nLab
LieAlg

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

Definition

The category LieAlgLie Alg is that whose objects are Lie algebras (𝔤,[,] 𝔤)(\mathfrak{g}, [-,-]_{\mathfrak{g}}) and whose morphisms are Lie algebra homomorphisms, that is linear maps ϕ:𝔤𝔥\phi\colon \mathfrak{g} \to \mathfrak{h} such that for all x,y𝔤x,y \in \mathfrak{g} we have

ϕ([x,y] 𝔤)=[ϕ(x),ϕ(y)] 𝔥. \phi( [x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_\mathfrak{h} \,.

If Lie algebras are expressed in terms of their Chevalley–Eilenberg algebras (and if restricted to finite-dimensional Lie algebras), this may equivalently be characterized as follows:

LieAlgLie Alg is the full subcategory of the opposite category of the category dgAlg of dg-algebras on those dg-algebras whose underlying graded algebra is a Grassmann algebra, i.e. of the form 𝔤\wedge^\bullet \mathfrak{g}.

Special objects

category: category

Revised on October 24, 2012 02:18:45 by Toby Bartels (64.89.53.111)