∞-Lie theory

Contents

Idea

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

Definition

Ordinary definition

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

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

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

$\varphi \left(\left[x,y{\right]}_{𝔤}\right)=\left[\varphi \left(x\right),\varphi \left(y\right){\right]}_{𝔥}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(C,\otimes ,1\right)$ with the symmetry $\tau$, a Lie algebra in $\left(C,\otimes ,1,\tau \right)$ is an object $L$ in $C$ together with a morphism $\left[,\right]:A\otimes A\to A$ such that the Jacobi identity

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

and antisymmetry

$\left[,\right]+\left[,\right]\otimes {\tau }_{L,L}=0$[,]+[,]\otimes\tau_{L,L} = 0

hold. If $k$ is the ring $ℤ$ of integers, then we say (internal) Lie ring, and if $k$ is a field and $C=\mathrm{Vec}$ then we say a Lie $k$-algebra. Other interesting cases are super-Lie algebras, which are the Lie algebras in the symmetric monoidal category ${ℤ}_{2}-\mathrm{Vec}$ 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.

See

Extra stuff, structure, properties

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

• extra property

• extra structure

• extra stuff

Examples

Examples of sequences of infinitesimal and local structures

first order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$←$ differentiationintegration $\to$
derivativeTaylor seriesgermsmooth function
tangent vectorjetgerm of curvecurve
Lie algebraformal grouplocal Lie groupLie group
Poisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

For instance chapter IV in vol III of

Revised on February 6, 2013 18:41:31 by Urs Schreiber (82.113.106.234)