ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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A finite complete filtered L∞ Maurer–Cartan equation

Example

Let $V^{1} = k t$ , $V^{2} = k c$ , and $V^{j} = 0$ in other degrees, so the shifted space $W = V [1]$ has $W^{0} = k t$ and $W^{1} = k c$ . Set $q_{1} = 0$ , define the graded-symmetric map $q_{2} (t ⊙ t) = 2 c$ , and let $q_{n} = 0$ for $n \geq 3$ . Give $W$ the filtration $F^{1} W = W$ , $F^{2} W = k c$ , and $F^{r} W = 0$ for $r \geq 3$ . This is a complete filtered L∞-algebra, and its Maurer–Cartan set consists only of $0$ .

Facts & Assumptions

Given: A characteristic-zero field $k$ and the operations and filtration specified in the Example.

[F1]

Shifted coalgebra definition of an L∞-algebra: degree-one symmetric operations on $W = V [1]$ define an L∞-algebra exactly when their coderivation squares to zero.

[F2]

Square-zero coderivations are equivalent to the shifted higher Jacobi identities: the square-zero condition is equivalent to the displayed unshuffle Jacobi identities.

[F3]

Complete filtered L∞-algebra: completeness is the bijection to compatible quotient cosets, and each $q_{n}$ must raise filtration by the sum of input filtration indices.

[F4]

The Maurer–Cartan series converges in a complete filtered L∞-algebra: the Maurer–Cartan series for $w \in F^{1} W^{0}$ has a unique limit in $W^{1}$ .

[F5]

Maurer–Cartan element of a complete filtered L∞-algebra: a Maurer–Cartan element is a degree-zero element of $F^{1} W$ where this series is zero.

[F6]

Field: $k$ is a field, so $λ^{2} = 0$ implies $λ = 0$ .

Verification

1.1givenF1F2constructalgebra

The only nonzero operation is the degree-one map $q_{2}$ ; it is symmetric because $t$ has degree zero. Any nonzero nested operation would apply an outer $q_{2}$ to an inner output in $k c$ , but $q_{2} (c, t) = q_{2} (t, c) = q_{2} (c, c) = 0$ . Thus every higher Jacobi expression in [F2] vanishes. By [F1]–[F2] these operations define an L∞-algebra.

2.1F3step 1.1algebra

The filtration is separated since $F^{3} W = 0$ . The compatible quotient cosets are determined by their representative in $W / F^{3} W = W$ ; conversely each $w \in W$ gives the compatible family $(w + F^{r} W)_{r \geq 1}$ . Hence the natural map in [F3] is bijective. Moreover $q_{2} (F^{1} W, F^{1} W) \subseteq k c = F^{2} W$ and $q_{2} (F^{2} W, F^{1} W) = q_{2} (F^{1} W, F^{2} W) = 0 \subseteq F^{3} W$ ; all other required filtration inclusions hold because $q_{1} = 0$ and $q_{n} = 0$ for $n \geq 3$ . This is complete filtered L∞ data.

3.1F4F5F6step 2.1algebra∎

Every degree-zero element has form $w = λ t$ . Its Maurer–Cartan series from [F4] is $q_{2} (w, w) / 2 = λ^{2} c,$ because all other operations vanish. It is zero exactly when $λ = 0$ by [F6]. Therefore the Maurer–Cartan set is ${0}$ .

Depends on

Used by

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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)