∞-Lie theory

# Contents

## 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

$d c = 0 \,.$

## 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 $\mathbf{B}^{n-1}\mathbb{R}$.

Revised on August 30, 2013 17:44:25 by Urs Schreiber (89.204.154.236)