Notes on the video about Lie algebra axioms

lie_algebra_axioms.txt

Those are the notes and TeX file for the video:

https://youtu.be/cCbn5x0z7pA 

# References

https://en.wikipedia.org/wiki/Lie_algebra#Definition_of_a_Lie_algebra

https://en.wikipedia.org/wiki/Jacobi_identity

https://en.wikipedia.org/wiki/Quasi-Lie_algebra

# TeX


${\mathbb V}$ ... ${\mathbb F}$-vector spaces

\section{Axioms}

$b: {\mathbb V}\times {\mathbb V}\to {\mathbb V}$, bilinear

$\bullet\ b(x, x) = 0$

$\bullet\ b(x, b(y, z)) + b(y, b(z, x)) + b(z, b(x, y)) = 0$

\subsection{Example}

$b(x,y) = x\cdot y - y\cdot x$ \hspace{1cm} where ${\mathbb V}$ are square matrices of fixed dimension.

$b(x, b(y, z)) = x\cdot y\cdot z - x\cdot z\cdot y - y\cdot z\cdot x + z\cdot y\cdot x$

\subsection{Immediate implications}

$b(y, x) = b(y, x) + b(x, x) + b(y, y) = b(x+y, x+y) - b(x, y) = - b(x,y)$

Note:

$b(y,x) = - b(x,y) \implies 2\cdot b(x,x) = 0$

$(2\neq 0)\land (b(y,x) = - b(x,y)) \implies b(x,x) = 0$

\subsubsection{Rewrite of the 2nd axiom using anti-symmetry}

Curry: ${\mathrm {ad}}_x:=y\mapsto b(x,y)$

Notation $D_x={\mathrm {ad}}_x$

Notation $x*y := b(x, y)$

Notation $[x,y] := b(x, y)$

$\bullet\ b(x, b(y, z)) + b(y, b(z, x)) + b(z, b(x, y)) = 0$

$\bullet\ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$

$\bullet\ b(x, b(y, z)) = b(b(x, y), z) + b(y, b(x, z))$

$\bullet\ {\mathrm {ad}}_x(y * z) = {\mathrm {ad}}_x(y) * z + y * {\mathrm {ad}}_x(z)$

$\bullet\ D_x(y * z) = D_x y* z + y * D_x z$