nLab
Lie algebra representation

Contents

Context

Representation theory

Lie theory

∞-Lie theory (higher geometry)

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

Idea

The representation/action of a Lie algebra 𝔤\mathfrak{g} on a vector space VV.

Definition

A Lie algebra homomorphism 𝔤end(V)\mathfrak{g} \to end(V) to the endomorphism Lie algebra of VV.

Properties

In terms of string diagrams / Jacobi diagrams

In string diagram-notation for Lie algebra objects internal to tensor categories, the Lie action property looks as follows:

ρ(f(x,y),z)=ρ(y,ρ(x,z))ρ(x,ρ(y,z)) f=[,]Liebracket ρ([x,y],z)=ρ(y,ρ(x,z))ρ(x,ρ(y,z))Lieactionproperty ρ=[,]adjointaction [[x,y],z]=[y,[x,z]]+[x,[y,z]]Jacobiidentity \begin{aligned} \Leftrightarrow & \;\;\;\;\; \rho(f(x,y),z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z)) \\ \underset{ {f = [-,-]} \atop {Lie\;bracket} }{ \Leftrightarrow } & \;\;\;\;\; \underset{ {Lie\;action\;property} }{ \underbrace{ \rho([x,y],z) \;=\; \rho(y,\rho(x,z)) - \rho(x,\rho(y,z)) } } \\ \underset{ {\rho = -[-,-]} \atop {adjoint\;action} }{ \Leftrightarrow } & \;\;\;\;\; \underset{ {Jacobi\;identity} }{ \underbrace{ [[x,y],z] \;=\; - [y,[x,z]] + [x,[y,z]] } } \end{aligned}

where the last line shows the equivalence to the Jacobi identity on the Lie algebra object itself in the case that the Lie action is the adjoint action.

In the language of Jacobi diagrams this is called the STU-relation. and is the reason behind the existence of Lie algebra weight systems in knot theory.

References

See also

Last revised on November 30, 2019 at 12:52:09. See the history of this page for a list of all contributions to it.