Part Iv.1. Lie Algebras and Co-commutative Co-algebras

Introduction 2 1. Lie algebras: recollections 3 1.1. The basics 3 1.2. Scaling the structure 3 1.3. Filtrations 4 1.4. The Chevalley complex 4 1.5. The functor of primitives 6 1.6. The enhanced adjunction 6 1.7. The symmetric Hopf algebra 8 2. Looping Lie algebras 9 2.1. Group-Lie algebras 10 2.2. Forgetting to group structure 10 2.3. Chevalley complex of group-Lie algebras 11 2.4. Chevalley complex and the loop functor 12 2.5. The tensor Hopf algebra 13 2.6. The co-symmetric algebra 15 2.7. Proof of Theorem 2.4.5 16 3. The universal enveloping algebra 17 3.1. Universal enveloping algebra: definition 17 3.2. Map from the tensor Hopf algebra 18 3.3. The PBW theorem 19 3.4. The Bar complex of the universal envelope 21 3.5. Modules for the Lie algebra 21 4. The universal envelope via loops 22 4.1. The main result 22 4.2. Proof of Theorem 4.1.2 23 4.3. Proof of Proposition 4.2.4 24 4.4. The map from the tensor Hopf algebra, revisited 24 Appendix A. Commutative co-algebras and bialgebras 27 A.1. Two incarnations of co-commutative bialgebras 27 A.2. Modules over co-commutative Hopf algebras 29 Appendix B. Actions of monoids and filtrations 31 B.1. Equivariance with respect to a monoid 31 B.2. Equivariance in algebraic geometry 31 B.3. The category of filtered objects 33 B.4. The associated graded 33 Appendix C. Proof of the PBW theorem 34 C.1. The monoidal category of symmetric sequences 34 C.2. The PBW theorem at the level of operads 35