν)).$ Since ${\pi }_{0}\left(\left(\mathrm{id}\otimes {\mathrm{tr}}_{L\left(\nu \right)}\right)\left(\gamma \right)\right)$ is a degree 1 element of $S\left(𝔥\right),$ ${S}^{1}{\left(𝔥\right)}^{{W}_{0}}=0$ and ${\pi }_{0}:Z\left(U\right)\to {\sigma }_{\rho }\left(S{\left(𝔥\right)}^{{W}_{0}}\right)$ is an isomorphism, it follows that $(id⊗trL(ν)) (γ) = 0.$

If $f=\sum _{\nu \in {𝔥}^{*}}{f}_{\nu }{X}^{\nu }$ is an element of $ℂ{\left[{𝔥}^{*}\right]}^{{W}_{0}}$ then $fsμ = f aμ+ρ aρ = 1 aρ f ∑w∈W0 det(w)w Xμ+ρ = 1 aρ ∑w∈W0 det(w)wf Xμ+ρ = 1 aρ ∑ν∈𝔥* ∑w∈W0 det(w)w fνXν Xμ+ρ = 1 aρ ∑ν∈𝔥* fν aν+μ+ρ = ∑ν∈𝔥* fν sν+μ. (Ss 4.1)$ Two special cases of (Ss 4.1) are $sγsμ = ∑ν∈P Kγν sν+μ, (Ss 4.2)$ and $∑w∈W0 Xwλ = s∅ ∑w∈W0 Xwλ = ∑w∈W0 swλ = ∑w∈W0 det(w-1) sw-1∘wλ = ∑w∈W0 det(w-1) sλ+w-1ρ-ρ. (Ss 4.3)$ Comparing coefficients of ${X}^{\nu }$ in (Ss 4.3) gives $δvλ,ν |(W0)λ| = ∑w∈W0 Kwλ,ν = ∑w∈W0 det(w) Kλ+wρ-ρ,ν (Ss 4.4)$ for some $v\in {W}_{0}.$

Baumann's identity

Identify Rep with For $l\in {ℤ}_{\ge 0}$ let $Ψl: ℂ[X]W0 → Z(Uh𝔤) sν ↦ (id⊗qtrL(ν)) ((ℛ21ℛ)l)$

[Bau, Thm. 1] For $l\in {ℤ}_{\ge 0}$ define $mλ(l) = ∑w∈W0 q2l ⟨wλ,ρ⟩ swλ.$ Then $Ψl (mλ(l)) = Ψ1 (mlλ(1/l)).$

 Proof. First note that $qtrL(μ) (Ψl(sγ)) = (qtrL(μ)⊗qtrL(γ)) ((ℛ21ℛ)l) = ∑ν∈P Kγν qtrL(μ+ν) ( ql ( ⟨μ+ν,μ+ν+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨γ,γ+2ρ⟩ ) ) = ∑ν∈P Kγν dimq(L(μ+ν)) ql ( ⟨μ+ν,μ+ν+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨γ,γ+2ρ⟩ ) = ∑ν∈P Kγν ev2ρ(aμ+ν+ρ) ev2ρ(aρ) ql ( ⟨μ+ν,μ+ν+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨γ,γ+2ρ⟩ )$ where the second equality uses (Ss 4.2). Taking ${\mathrm{qtr}}_{L\left(\mu \right)}$ of the LHS of the Baumann identity is $qtrL(μ) (Ψl (mλl)) = qtrL(μ) ∑w∈W0 det(w) q2l⟨λ,wρ⟩ Ψl (sλ+wρ-ρ) = ∑w∈W0 det(w) q2l⟨λ,wρ⟩ ∑ν∈P Kλ+wρ-ρ,ν ev2ρ(aμ+ν+ρ) ev2ρ(aρ) ql( ⟨μ+ν,μ+ν+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨λ+wρ-ρ,λ+wρ-ρ+2ρ⟩ ) = ∑ν∈P ev2ρ(aμ+ν+ρ) ev2ρ(aρ) ∑w∈W0 det(w) Kλ+wρ-ρ,ν q2l⟨λ,wρ⟩ ql( ⟨μ+ν,μ+ν+2ρ⟩ - ⟨μ,μ+2ρ⟩ - ⟨λ+wρ-ρ,λ+wρ-ρ+2ρ⟩ ) = ∑ν∈P ev2ρ(aμ+ν+ρ) ev2ρ(aρ) ql( ⟨ν,ν⟩ - ⟨λ,λ⟩ +2 ⟨ν,μ+ρ⟩ ) ∑w∈W0 det(w) Kλ+wρ-ρ,ν = ∑w∈W0 ev2ρ(aμ+wλ+ρ) ev2ρ(aρ) q2l ⟨wλ,μ+ρ⟩ ,$ where the last equality uses (Ss 4.4). Taking ${\mathrm{qtr}}_{L\left(\mu \right)}$ of the RHS of the Baumann identity is $qtrL(μ) ( Ψ1(mlλ(1/l)) ) = qtrL(μ) Ψ1 ∑v∈W0 det(v) q2 ⟨λ,vρ⟩ slλ+vρ-ρ = ∑v∈W0 det(v) q2 ⟨λ,vρ⟩ dimq (L(μ)) ev2(μ+ρ) (slλ+vρ-ρ),$ by Turaev-Wenzl. Thus $qtrL(μ) (Ψl (mλl)) = ∑v∈W0 det(v) q2 ⟨λ,vρ⟩ ev2ρ(aμ+ρ) ev2ρ(aρ) ev2(μ+ρ)(alλ+vρ) ev2(μ+ρ)(aρ) = 1 ev2ρ(aρ) ∑v∈W0 det(v) q2 ⟨λ,vρ⟩ ev2(μ+ρ)(alλ+vρ) = 1 ev2ρ(aρ) ∑v∈W0 q2 ⟨v-1λ,ρ⟩ ev2(μ+ρ)(alv-1λ+ρ) = ∑w∈W0 q2 ⟨wlλ,μ+ρ⟩ ev2ρ (awλ+μ+ρ) ev2ρ (aρ) ,$ since $∑y∈W0 q2⟨yλ,ρ⟩ ev2(μ+ρ) (alyλ+ρ) = ev2(μ+ρ) ∑y,w∈W0 q2⟨yλ,ρ⟩ det(w) Xlwyλ+wρ = ∑y,w∈W0 q2⟨wyλ,wρ⟩ det(w) q2⟨μ+ρ,lwyλ⟩ q2⟨μ+ρ,wρ⟩ = ∑x,w∈W0 det(w) q2⟨μ+ρ,lxλ⟩ q2⟨xλ+μ+ρ,wρ⟩ = ∑x∈W0 q2⟨μ+ρ,xlλ⟩ ev2ρ (axλ+μ+ρ).$ Thus $qtrL(μ) (Ψl(mλl)) = qtrL(μ) (Ψ1(mlλ(1/l)))$ which completes the proof of the Baumann identity (THOUGH NOT THE STATEMENT THAT THE ${B}_{L\left(\lambda +w\rho -\rho \right)}^{\left(l\right)}$ ARE CHARACTERIZED BY THIS IDENTITY. $\square$

It follows from (Ss 3.4)??? that the quantum dimension is $dimq (L(ν)) = ev2ρ (sν) = ev2ρ ( aν+ρ aρ ) = ∑w∈W det(w) q ⟨w(ν+ρ),2ρ⟩ ∑w∈W det(w) q ⟨wρ,2ρ⟩ = ∑w∈W det(w) q ⟨2(ν+ρ),wρ⟩ ∑w∈W det(w) q ⟨2ρ,wρ⟩ = ev2(ν+ρ)(aρ) ev2ρ(aρ) = ∏α∈R+ ev2(ν+ρ) (Xα/2 - X-α/2) ev2ρ (Xα/2 - X-α/2) = ∏α∈R+ q ⟨ν+ρ,α⟩ - q -⟨ν+ρ,α⟩ q ⟨ρ,α⟩ - q -⟨ρ,α⟩ .$ Hence $(id⊗qtrL(ν)) ((ℛ21ℛ)0) = ∏α∈R+ [⟨ν+ρ,α⟩] [⟨ρ,α⟩] , where [k] = qk-q-k q-q-1 ,$ and the Weyl dimension formula is $(id⊗qtrL(ν)) (γ0) = dim(L(ν)) = ev0(sν) = limq→1 ev2ρ(sν) = ∏α∈R+ ⟨ν+ρ,α⟩ ⟨ρ,α⟩ .$

References [PLACEHOLDER]

[DRV] Z. Daugherty, A. Ram, R. Virk, Appendices to Affine and degenerate affine BMW algebras: Actions on tensor space.