Given a principal ideal domain $R$ of characteristic zero, containing $1/2$, and a connected differential non-negatively graded free finite type $R$-module $V$, we prove that the natural arrow $𝕃FH\left(V\right)\to FH𝕃\left(V\right)$ is an isomorphism of graded Lie algebras over $R$, and deduce thereby that the natural arrow $UFH𝕃\left(V\right)\to FHU𝕃\left(V\right)$ is an isomorphism of graded cocommutative Hopf algebras over $R$; as usual, $F$ stands for free part, $H$ for homology, $𝕃$ for free Lie algebra, and $U$ for universal enveloping algebra. Related facts and examples are also...

Let $\left(𝕃\right(V),d)$ be a free graded connected differential Lie algebra over the field $\mathbb{Q}$ of rational numbers. An ideal $I$ in the Lie algebra $H\left(𝕃\right(V),d)$ is called if, for every cycle $\alpha \in 𝕃\left(V\right)$ such that $\left[\alpha \right]$ belongs to $I$, the kernel of the map $H\left(𝕃\right(V),d)\to H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$, $d\left(x\right)=\alpha$, is contained in $I$. We show that the center of $H\left(𝕃\right(V),d)$ is a nice ideal and we give in that case some informations on the structure of the Lie algebra $H\left(𝕃\right(V\oplus \mathbb{Q}x),d)$. We apply this computation for the determination of the rational homotopy Lie algebra $L_X={\pi }_{*}\left(\Omega X\right)\otimes \mathbb{Q}$ of a simply connected space $X$. We deduce...

Let $A={⨁}_k{A}_k$ and $B={⨁}_k{B}_k$ be graded Lie algebras whose grading is in $𝒵$ or ${𝒵}_2$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha$ and $\beta$ satisfy equations which look “derivatively knitted"; then $A\oplus B:={⨁}_{k,l}(A_k\oplus B_l)$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A{\oplus }_{(\alpha ,\beta )}B$. This graded Lie algebra is called the knit product of $A$ and $B$. The author investigates the general situation for any graded Lie subalgebras $A$ and...