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Q1: This algebra is just the Manin black product of $A$ and $A^!$ (in other words, the Koszul dual of the Segre product of $A$ and $A^!$), and hence it is Koszul. (As requested, the Segre product of t …

I figured out that in full generality this problem has no chance of leading to a different algebraic structure for which the given one is a quasi-classical limit (like it is for associative/Poisson): …

This is too long for a comment. The reason for my question in comments was as follows. In some cases (Gerstenhaber algebras, Poisson algebras, etc.) an operad describing a certain algebraic structure …

Well, for your question 1 you presumably may ask yourself first about a relationship between (shifted) Lie algebras and Lie 2-algebras. Lie 2-algebras of Hanlon and Wachs can be viewed as $L_\infty$-a …

The bialgebra pairing condition implies $(1,b_1b_2)=(\Delta(1),b_1\otimes b_2)=(1,b_1)(1,b_2)$, so in particular $(1,1)=(1,1)^2$, and $(1,1)$ is zero or one. Also by a similar calculation, $(b_1b_2,1 …

I personally like this note of John Baez: http://math.ucr.edu/home/baez/braids/node4.html

First of all, you might want to look at the MO question How to define the equivalence of Maurer-Cartan elements in an $L_\infty$-algebra?, since of course this is effectively what you need (morphisms …

It really sounds like you'd want to look at works of Pierre Vogel, starting from The universal Lie algebra, and also check recent results that use the same setup, like this one.