The quantum double
Last updates: 16 July 2011
The quantum double
([D1] §13) Let be a finited dimensional Hopf algebra and let
denote the Hopf algebra except with the opposite comultiplication. Then there exists a unique quasitriangular Hopf algebra
as Hopf subalgebras.
is the image of the canonical element of
i.e. if is a basis of and
ei is the dual basis in then
The linear map
2.2 Remark. If is infinite dimensional then one may be abl to apply the theorem if there is a suitable way of completing the tensor product so that the element is a well defined element of the completion
Proof of Theorem 1.1.
2.3 Let the algebra be the Hopf algebra with basis
and multiplication, comultiplication, and skew antipode given by
The unit and counit will ge given by
Recall that the skew antipode is the inverse of the antipode of and is the antipode for the Hopf algebra which is the same as the algebra except with the opposite comultiplication.
2.4 The algebra has basis
which is dual to the basis
of and has multiplication and comultiplication given by
Then the algebra
and has multiplication given by
and comultiplication given by
Alternatively, we could have chosen to use the basis
instead of the basis
It is clear from (LABEL
) that we need to describe a product
in terms of the basis
. The relation is
This relation is derived as follows.
Let us expand the left hand factor of this inner product.
The right hand factor of the inner product expands in the form
Now let us evaluate the inner product. Thi inner product picks out only the terms when
and this term appears with coefficient
It follows that
The multiplication rule follows.
2.6 We shall need the following calculation in our proof that is quasitriangular. We shall need the identities in §4 of the notes on co-Poisson Hopf algebras.
Now we prove that satisfies the condition (1.2a) for a quasitriangular Hopf algebra.
A similar calculation on the right hand side gives
2.8 It remains to prove the identities
Similarly we have that
This completes the proof of Theorem 1.1.
The quantum double seems to have appeared first in the following paper.
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.
Some further proofs and hints appear in the following.
V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in
Leningrad Math. J. 1 (1990), 321–342.
N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).