The canonical Poisson bracket on a polynomial algebra in $2n$ variables

For a commutative ring R and a finite type σ, we equip MvPolynomial (σ ⊕ σ) R with its canonical Poisson bracket $${f, g} = \sum_{i} \left(\frac{\partial f}{\partial x_i}\frac{\partial g}{\partial y_i}

\frac{\partial f}{\partial y_i}\frac{\partial g}{\partial x_i}\right),$$ where $x_i$ (resp. $y_i$) denotes the variable indexed by Sum.inl i (resp. Sum.inr i). This is the polynomial algebra underlying the n-th canonical Poisson algebra $P_n(R)$ when σ = Fin n.

Classical sources index the same object by $2n$ variables $X_1, \dots, X_{2n}$ with bracket ${f, g} = \sum_{i=1}^n (\partial f/\partial X_i \cdot \partial g/\partial X_{i+n}

\partial f/\partial X_{i+n} \cdot \partial g/\partial X_i)$; here Sum.inl i plays the role of $X_i$ and Sum.inr i the role of $X_{i+n}$ (see e.g. AvdE07, Notations 1).