nLab LieAlg

Redirected from "LieAlgebras".
Contents

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

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

Last revised on September 1, 2024 at 14:17:59. See the history of this page for a list of all contributions to it.