nLab curved L-infinity algebra

Idea

A curved $L_\infty$-algebra (e.g. Markl 11, p. 100) is just like an ordinary L-∞ algebra, but possibly including also a 0-ary bracket, i.e. a constant. Conversely, an ordinary L-∞ algebra is a curved $L_\infty$-algebra for which the 0-ary operation happens to be zero.

Accordingly, a “strong homotopy homomorphism” of curved $L_\infty$-algebras is defined just as for ordinary $L_\infty$-algebras, but allowing also for a 0-ary component. Notice that such “curved sh-maps” may be non-trivial even between ordinary $L_\infty$-algebras (amplified e.g. in Mehta & Zambon 12, below (2)).

The dual Chevalley-Eilenberg algebras automatically capture curved $L_\infty$-algebras unless one imposes a constraint: the non-curved $L_\infty$-algebras correspond to the augmented CE-algebras. Similarly in the dg-coalgebra description the restriction to non-curved $L_\infty$-algebras requires co-augmentation or else (this is what is commonly used) non-unital dg-coalgebras.

Definition

Where an ordinary L-infinity algebra is a $\mathbb{Z}$-graded vector space $\mathfrak{g}$ equipped for all $n \in \mathbb{N}$, $n \geq 1$ with $n$-ary brackets:

$$l_n \;\colon\; \mathfrak{g}^{\otimes^n} \longrightarrow \mathfrak{g}$$

out of the tensor product of $n$-copies of $\mathfrak{g}$, subject to some conditions, for a curved $L_\infty$-algebra also a component

$$l_0 \;\colon\; \mathfrak{g}^{\otimes^0} \simeq \mathbb{R} \to \mathfrak{g}$$

is allowed. Since an $\mathbb{R}$-linear map out of $\mathbb{R}$ is uniquely fixed by a single element (the image of $1 \in \mathbb{R}$), this is “a constant”, called the curvature of the curved $L_\infty$-algebra.

Now the strong homotopy Jacobi identity

(1)$$\sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} \sum_{\sigma \in UnShuff(i,j)} \chi(\sigma,v_1, \cdots, v_{n}) (-1)^{i(j-1)} l_{j} \left( l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right), v_{\sigma(i+1)} , \cdots , v_{\sigma(n)} \right) = 0 \,,$$

implies in particular that

$$l_1 \circ l_1 = \pm l_2(l_0, -)$$

hence that the unary operation $l_1$ no longer necessarily squares to zero (no longer defines a chain complex $(\mathfrak{g}, l_1)$) but to the binary bracket with the curvature.

Related concepts

• curved A-infinity algebra

• curved dg-algebra

References

• Martin Markl, Deformation Theory of Algebras and Their Diagrams, Regional Conference Series in Mathematics Number 116, American Mathematical Society (2011)

• Andrey Lazarev, Travis Schedler, Curved infinity-algebras and their characteristic classes, J Topology (2012) 5 (3): 503-528 (arXiv:1009.6203)

• Rajan Mehta, Marco Zambon: $L_\infty$-Actions, Differential Geometry and its Applications 30 (2012) 576-587 [arXiv:1202.2607, doi:10.1016/j.difgeo.2012.07.006]

On Lie integration of curved $L_\infty$-algebras:

• Victor Roca i Lucio: The integration theory of curved absolute $\mathcal{L}_\infty$ algebras [arXiv:2209.10282]