nLab Lie algebra representation

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Idea

A Lie algebra representation is a Lie algebra homomorphism $\rho \;\colon\;\mathfrak{g} \to end(V)$ to the endomorphism Lie algebra of $V$ .

This means equivalently that

\array{ \mathfrak{g} \otimes V & \overset{\rho}{\longrightarrow} & V \\ (x,v) &\mapsto& \rho_x(v) }

is a bilinear map such that for all $x_i \in \mathfrak{g}$ and $v \in V$ we have the Lie action property:

(1)

\rho_{x_1}(\rho_{x_2}(v)) - \rho_{x_2}(\rho_{x_1}(v)) \;=\; \rho_{[x_1,x_2]}(v)

\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}

Here 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.