I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka.

When discussing connections between how algebraists might find Lie algebras (as the primitives of Hopf algebras) and how geometers might find Lie algebras (as the tangent space at identity of a Lie group) Tanaka writes:

A cooler link arises when a geometer looks at distributions near the identity of G (which are dual to 'functions on G') rather than functions themselves. This isn't so obviously the right thing to look at in the finite groups example, but if you believe that functions on a Lie group G are like de Rham forms on G, then you'd believe that something like 'the duals to functions on G' (which are closer to vector fields) would somehow safeguard the Lie algebra structure. The point being, you should expect to find Lie structures to arise from things that look like 'duals to functions on a group'. So one should take 'distributions' to be the Hopf algebra in question, and look at its primitives to find the Lie algebra of 'vector fields.'

This is the paragraph that I'd very much like to understand. Unfortunately I have very little background in classical Lie theory and so do not understand the motivation very deeply. Also my knowledge of distributions is even more limited.

My first question must then be:

Can someone expand slightly on this paragraph or perhaps point in a fast-track direction towards some literature that might help me?

Tanaka seems to be motivating the part of Quillen's construction which takes a simplicial group $G$ and associates the completion $\widehat{\mathbb Q G}$ of the Hopf algebra $\mathbb Q G$ (where $\Delta x = x \otimes x$ in $\mathbb Q G$).

This then allows for my main question:

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity on $G$?

Are distributions supported at the identity like differential operators?

What role is the completion playing here?



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