nLab
line Lie n-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

Definition

Definition

For n,n1n \in \mathbb{N}, n \geq 1 the line Lie nn-algebra

b n1kL CEdgAlg op b^{n-1} k \in L_\infty \stackrel{CE}{\hookrightarrow} dgAlg^{op}

is the L-∞ algebra whose Chevalley-Eilenberg algebra

CE(b n1)=( 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 CC in degree kk equipped with the trivial differential

dc=0. d c = 0 \,.

Properties

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

    μ:𝔤b n1. \mu : \mathfrak{g} \to b^{n-1}\mathbb{R} \,.

    More generally, for 𝔤\mathfrak{g} an L-∞ algebra, a degree-nn cocycle in ∞-Lie algebra cohomology is given by such a morphism.

  • There is a unique (up to rescaling) indecomposable invariant polynomial on b n1b^{n-1} \mathbb{R}, given by the shifted copy of the generator cc in the Weil algebra W(b n1)W(b^{n-1}\mathbb{R}).

    Equivalently, we have

    inv(b n1)=CE(b n). inv(b^{n-1}\mathbb{R}) = CE(b^n \mathbb{R}) \,.
  • The Lie integration (see there) of b n1b^{n-1}\mathbb{R} is the line Lie n-group B n1\mathbf{B}^{n-1}\mathbb{R}.

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