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

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

is the L-∞ algebra whose Chevalley-Eilenberg algebra

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

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

$$d c = 0 \,.$$

Example

For $n = 2$ then the line Lie 2-algebra is the Lie 2-algebra that comes from the differential crossed module $(\mathbb{R} \to 0)$.

Properties

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

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

  More generally, for $\mathfrak{g}$ 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 $b^{n-1} \mathbb{R}$, given by the shifted copy of the generator $c$ in the Weil algebra $W(b^{n-1}\mathbb{R})$.

  Equivalently, we have

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

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

Related concepts

• translation Lie algebra

• super translation Lie algebra