nLab line Lie-n algebra



\infty-Lie theory

∞-Lie theory (higher geometry)


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\infty-Lie groupoids

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\infty-Lie algebroids

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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.



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 \,.

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


  • 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}.

Last revised on July 30, 2018 at 16:05:08. See the history of this page for a list of all contributions to it.