We show that one can construct a classical affine W-algebra via a classicalBRST complex. This definition clarifies that classical affine W-algebras can beconsidered as quasi-classical limits of quantum affine W-algebras. We also give a definition of a classical affine fractional W-algebra as aPoisson vertex algebra. As in the classical affine case, a classical affinefractional W-algebra has two compatible $\lambda$-brackets and is isomorphic toan algebra of differential polynomials as a differential algebra. When aclassical affine fractional W-algebra is associated to a minimal nilpotent, wedescribe explicit forms of free generators and compute $\lambda$-bracketsbetween them. Provided some assumptions on a classical affine fractionalW-algebra, we find an infinite sequence of integrable systems related to thealgebra, using the generalized Drinfel'd and Sokolov reduction.
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