nLab
line Lie n-algebra

Idea

A line Lie $n$-algebra over a ground field $k$ is the Lie n-algebra analog of the abelian (trivial) 1-dimensional Lie algebra on $k$.

Definition

Definition

For $n\in \mathbb{N},n\ge 1$ the line Lie $n$-algebra

$$b^{n-1}k\in L_{\mathrm{\infty }}\stackrel{\mathrm{CE}}{\hookrightarrow }{\mathrm{dgAlg}}^{\mathrm{op}}$$b^{n-1} k \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op}

is the L-∞ algebra whose Chevalley-Eilenberg algebra

$$\mathrm{CE}(b^{n-1}ℝ)=({\wedge }^{•}⟨c⟩,d=0)$$CE(b^{n-1} \mathbb{R}) = (\wedge^\bullet \langle c\rangle , d = 0)

is the free graded-commutative algebra on a single generator $C$ in degree $k$ equipped with the trivial differential

$$dc=0.$$d c = 0 \,.

Properties

• For $𝔤$ a Lie algebra a cocycle $\mu$ in degree $n$-Lie algebra cohomology on $𝔨$ is equivalently a morphism of L-∞ algebras

  $$\mu :𝔤\to b^{n-1}ℝ.$$\mu : \mathfrak{g} \to b^{n-1}\mathbb{R} \,.

  More generally, for $𝔤$ an L-∞ algebra, a degree-$n$ cocycle in ∞-Lie algebra cohomology is given by such a morphism.

• There is a unique (up to rescaling) indecomposable invariant polynomial on $b^{n-1}ℝ$, given by the shifted copy of the generator $c$ in the Weil algebra $W(b^{n-1}ℝ)$.

  Equivalently, we have

  $$\mathrm{inv}(b^{n-1}ℝ)=\mathrm{CE}(b^nℝ).$$inv(b^{n-1}\mathbb{R}) = CE(b^n \mathbb{R}) \,.

• The Lie integration (see there) of $b^{n-1}ℝ$ is the line Lie n-group $B^{n-1}ℝ$.