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5. The Gauge Algebra is an sh-Lie Algebra We now restrict our attention to the constant maps in Homk ( ∗ , ) and show that our algebraic structure on Homk ( ∗ , ) induces an sh-Lie structure on the graded space L = { , }. Throughout this section, we assume the BBvD hypothesis and that δˆ is injective, so Theorem 1 holds and consequently the bracket on Homk ( ∗ , ) defined by ˆ 1 , π2 ) ˆ 2 ) − π2 δ(π ˆ 1 ) + C(π [π1 , π2 ] := π1 δ(π satisfies the Jacobi identity. By definition, [δ(ξ ), δ(η)] = δ(ξ ) δ(η) − δ(η) δ(ξ ), while the definition of C gives ˆ [δ(ξ ), δ(η)] = δC(ξ, η) ∈ Hom( ∗ , ), so our commutator relation is δ(ξ ) δ(η) − δ(η) ˆ δ(ξ ) = δ(C(ξ, η)).

If V is any vector space, the fine bornology Fine(V) is the smallest admissible bornology: a subset is small iff it is contained in the disked hull of a finite number of points of V. In particular, any small subset is contained in a finite-dimensional subspace of V. A bornological space with fine bornology is always complete because finite-dimensional spaces are complete. 4. A useful way to construct a bornology on V is to start from a collection U of subsets not satisfying the axioms of a bornology, and then to consider the smallest bornology S(V) containing U.

Consequently, in conformity with our conventions in Sect. 7, we can identify δ(c) with the unique element of Hom(∧∗ , ) whose value at diag(∂φ), for φ ∈ 0 , gives δ(c)(φ) as defined by Ikeda. The usual Lie bracket of the vector fields δ(c1 ) and δ(c2 ) as defined by Eq. 8 corresponds to our Lie structure on Hom(∧∗ , ). Using his brackets, Ikeda finds that the ψ component of [δ(c1 ), δ(c2 )](φ) is given by [δ(c1 ), δ(c2 )](ψ) = δ(c3 (ψ))(ψ), where πA (c3 (ψ)) = ∂WBD (ψ)c1B c2D . ∂TA (9) We see that the Lie bracket of [δ(c1 ), δ(c2 )] is not of the form δ(c), where c is a gauge parameter independent of the fields φ but rather the gauge parameter c3 depends on c1 , c2 and on the field ψ.