Quasitriangular Hopf algebras

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

Last updates: 11 July 2011

Quasitriangular Hopf algebras

1.1 Let A be a Hopf algebra with coproduct Δ and antipode S. Let σ:AAAA be the map given by σ(ab)=ba for all a,bA. Define Δ to be the opposite coproduct given by Δ=σΔ. Then A with coproduct Δ and antipode S-1 is also a Hopf algebra. This follows by applying S-1 to the defining relation for the antipode a a(1) S(a(2))= a S(a(1)) a(2) =ϵ(a), for all aA and using the fact that S (and therefore S-1) is an antihomomorphism.

1.2 A pair (A,R) consisting of a Hopf algebra A and an invertible element RAA is called quasitriangular if

  1. Δ(a)= RΔ(A)R-1, for all aA,
  2. (Δid)(R)= R12R23,
  3. (idR)= R13R12,
where, if R= i aibi then R12= i aibi1, R13= i ai1bi, R23= i 1aibi1, etc.

([D] Prop. 3.1) If (A,R) is a quasitriangular Hopf algebra then

  1. R12 R13 R23 = R23 R13 R12.
  2. R ˇ 12 R ˇ 23 R ˇ 12 = R ˇ 23 R ˇ 12 R ˇ 23 ,
    where R ˇ ij = σ LRij End(AA), and σ,LR End(AA) are given by σ(ab)= ba and left multiplication by R respectively.
  3. (ϵid)(R) =1= (idϵ)(R)
  4. (Sid)(R) = (idS-1)(R) = R-1.
  5. (SS)(R)=R.



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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