Last updates: 10 April 2011
[D1 Thm. 1] Let be a Lie algebra and let be a map that turns into a co-Poisson Hpf algebra with the standard multiplication and comultiplication. Then , and the map indused by turns into a Lie bialgebra. Conversely, if is a Lie bialgebra, then there is a uniqe map which is an extension of and which turns into a co-Poisson Hopf algebra.
is a Lie bialgebra. Let and let
Computation 1. The co-Poisson homomorphism condition gives us that
Computation 2. Now let us analyse the condition that is a co-Poisson map. On the other hand this must also be equal to
Computation 3. The following fact will also be useful:
: Suppose that is a Lie bialgebra. We must show that there is a unique extension to a map such that
2) Let and let us assume that we know that Consider the following On the other hand we have that Thus we need only show that The computation shows that is a Poisson bracket. Dualizing this computation will show that satisifies the appropriate condition if and do.
3) It remains to show that is a Lie cobracket. Since is a Lie bracket on , the skew symmetry condition implies that of . Now assume that and that we know that Since is cocommutative it follows that Now it follows from computation 1 that satisfies the skew-symmetry condition. We must check that satisfies the co-Jacobi identity. Let and let us assume that we know that We need to show that . It follows from that Thus it is sufficient to show that satisfies the co-Jacobi identity. This follows (dually) from the fact that satisfies the Jacobi identity.
: Suppose that is a co-Poisson Hopf algebra with the usual multiplication and comultiplication. We need to show two things:
1) First let us show that . Since , the co-Poisson homomorphism condition gives that It follows that if .
Now suppose that . Then , and computation 2 becomes It follows that is a primitive element of and thus is an element of . The result follows by noting that since must satisfy the skew symmetry condition for a Lie algebra.
2) Suppose that . Then and . It follows from computation 1 above that So satisfies the -cocycle condition.Thus is a bialgebra.
[Bou] N. Bourbaki, Lie groups and Lie algebras, Chapters I-III, Springer-Verlag, 1989.
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.
[J] N. Jacobson, Lie algebras, Interscience Publishers, New York, 1962.