nLab
String Lie 2-algebra

Idea

The string Lie 2-algebra is the infinitesimal approximation to the string 2-group.

In the strict sense of the word, the String Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ is a shifted ∞-Lie algebra central extension

$$0\to b𝔲(1)\to \mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)\to \mathrm{𝔰𝔬}(n)\to 0$$0 \to \mathbf{b} \mathfrak{u}(1) \to \mathfrak{string}(n) \to \mathfrak{so}(n) \to 0

of the Lie algebra $\mathrm{𝔰𝔬}(n)$ by the Lie 2-algebra $b𝔲(1)$.

This central extension is controled by the canonical (up to normalization) Lie algebra 3-cocycle $\mu$ on $\mathrm{𝔰𝔬}(n)$.

When $\mu$ is normalized such that it represents the image in deRham cohomology of the generator of the integral cohomology $H^3(X,\mathrm{Spin}(n))$ then the corresponding String Lie 2-algebra is the Lie 2-algebra of the String Lie 2-group.

More generally, for $𝔤$ an ∞-Lie algebra and $\mu \in \mathrm{CE}(𝔤)$ an $\mathrm{\infty }$-Lie algebra cocycle (a closed element in the Chevalley–Eilenberg algebra of $𝔤$) of degree $k$, there is a corresponding shifted central extension

$$0\to b^{k-2}𝔲(1)\to {𝔤}_{\mu }\to 𝔤\to 0.$$0 \to \mathbf{b}^{k-2} \mathfrak{u}(1) \to \mathfrak{g}_\mu \to \mathfrak{g} \to 0 \,.

For instance the supergravity Lie 3-algebra is such an extension of the super Poincare Lie algebra by a super Lie algebra 4-cocycle.

Definition

As with any L-∞ algebra, we may define the String Lie 2-algebra $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ equivalently in terms of its Chevalley–Eilenberg algebra.

Write $𝔤:=\mathrm{𝔰𝔬}(n)$ in the following. The Chevalley–Eilenberg algebra $\mathrm{CE}(𝔤)$ of $𝔤$ has a degree 3 element

$$\mu =⟨-,[-,-]⟩,$$\mu = \langle -, [-,-]\rangle \,,

well defined up to normalization ($⟨-,⟩$ is the canonical bilinear symmetric invariant form on $𝔤$ and $[-,-]$ the Lie bracket), which is closed

$$d_{𝔤}\mu =0.$$d_{\mathfrak{g}} \mu = 0 \,.

Hence this is the canonical (up to normalization) 3-cocycle in the Lie algebra cohomology of $𝔤$.

The Chevalley–Eilenberg algebra of $\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n)$ is

$$\mathrm{CE}(\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}(n))=({\wedge }^{•}{𝔤}^{*}\oplus ⟨b⟩,d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}),$$CE(\mathfrak{string}(n)) = (\wedge^\bullet \mathfrak{g}^* \oplus \langle b\rangle, d_{\mathfrak{string}}) \,,

where

• $b$ is a single new generator in degree 2;

• the differental $d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}$ coincides with $d_{𝔤}$ on ${𝔤}^{*}$:

  $$d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}{\mid }_{{𝔤}^{*}}=d_{𝔤}$$d_{\mathfrak{string}} |_{\mathfrak{g}^*} = d_{\mathfrak{g}}

• on the new generator it is defined by

  $$d_{\mathrm{𝔰𝔱𝔯𝔦𝔫𝔤}}:b\mapsto \mu .$$d_{\mathfrak{string}} : b \mapsto \mu \,.

That the differential defined this way is indeed of degree +1 and squares to 0 is precisely the fact that $\mu$ is a degree 3-cocycle of $𝔤$.

References

In one incarnation or other the String Lie 2-algebra has been considered in literature of dg-algebras, but its Lie theoretic interpretation as a Lie 2-algebra has been made fully explicit only in

• John Baez, Alissa Crans?, Lie 2-algebras .

After its relation to the String Lie 2-group under Lie integration was established by in

• André Henriques, Integrating $L_{\mathrm{\infty }}$-algebras

and

• Crans, Baez, Schreiber, Stevenson, From loop groups to 2-groups,

it was reconsidered in a wider $\mathrm{\infty }$-Lie theoretic context in

• Sati, Schreiber, Stasheff, $L_{\mathrm{\infty }}$-algebra connections

where also the relation to the supergravity Lie 3-algebra and other structures is discussed.