Cohomology and Rigidity Theorems
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.
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