# nLab Lie algebra

Contents

### Context

#### Lie theory

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

## Idea

A Lie algebra is the infinitesimal approximation to a Lie group.

## Definition

### Ordinary definition

A Lie algebra is a vector space $\mathfrak{g}$ equipped with a bilinear skew-symmetric map $[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}$ which satisfies the Jacobi identity:

$\forall x,y,z \in \mathfrak{g} : \left[x,\left[y,z\right]\right] + \left[z,\left[x,y\right]\right] + \left[y,\left[z,x\right]\right] = 0 \,.$

A homomorphism of Lie algebras is a linear map $\phi : \mathfrak{g} \to \mathfrak{h}$ such that for all $x,y \in \mathfrak{g}$ we have

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

This defines the category LieAlg of Lie algebras.

### Internal to a general linear category

The notion of Lie algebra may be formulated internal to any linear category. This general definition subsumes variants of Lie algebras such as super Lie algebras.

Consider a commutative unital ring $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(\mathcal{C},\otimes,1)$ with braiding $\tau$.

A Lie algebra object in $(\mathcal{C},\otimes,1,\tau)$ is

1. an object

$L \in \mathcal{C}$
2. morphism (the Lie bracket)

$[-,-] \;\colon\; L \otimes L \to L$

such that the following conditions hold:

1. $\left[-,\left[-,-\right]\right] + \left[-,\left[-,-\right]\right] \circ(id_L\otimes\tau_{L,L}) \circ(\tau\otimes id_L) + \left[-,\left[-,-\right]\right] \circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0$
2. skew-symmetry:

\begin{aligned} & \phantom{+} [-,-] \\ & + [-,-]\circ \tau_{L,L} \\ & = \phantom{+} 0 \end{aligned}

Equivalently, Lie algebra objects are the algebras over an operad over a certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.

Examples of types of Lie algebra objects:

If $k$ is the ring $\mathbb{Z}$ of integers and $\mathcal{C} =$ $k$Mod = Ab is the category of abelian groups equipped with the tensor product of abelian groups, then a Lie algebra object is called a Lie ring.

If $k$ is a field and $\mathcal{C} =$ Vect is the category of vector spaces over $k$ equipped with the tensor product of vector spaces then a Lie algebra object is an ordinary_Lie k-algebra.

If $k$ is a field and $\mathcal{C}$ = sVect is the category of super vector spaces over $k$, then a Lie algebra object is a super Lie algebra.

### General abstract perspective

Lie algebras are equivalently groups in “infinitesimal geometry”.

For instance in synthetic differential geometry then a Lie algebra of a Lie group is just the first-order infinitesimal neighbourhood of the unit element (e.g. Kock 09, section 6).

More generally in geometric homotopy theory, Lie algebras, being 0-truncated L-∞ algebras are equivalently “infinitesimal ∞-group geometric ∞-stacks” (e.g. here-topos#FormalModuliProblems)), also called formal moduli problems (see there for more).

## Extra stuff, structure, properties

Notions of Lie algebras with extra stuff, structure, property includes

• extra property

• extra structure

• extra stuff

See

## Examples

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

Textbook accounts:

Discussion with a view towards Chern-Weil theory:

Discussion in synthetic differential geometry is in

• Anders Kock, section 6 of Synthetic Geometry of Manifolds, 2009 (pdf)