Contents

# 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.

### In a general linear category

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

Given a commutative unital ring $k$, and a (strict for simplicity) symmetric monoidal $k$-linear category $(C,\otimes,1)$ with the symmetry $\tau$, a Lie algebra in $(C,\otimes,1,\tau)$ is an object $L$ in $C$ together with a morphism $[-,-]: A\otimes A\to A$ such that the Jacobi identity

$\left[-,\left[-,-\right]\right]+\left[-,\left[-,-\right]\right]\circ(id_L\otimes\tau_{L,L})\circ(\tau\otimes id_L)+\left[-,\left[-,-\right]\right]\circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0$

and antisymmetry

$[-,-]+[-,-]\circ \tau_{L,L} = 0$

hold. If $k$ is the ring $\mathbb{Z}$ of integers, then we say (internal) Lie ring, and if $k$ is a field and $C=Vec$ then we say a Lie $k$-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category $\mathbb{Z}_2-Vec$ of supervector spaces and the Lie algebras in the Loday-Pirashvili tensor category of linear maps.

Alternatively, Lie algebras are the algebras over certain quadratic operad, called the Lie operad, which is the Koszul dual of the commutative algebra operad.

### 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
• A. L. Onishchik (ed.) Lie Groups and Lie Algebras

• I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

• II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

• Eckhard Meinrenken, Lie groups and Lie algebas, Lecture notes 2010 (pdf)

Discussion with a view towards Chern-Weil theory is in chapter IV in vol III of

Discussion in synthetic differential geometry is in

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