nLab curved L-infinity algebra



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Rational homotopy theory



A curved L 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 L_\infty-algebra for which the 0-ary operation happens to be zero.

Accordingly, a “strong homotopy homomorphism” of curved L L_\infty-algebras is defined just as for ordinary L 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 L_\infty-algebras (amplified e.g. in Mehta-Zambon 12, below (2)).

The dual Chevalley-Eilenberg algebras automatically capture curved L L_\infty-algebras unless one imposes a constraint: the non-curved L L_\infty-algebras correspond to the augmented CE-algebras. Similarly in the dg-coalgebra description the restriction to non-curved L 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 𝔤\mathfrak{g} equipped for all nn \in \mathbb{N}, n1n \geq 1 with nn-ary brackets:

l n:𝔤 n𝔤 l_n \;\colon\; \mathfrak{g}^{\otimes^n} \longrightarrow \mathfrak{g}

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

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

is allowed allowed. Since an \mathbb{R}-linear map out of \mathbb{R} is uniquely fixed by a single element (the image of 11 \in \mathbb{R}), this is “a constant”, called the curvature of the curved L L_\infty-algebra.

Now the strong homotopy Jacobi identity

(1) i,ji+j=n+1 σUnShuff(i,j)χ(σ,v 1,,v n)(1) i(j1)l j(l i(v σ(1),,v σ(i)),v σ(i+1),,v σ(n))=0, \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 1l 1=±l 2(l 0,) l_1 \circ l_1 = \pm l_2(l_0, -)

hence that the unary operation l 1l_1 no longer necessarily squares to zero (no longer defines a chain complex (𝔤,l 1)(\mathfrak{g}, l_1)) but to the binary bracket with the curvature.


Last revised on September 20, 2022 at 11:04:29. See the history of this page for a list of all contributions to it.