Lie Theory

We want to define some kind of multiplication on the tangent space \(T_eG\) of Lie group \(G\) at the identity element \(e\).

For \(A, B \in T_{e}G\) and its Lie group elements \(a = e^{tA},b = e^{sB} \in G\), \( AB = \not\in T_eG,\) in general. Let me give a general reasoning for that: We know that \( aB\in T_{a}G \) using the straightforward differential of linear operation. Now if we try to take the derivative of \(B\) with respect to \(t\), a component outside of tanget space can arise as the tangent spaces can be in different orientation in higher dimensional space..

Meanwhile the adjoint action \(\mathrm{Ad}_b(A)\) always procedues a valid Lie algebra element: \[ \mathrm{Ad}_b(A) := bAb^{-1}\in T_eG \] because

\begin{align*} b &e^{t A} b^{-1} \Big|_{t=0} = e, \\ b &e^{t A}b^{-1} \in G, \\ \dv{t} &b e^{t A}b^{-1} \Big|_{t=0} = bAb^{-1} \in T_e G, \end{align*}

From the fact that Lie algebra is closed under addition and scalar multiplication, we can see that \(BA - AB\) is also in Lie algebra:

\begin{align*} \forall s, &e^{sB} A e^{-sB} \in T_eG, \\ \dv{s} &e^{sB} A e^{-sB} \Big|_{t=0} = BA - AB \in T_eG \end{align*}

This quantity represent the rate of change of of the vector field \(A\) in the representation in the direction given by \(B\) at each point.

We define it to be the Lie bracket. \[ [B,A] := BA - AB. \]

Distribution Rule \[ [X,[Y,Z]] = [[X, Y], Z] + [Y, [X, Z]] \] which is the generalization of the adjoint operation.² \[ [[X,Y],Z] = [X, [Y, Z]] - [Y, [X, Z]] \]

• \(\Delta_+\)
• Roots that are on the upper half, including the ones on the positive axes.

• The one that corresponds to the actual basis elements within a Lie algebra.

\(i\quad j\)	\(d_i\)	\(a_{ij}\)	\(\langle \alpha_i, \alpha_j\rangle = d_ia_{ij}\)
\(i = j\)	---	2	\(2d_i\)
\(\bullet \quad\bullet\)	---	0	0
\(\bullet—\bullet\)	\(d_i = d_j\)	-1	\(-d_i\)
\(\bullet\!\mathord{\Longrightarrow}\!\bullet\)	2	-1	-2
\(\bullet\!\mathord{\equiv\!\!>\!\!\!\!\!\!\equiv\!\equiv}\!\bullet\)	3	-1	-3
\(\bullet\!\mathord{\Longleftarrow}\!\bullet\)	1	-2	-2
\(\bullet\!\mathord{\equiv\!\!<\!\!\!\!\!\!\equiv\!\equiv}\!\bullet\)	1	-3	-3

• Dynkin diagram determines a root system
• Note \(\|\alpha_i\| = \sqrt{2d_i}\)
• \(a_{ij}\) is the entries of the Cartan matrix and can equivalently be defined: \[ a_{ji} = 2\frac{\langle \alpha_i, \alpha_j\rangle}{\langle \alpha_j, \alpha_j\rangle} \]

• \(\mathfrak{g}_2\)
• The Cross Product and the Exceptional G2 - YouTube
• The result of the 7-Dimensional cross product generally does not follow the transformation rule.
• But there's subgroup of the \(\mathfrak{so}(7)\), which contains 14 families of two-plane rotation that preserves the relationship. This is the G2.
• The existence is based on the multiplicative property of the octonion. G2 is the automorphism group of octonions.