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Filtered L∞-morphisms preserve Maurer–Cartan elements and compose on solutions

Statement

Let $f : W \to Z$ be a filtered L∞-morphism between complete filtered algebras over a characteristic-zero field. For every $w \in F^{1} W^{0}$ , the Taylor series $f_{*} (w) = \sum_{n \geq 1} f_{n} (w^{n}) / n!$ converges in $F^{1} Z^{0}$ , and its curvature satisfies

$c_{r} (f_{*} (w)) = \sum_{n \geq 0} \frac{1}{n!} f_{n + 1} (w^{n}, c_{q} (w)), c_{q} (w) = \sum_{n \geq 1} \frac{q_{n} (w^{n})}{n!} .$

Consequently $f_{*}$ defines an actual continuous map $MC (W) \to MC (Z)$ , with Maurer–Cartan elements as in Maurer–Cartan element of a complete filtered L∞-algebra. If $g : Z \to T$ is another filtered L∞-morphism, the composite $h = g \circ f$ has Taylor coefficients

$h_{n} (x_{1}, \dots, x_{n}) = \sum_{s = 1}^{n} \frac{1}{s!} \sum_{(B_{1}, \dots, B_{s})} ϵ (B_{1}, \dots, B_{s}; x) g_{s} (f_{∣ B_{1} ∣} (x_{B_{1}}), \dots, f_{∣ B_{s} ∣} (x_{B_{s}})),$

where blocks and signs are as in Filtered L∞-morphism. The composite is filtered, and

$(g \circ f)_{*} (w) = g_{*} (f_{*} (w)) .$

Identity morphisms act as identities. These are equalities of convergent elements, not merely formal Taylor expressions; no quasi-isomorphism hypothesis is needed.

Facts & Assumptions

Given: Complete filtered shifted L∞-algebras over one characteristic-zero field and the stated filtered morphisms.

[F1]

Filtered Taylor maps extend to the completed coaugmented symmetric coalgebra supplies the completed coaugmented coalgebras, exponentials, curvature and marked-input identities, and continuous Taylor extensions.

[F2]

Maurer–Cartan element of a complete filtered L∞-algebra defines $MC (W)$ by $w \in F^{1} W^{0}$ and $c_{q} (w) = 0$ .

Proof

1.1F1algebra

The series defining $u = f_{*} (w)$ converges in $F^{1} Z^{0}$ by [F1]. Since $F Q = R F$ , its exponential satisfies $c_{r} (u) e^{u} = R (e^{u}) = R F (e^{w}) = F Q (e^{w}) = F (c_{q} (w) e^{w})$ . The marked-input identity of [F1] rewrites the last expression as $(\sum_{n \geq 0} f_{n + 1} (w^{n}, c_{q} (w)) / n!) e^{u}$ . Multiply by $e^{- u}$ to obtain the curvature formula. This series converges since its $n$ th term has filtration at least $n + 1$ .

1.2F1algebra

The continuous coalgebra map $G F$ intertwines coderivations because $G F Q = G R F = S G F$ , where $S$ is the coderivation on $T$ . Its length-one projection on $S^{n} W$ applies $g_{s}$ to the length- $s$ part of the partition formula for $F$ , giving exactly the stated formula for $h_{n}$ . Every term has degree zero and additive filtration, because first the $f$ blocks and then $g_{s}$ add the input filtration indices. Thus $h$ is a filtered morphism.

2.1F1F2step 1.1algebra

If $c_{q} (w) = 0$ , every term on the right vanishes, so $c_{r} (f_{*} (w)) = 0$ and [F2] gives the asserted map on Maurer–Cartan elements. For $w - v \in F^{p} W$ with $w, v \in F^{1} W^{0}$ , the multilinear telescoping expansion of $f_{n} (w^{n}) - f_{n} (v^{n})$ has in every term one input in $F^{p} W$ and all other inputs in $F^{1} W$ , so it lies in $F^{p + n - 1} Z$ . Hence $f_{*} (w) - f_{*} (v) \in F^{p} Z$ . This proves continuity in the filtration topology.

2.2F1step 1.2algebra

For repeated degree-zero $w$ , the coefficient formula becomes $h_{n} (w^{n}) = \sum_{s = 1}^{n} (1 / s!) \sum_{i_{1} + \dots + i_{s} = n, i_{j} \geq 1} (n! / (i_{1}! \dots i_{s}!)) g_{s} (f_{i_{1}} (w^{i_{1}}), \dots, f_{i_{s}} (w^{i_{s}}))$ . After dividing by $n!$ and summing over $n$ , the result is $\sum_{s \geq 1} (1 / s!) g_{s} ((\sum_{i \geq 1} f_{i} (w^{i}) / i!)^{s}) = g_{*} (f_{*} (w))$ . The regrouping is finite modulo $F^{N}$ , since each input has filtration at least one and all terms with input count at least $N$ vanish there. Equivalently $G F (e^{w}) = G (e^{f_{*} (w)}) = e^{g_{*} (f_{*} (w))}$ , and projection to $T$ gives the same result.

3.1step 1.1step 2.1step 1.2step 2.2algebra∎

The identity Taylor family has only its linear term nonzero, so its Taylor pushforward is $w$ itself. Composition is associative because composition of the continuous coalgebra maps is associative; it can also be read from refinement of the finite set partitions modulo each filtration quotient. Together with the preceding calculations this proves all the claims on actual solution sets.

Depends on

Used by

Filtered L∞-morphisms preserve completed polynomial gauge equivalence Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Vasily Dolgushev and Christopher Rogers, A Version of the Goldman–Millson Theorem for Filtered L-infinity-Algebras, §§1–2 (standard reference, not scraped)
Ezra Getzler, Lie theory for nilpotent L-infinity-algebras, §4 (standard reference, not scraped)