I just stepped into this definition thorugh this stackexchange question and entered to skim the first paper to pop up in google and I have absolute zero intuition for how to interpret intuitively the $t$ (the infinitesimal braiding) or what is the canonical example I should have in mind.
I feel comfortable with braided monoidal categories and pre-additive/$Ab$-enriched categories.
I don't know very much about lie algebras for instance though, which I feel would give me an answer but I didn't find anything quickly regarding this.
Hoping someone can illuminate my picture on the topic possibly with an easy example. If it can be taken all the way back to basics, even better.
Here's the definition as a quick recap or reference.
Consider a pre-additive braided monoidal category $(M,\otimes,I,\tau)$.
An infinitesimal braiding is a natural transformation $t : \otimes \to \otimes$ such that
$$ t_{x,y\otimes z} = t_{x,y}\otimes \mathrm{id}_z + (\tau^{-1}_{x,y}\otimes \mathrm{id}_z) \circ (\mathrm{id}_y\otimes t_{x,z}) \circ (\tau_{x,y}\otimes \mathrm{id}_z) $$
$$ t_{x\otimes y,z} = \mathrm{id}_x\otimes t_{y,z} + (\mathrm{id}_x\otimes\tau^{-1}_{y,z}) \circ (t_{x,z}\otimes \mathrm{id}_y) \circ (\mathrm{id}_x\otimes \tau_{y,z}) $$
Thanks in advance!
