(Contravariant) Koszul duality for DG algebras

A DG algebras $A$ over a field $k$ with $H(A)$ connected and $H_{<0}(A)=0$ has a unique up to isomorphism DG module $K$ with $H(K)\cong k$. It is proved that if $H(A)$ is degreewise finite, then RHom_A(?,K): D^{df}_{+}(A)^{op} \equiv D_{df}^{+}}(RHom_A(K,K)) is an exact equivalence of derived categories of DG modules with degreewise finite-dimensional homology. It induces an equivalences of $D^{df}_{b}(A)^{op}$ and the category of perfect DG $RHom_A(K,K)$-modules, and vice-versa. Corresponding statements are proved also when $H(A)$ is simply connected and $H^{<0}(A)=0$.Comment: 33 page

Invariants of Lie algebras extended over commutative algebras without unit

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.Comment: v3: added a footnote on p.10 about a wrong derivation of the correct statemen