Zappa-Szép product of free Lie algebras

There is a perfect description for Zappa-Szép product of groups in Has this "generalized semidirect product" been studied? . Has it been studied for free Lie algebras?

I note that, the Zappa-Szép product of Lie algebras is defined as follows: Let $A$ and $B$ be Lie algebras. A derivatively knitted pair of representations $(α, β)$ for $(A, B)$ are Lie algebra homomorphisms $\alpha: A \to \operatorname{End}(B)$ and $\beta: B \to \operatorname{End}(A)$ such that:

$α(a)[b_{1}, b_{2}] = [α(a)b_{1}, b_{2}] + [b_{1}, α(a)b_{2}]- α(β(b_{1})a)b_{2} - α(β(b_{2})a)b_{1}$

$β(b)[a_{1}, a_{2}] = [β(b)a_{1}, a_{2}] + [a_{1}, β(b)a_{2}] -β(α(a_{1})b)a_{2} -β(α(a_{2})b)a_{1}$

1 Answer

Most probably, yes. There is a wonderful quote of Matt Brin:

"It turns out that the Zappa-Szép product has the remarkable property that it seems to require no hypotheses at all. Group-like properties such as associativity, fullness of the multiplication, identities, and inverses can be assumed or removed at random and the Zappa-Szép product can still be discussed and used at some level."

For a concrete starting point, see short this article* on "knit products" of graded Lie algebras.

*Peter W. Michor, Knit Products of Graded Lie Algebras and Groups, Proceedings of the Winter School on Geometry and Physics, Srni 1988 Suppl. Rendiconti Circolo Matematico di Palermo, Ser. II, 22 (1989), 171–175

• Nice article by Peter Michor. – Dietrich Burde Oct 25 '17 at 18:37