Abstract

In this chapter we prove two rigidity theorems, both needed in Chapter XIX. The first one is classical: it asserts that any formal deformation of the enveloping algebra of a semisimple Lie algebra is trivial. The proof is based on the vanishing of certain cohomology groups. The second rigidity result is due to Drinfeld [Dri89b] [Dri90]. It states that if A and A’ are quantum en-veloping algebras with the same underlying cocommutative bialgebras and the same universal R-matrices, then there exists a gauge transformation from A to A’. The proof again relies on some cohomological considerations, this time involving the cobar complex of a symmetric coalgebra.

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.