After establishing the notation and conventions (especially for signs), we give the formal definition of sh (strongly homotopy) Lie structure and verify the equivalence with a formulation in terms of a “nilpotent” operator on \(\wedge sV\). Comparison with the physics literature calls attention to some further subtleties of signs. Then we establish the sh analog of the familiar fact that commutators in an associative algebra form a Lie algebra. We next point out the relevance of these structures to \((N+1)\)-point functions in physics. We remark on the distinctions between these structures on the cohomology level and at the underlying form level and conclude with the basic theorem of homological perturbation theory relating higher-order bracket operations on cohomology to strict Lie algebra structures on forms.

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