I'm having difficulty following computations in the paper "KdV and W-flows" by Zuber.

On pg. 2, what would be the conserved quantity $I_4$, related to the conservation laws of the KdV hierarchy? (Off-chance connection to face polynomials of associahedra.)

On pg. 3, how to interpret the Poisson bracket in Eqn. 2.8 to explicitly derive the RHS? ($u_n$ as defined by Eqn. 2.7 is a scalar, so the bracket makes no sense to me. This is supposedly an analogue to the celebrated Virasoro central extension of the Witt Lie algebra.)


As for $I_4$ and $I_j$ for all $j\geqslant 4$ up to overall sign factors, see e.g. equation (1.9) of these lecture notes with $u=-r$. Also you may wish to look at this link.

Regarding the bracket at p.3, the point is that in fact the brackets considered by the author are defined on functionals, which $u_n$ indeed are, and this is the proper way to interpret formulas like the one immediately after (2.4); for Poisson brackets on functionals, including in particular the KdV case, see e.g. Chapter 7 of Applications of Lie groups to differential equations by Peter Olver.


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