## Explicit Poisson bracket on total complex of n-commutative cosimplicial monoid

### Michael Batanin – 5 June 2019

This is the third talk of a series. In two previous talks I showed how to construct a E_{n+1}-algebra structure on a total complex of an n-commutative cosimplicial monoid.
In this talk I'll explain that this action can be made combinatorially explicit. More specifically I'll prove a formula for Poisson bracket of degree - n in terms of a sum over all noncommutative liftings of smooth Delannoy paths on a pxq -lattice. Except for application to deformation theory of tensor categories this result can be very useful for constructing of explicit small models of E_{n+1}-algebras. I conjecture that there should be a Quillen equivalence between n-commutative cosimplicial R-algebras and E_{n+1}-algebras over a commutative ring R.

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