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Without a complete filtration an infinite Maurer–Cartan sum may be undefined

Statement refuted

The unfiltered direct-sum L∞ data alone always makes $\sum_{n \geq 1} q_{n} (w, \dots, w) / n!$ an element of the underlying graded space for every degree-zero $w$ .

Facts & Assumptions

Given: A characteristic-zero field $k$ and the shifted L∞ convention of Shifted coalgebra definition of an L∞-algebra.

[F1]

Shifted coalgebra definition of an L∞-algebra: an L∞-structure is a degree-one coderivation with square zero on the algebraic direct sum $S^{c} (W)$ .

[F2]

Square-zero coderivations are equivalent to the shifted higher Jacobi identities: it suffices to verify the shifted unshuffle Jacobi identities to verify $Q^{2} = 0$ .

[F3]

Koszul sign rule for graded tensor products and permutations: the unshuffle signs are determined by the graded degrees; all nonzero inputs below have degree zero.

[F4]

Field, The characteristic of a ring: the least $n \geq 1$ with $n \cdot 1_{R} = 0$ when one exists, and $0$ otherwise, and Every field is a commutative ring with $1 \neq 0$ ; it is an integral domain, and it is a commutative division ring: the field has characteristic zero, so each $n!$ is nonzero and invertible.

Counterexample

Take $V^{1} = k t$ , $V^{2} = ⨁_{j \geq 1} k e_{j}$ , and $V^{j} = 0$ otherwise, so $W = V [1]$ has $W^{0} = k t$ and $W^{1} = ⨁_{j \geq 1} k e_{j}$ . For each $n \geq 1$ define a graded-symmetric map $q_{n} : S^{n} (W) \to W$ by

$q_{n} (t^{⊙ n}) = n! e_{n},$

and set it to zero on every symmetric basis tensor containing any $e_{j}$ . This defines an L∞-structure on the algebraic direct sum $S^{c} (W)$ .

1.1givenF1F3construct

Each $q_{n}$ has degree $+ 1$ , since its only nonzero input has degree zero and its output $e_{n}$ has degree one. It is graded-symmetric because $t$ has degree zero and all tensors containing an $e_{j}$ map to zero.

2.1F1F2step 1.1

Every composition $q_{m} (q_{n} (-), -)$ vanishes: a nonzero inner value is a multiple of $e_{n}$ , and every outer operation is zero when any input contains an $e_{j}$ . Thus all Jacobi sums in [F2] vanish, and the specified operations do define an L∞-algebra.

3.1F4step 1.1algebra∎

For $w = t$ , the $n$ th Maurer–Cartan summand is $q_{n} (t^{⊙ n}) / n! = e_{n}$ , so the proposed sum is $\sum_{n \geq 1} e_{n}$ . Elements of $W^{1} = ⨁_{j \geq 1} k e_{j}$ have finite support, whereas the partial sums have nonzero support in every index from $1$ through $n$ and never stabilize. Therefore $\sum_{n \geq 1} e_{n}$ is not an element of $W$ ; without a complete filtration or other completion, this Maurer–Cartan expression is undefined.

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Sources

Jean-Louis Loday and Bruno Vallette, Algebraic Operads, §13.2 (standard reference, not scraped)
Tom Lada and Jim Stasheff, Introduction to sh Lie algebras for physicists, sh Lie Structures (standard reference, not scraped)