After defining cohomologically higher-order BRST and anti-BRST operators for a compact simple algebra 𝒢, the associated higher-order Laplacians are introduced and the corresponding supersymmetry algebra Σ is analyzed. These operators act on the states generated by a set of fermionic ghost fields transforming under the adjoint representation. In contrast with the standard case, for which the Laplacian is given by the quadratic Casimir, the higher-order Laplacians W are not, in general, given completely in terms of the Casimir–Racah operators, and may involve the ghost number operator. The higher-order version of the Hodge decomposition is exhibited. The example of is worked out in detail, including the expression of its higher-order Laplacian W.
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