∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
and
rational homotopy theory (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)
Examples of Sullivan models in rational homotopy theory:
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.
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:
out of the tensor product of $n$-copies of $\mathfrak{g}$, subject to some conditions, for a curved $L_\infty$-algebra also a component
is allowed 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
implies in particular that
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.
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)
Last revised on September 20, 2022 at 11:04:29. See the history of this page for a list of all contributions to it.