The quantum double

([D1] §13) Let $A$ be a finited dimensional Hopf algebra and let ${A}^{\ast opp}$ denote the Hopf algebra ${A}^{\ast }$ except with the opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra $D\left(A,R\right)$ such that

1. $D\left(A\right)$ contains $A$ and ${A}^{\ast opp}$ as Hopf subalgebras.
2. $R$ is the image of the canonical element of $A\otimes {A}^{\ast opp}$ under $A\otimes {A}^{\ast opp}\to D\left(A\right)\otimes D\left(A\right),$ i.e. if ${e}_{i}$ is a basis of $A$ and ei is the dual basis in ${A}^{\ast opp}$ then $R=∑ei⊗ei∈ D(A)⊗D(A).$
3. The linear map $A⊗A∗opp → D(A) a⊗b → ab$ is bijective.

2.2 Remark. If $A$ is infinite dimensional then one may be abl to apply the theorem if there is a suitable way of completing the tensor product $D\left(A\right)\otimes D\left(A\right)$ so that the element $R=\sum {e}_{i}\otimes {e}^{i}$ is a well defined element of the completion $D\left(A\right)\stackrel{ˆ}{\otimes }D\left(A\right)$.

 Proof of Theorem 1.1.
$\square$

2.3 Let the algebra $A$ be the Hopf algebra with basis $\left\{{e}_{r}\right\}$ and multiplication, comultiplication, and skew antipode given by $eres = ∑ t mrst et, Δ(et) = ∑ r,s μtrs er⊗es, σ(et) = ∑ r σtr er.$ The unit and counit will ge given by $1={\sum }_{t}{E}^{t}{e}_{t}$ and $ϵ\left({e}_{r}\right)={e}_{r}$. Recall that the skew antipode is the inverse ${S}^{-1}$ of the antipode of $A$ and is the antipode for the Hopf algebra ${A}^{opp}$ which is the same as the algebra $A$ except with the opposite comultiplication.

2.4 The algebra ${A}^{\ast opp}$ has basis $\left\{{e}^{r}\right\}$ which is dual to the basis $\left\{{e}_{r}\right\}$ of $A$ and has multiplication and comultiplication given by $eres = ∑ t μtrs et, Δ(et) = ∑ rs es⊗er.$ Then the algebra $A\otimes {A}^{\ast opp}$ has basis $\left\{{e}^{r}{e}_{s}\right\}$ and has multiplication given by

 $( eres⊗epeq ) ( ekel⊗emen ) = ( eres ekel ⊗ epeq emem )$ LABEL
and comultiplication given by $Δ(eres) = Δ(er) Δ(es) = ( ∑ u,v muvr ev⊗eu ) ( ∑ p,q μspq ep⊗eq ) = ∑ u,v,p,q muvr μspq evep⊗eueq.$ Alternatively, we could have chosen to use the basis $\left\{{e}_{r}{e}^{s}\right\}$ instead of the basis ${e}^{r}{e}_{s}$. It is clear from (LABEL) that we need to describe a product ${e}_{s}{e}^{k}$ in terms of the basis ${e}^{p}{e}_{q}$. The relation is $eres= ∑ α,β,γ,δ,p μrγβα σαp m pδγs eδeβ.$ This relation is derived as follows. $⟨ eveb , ejel ⟩ = ⟨ eveb , m∘σ (el⊗ej) ⟩ = ⟨ σ∘Δ(eveb) , el⊗ej ⟩ = ⟨ Δ′(eveb) , el⊗ej ⟩ = ⟨ RΔ(eveb) R-1 , el⊗ej ⟩ = ⟨ RΔ(eveb) (id⊗S-1) (R) , el⊗ej ⟩by (1.3e) = ⟨ R⊗Δ(eveb) ⊗(id⊗S-1) (R) , (Δ⊗)2 (el⊗ej) ⟩.$ Let us expand the left hand factor of this inner product.
 $R⊗Δ(eveb) ⊗ (id⊗S-1)(R) = ∑ k,p ek⊗ek⊗ Δ(eveb)⊗ (id⊗S-1)(R) (ep⊗ep) = ∑ k,p,q r,s,t,u ek⊗ek⊗ mrsv μbut ereu⊗eset ⊗ep⊗ σqp eq.$ LABEL2
The right hand factor of the inner product expands in the form $(Δ⊗)2 (el⊗ej) = ( Δ⊗⊗id⊗ ) ∘ Δ⊗ ( el⊗ej ) = ( Δ⊗⊗id⊗ ) ( ∑ x,y,w,z mxyl μjwz ey⊗ew⊗ex⊗ ez ) = ∑ x,y,w,z m,n,c,d mmny μwcd mxyl μjwz en⊗ec⊗em ⊗ed⊗ex⊗ez = ∑ m,n,x c,d,z mxmnl μjcdz en⊗ec⊗em ⊗ed⊗ex⊗ez.$ Now let us evaluate the inner product. Thi inner product picks out only the terms when $m=n,k=c,v=m, b=d,p=x,q=z,$ and this term appears with coefficient $mxmml μjcdz σqp = mpvkl μjkbq σqp .$ It follows that $⟨ eveb , ejel ⟩= ∑ p,q,k mpvkl μjkbq σqp .$ The multiplication rule follows.

2.6 We shall need the following calculation in our proof that $D\left(A\right)$ is quasitriangular. We shall need the identities in §4 of the notes on co-Poisson Hopf algebras. $∑ γ,α,p,s μaγβαs σαp mspδγv = ∑ γ,α,n,k,s,p μaγβn μnαs σαs mspk mkδγv by 4.1 and 4.4 = ∑ γ,α,n,k μaγβn εnEk mkδγv by 4.11 = ∑ γ,α,n,k μaγβn εnEk δδn mmγv by 4.2 = ∑ γ,n μaγβn εn Ek mδγv = ∑ γ,n,k μaγk μkβn εn mδγv by 4.4 = ∑ γ,k μaγk δkβ mδγv by 4.5 = ∑ γ μaγβ mδγv .$

2.7 Now we prove that $A\otimes {A}^{\ast opp}$ satisfies the condition (1.2a) for a quasitriangular Hopf algebra. $( ( σ∘Δ ) ( ev eb ) )R = ∑ δ,γ,r,s,m mδγv μbrs ( eδes⊗ eγer ) ( em⊗em ) = ∑ δ,γ,r,s,m mδγv μbrs ( eδesem⊗ eγerem ) = ∑ δ,γ,r,s,m,λ mδγv μbrs msmλ ( eδeλ⊗ eγerem ) = ∑ δ,γ,r,s,m,λ u,t,α,p,β mδγv μbrs μrutα σαp mpβum msmλ ( eδeλ⊗ eγeβet ) = ∑ δ,γ,r,s,m,λ u,t,α,p,β,a mδγv μaγβ μrutα σαp mpβum msmλ ( eδeλ⊗ eaet ) = ∑ δ,γ,s,m,λ u,t,α,p,β,a mδγv μaγβ μbutαs σαp mspβuλ ( eδeλ⊗ eaet ) = ∑ δ,γ,s,λ u,t,α,p,β,a mδγv μaγβ μbutαs σαp mspβuλ ( eδeλ⊗ eaet ) = ∑ δ,γ,u,t,β,a,λ mδγv μaγβ μbut mβuλ ( eδeλ⊗ eaet ).$ A similar calculation on the right hand side gives $RΔ(eveb) = ∑ r,s,t,u,m msrv μbut ( em⊗em ) ( ereu⊗ eset ) = ∑ r,s,t,u,m msrv μbut ( emereu⊗ emeset ) = ∑ r,s,t,u,m,a msrv μbut μams ( emereu⊗ eaet ) = ∑ r,s,t,u,m,a α,β,γ,δ,p msrv μbut μams μmγβα σαp mpδγr ( edeβeu⊗ eaet ) = ∑ r,s,t,u,m,a α,β,γ,δ,p,λ msrv μbut μams μmγβα σαp mpδγr mβuλ ( eδeλ⊗ eaet ) = ∑ s,t,u,m,a α,β,γ,δ,p,λ μbut μaγβαs σαp mspδγv mβuλ ( eδeλ⊗ eaet ) = ∑ δ,γ,u,t,β,aλ mδγv μaγβ μbut mβuλ ( eδeλ⊗ eaet ).$

2.8 It remains to prove the identities $\left(id\otimes \Delta \right)\left(R\right)={R}^{13}{R}^{12}$ and $\left(\Delta \otimes id\right)\left(R\right)={R}^{13}{R}^{23}$. $( id⊗Δ ) ( R ) = ∑ k ek⊗ Δ(ek) = ∑ k,r,s mrsk ek⊗es⊗er = ∑ r,s eres⊗ es⊗er = ∑ r,s ( er⊗1⊗er ) ( es⊗es⊗1 ) = R13R12.$ Similarly we have that $( Δ⊗id ) ( R ) = ∑ k Δ(ek)⊗ek = ∑ k,r,s μkrs er⊗esek = ∑ r,s er⊗es⊗eres = ∑ r,s ( er⊗1⊗er ) ( 1⊗es⊗es ) = R13R23$ This completes the proof of Theorem 1.1.

References

The quantum double seems to have appeared first in the following paper.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

Some further proofs and hints appear in the following.

[D1] V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in Leningrad Math. J. 1 (1990), 321–342. MR1025154

[Re] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).