Last updates: 28 June 2011
1.1 Recall that
is the Hopf algebra given by generators with multiplication and comultiplication given by
2.2 Recall that there is a grading on
given by setting and
. This grading induces a grading on
and . The inner product between and
respects this grading. It follows that
for all since the left hand component of the inner product has degree zero and the right hand component of the inner product has degree . Using this observation we have that
Let us evaluate this inner product
and recall that
It follows easil by degree counts and arguments as above that
Thus, the bases
are dual bases in
2.3 It follows that the universal -matrix for
is given by
2.4 We shall write the universal -matrix in terms of the generators
so that we may project this -matrix modulo the ideal in
generated by in order to get an -matrix for
Recall that in the notations of section 1 we had
it follows that
and we have
In a similar fashion, using the fact that
we have that
Now we may rewrite the
-matrix as follows
Now we may project modulo the ideal of
and write the universal
in the form
The fact that
is a quotient of the double of
is stated in §13 of [D
] along with the formula for the universal
N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.
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.
The theorem that a Lie bialgebra structure determinse a co-Poisson Hopf algebra structure on its enveloping algebra is due to Drinfel'd and appears as Theorem 1 in the following article.
V.G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dokl. 32 (1985), 254–258.
Derivations of the double and the universal -matrix for
appear in [R] and [B]. Our derivation of
follows the example given in [B].
The universal R-matrix for and beyond!, Comm. Math. Phys. 127 (1990), 109–128.
An analogue of P.B.W. theorem and the universal R-matrix for Uhsl(N+1)., Quantum groups (Clausthal, 1989),
Comm. Math. Phys. 124 (1989), no. 2, 307–318.
H.-D. Doebner, Hennig, J. D. and W. Lücke,
Mathematical guide to quantum groups, Quantum groups (Clausthal, 1989),
Lecture Notes in Phys., 370, Springer, Berlin, 1990, pp. 29–63.
There is a very readable and informative chapter on Lie algebra cohomolgy in the forthcoming book
R. Howe, R. Goodman, and N. Wallach,
Representations and Invariants of the Classical Groups, manuscript, 1993.
The following book is a standard introductory book on Lie algebras; a book which remains a standard reference. This book has been released for a very reasonable price by Dover publishers. There is a short description of Lie algebra cohomology in Chapt. III §11, pp.93-96.
N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.
A very readable and complete text on Lie algebra cohomology is
Lie groups, Lie algebras and cohomology, Mathematical Notes 34, Princeton University Press, 1998.