The quantum double

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

Last updates: 16 July 2011

The quantum double

([D1] §13) Let A be a finited dimensional Hopf algebra and let Aopp denote the Hopf algebra A except with the opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra D(A,R) such that

  1. D(A) contains A and Aopp as Hopf subalgebras.
  2. R is the image of the canonical element of AAopp under AAopp D(A)D(A), i.e. if ei is a basis of A and ei is the dual basis in Aopp then R=eiei D(A)D(A).
  3. The linear map AAopp D(A) ab 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(A)D(A) so that the element R=eiei is a well defined element of the completion D(A) ˆ D(A) .

Proof of Theorem 1.1.

2.3 Let the algebra A be the Hopf algebra with basis er and multiplication, comultiplication, and skew antipode given by eres = t mrst et, Δ(et) = r,s μtrs eres, σ(et) = r σtr er. The unit and counit will ge given by 1=tEtet and ϵ(er)=er . Recall that the skew antipode is the inverse S-1 of the antipode of A and is the antipode for the Hopf algebra Aopp which is the same as the algebra A except with the opposite comultiplication.

2.4 The algebra Aopp has basis er which is dual to the basis er of A and has multiplication and comultiplication given by eres = t μtrs et, Δ(et) = rs eser. Then the algebra AAopp has basis eres and has multiplication given by

( eresepeq ) ( ekelemen ) = ( eres ekel epeq emem ) LABEL
and comultiplication given by Δ(eres) = Δ(er) Δ(es) = ( u,v muvr eveu ) ( p,q μspq epeq ) = u,v,p,q muvr μspq evepeueq. Alternatively, we could have chosen to use the basis eres instead of the basis eres . It is clear from (LABEL) that we need to describe a product esek in terms of the basis epeq . The relation is eres= α,β,γ,δ,p μrγβα σαp m pδγs eδeβ. This relation is derived as follows. eveb , ejel = eveb , mσ (elej) = σΔ(eveb) , elej = Δ(eveb) , elej = RΔ(eveb) R-1 , elej = RΔ(eveb) (idS-1) (R) , elej by (1.3e) = RΔ(eveb) (idS-1) (R) , (Δ)2 (elej) . Let us expand the left hand factor of this inner product.
RΔ(eveb) (idS-1)(R) = k,p ekek Δ(eveb) (idS-1)(R) (epep) = k,p,q r,s,t,u ekek mrsv μbut ereueset ep σqp eq. LABEL2
The right hand factor of the inner product expands in the form (Δ)2 (elej) = ( Δid ) Δ ( elej ) = ( Δid ) ( x,y,w,z mxyl μjwz eyewex ez ) = x,y,w,z m,n,c,d mmny μwcd mxyl μjwz enecem edexez = m,n,x c,d,z mxmnl μjcdz enecem edexez. 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(A) 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 AAopp satisfies the condition (1.2a) for a quasitriangular Hopf algebra. ( ( σΔ ) ( ev eb ) )R = δ,γ,r,s,m mδγv μbrs ( eδes eγer ) ( emem ) = 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 ( emem ) ( 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 ( idΔ ) ( R )= R13R12 and ( Δid ) ( R )= R13R23 . ( idΔ ) ( R ) = k ek Δ(ek) = k,r,s mrsk ekeser = r,s eres eser = r,s ( er1er ) ( eses1 ) = R13R12. Similarly we have that ( Δid ) ( R ) = k Δ(ek)ek = k,r,s μkrs eresek = r,s ereseres = r,s ( er1er ) ( 1eses ) = R13R23 This completes the proof of Theorem 1.1.


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).

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