Lie Theory

1. Lie Group

A continuous group that is also a manifold.¹
- e.g. \(O(n),\ SO(n),\ U(n),\ SU(n)\)
The composition and inversion are smooth for some parametrization.
It is locally Euclidean.

Lie group can be reduced to Lie algebra through logarithmic map, in which it is easier to do calculations.

Lie algebra is the vector field toward the transformation direction on the vector space on which the transformation is applied.²

Lie algebra is the tangent space of a Lie group at the identity.³^,⁴
Given two curves \(A(t)\), \(B(t)\) on Lie group \(G\) such that \(A(0)=B(0)=1\), \(A'(0)=a\in T_eG\) and \(B'(0)=a\in T_eG\). And one find \((A(t)B(t))'(0)=a+b\in T_eG\).

Let us calculate group multiplication of Lie group, within Lie algebra.
Its properties comes from the properties of group multiplication.
It is kind of a second derivative.
It is defined to satisfy the Jacobi identity:
- \[ [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y]] = 0 \]
We want to define some kind of multiplication on \(T_eG\)
- but \(ab=A'(0)B'(0)=\partial_t\partial_s(A(t)B(s))|_{t=0,s=0}\not\in T_eG\), because \(\partial_t (A(s)B(t))|_{t=0}\in T_{A(s)}G\).
- Define \(C_s(t)=A(s)B(t)A^{-1}(s)\) and take derivative, we have \(\partial_t C_s(t)|_{t=0}=A(s)bA^{-1}(s)\in T_eG\) and \(\partial_s\partial_t C_s(t)|_{t=0,s=0}=ab-ba \in T_eG\). We make it a definition: \([a,b]\equiv ab-ba\).
- For an element of Lie algebra \(A\), \(gAg^{-1} = e^{tB}Ae^{-tB}\) is also in the Lie algebra, which is the vector field \(A\) but conjugated along a different axis. The derivative evaluated at \(t=0\) is \([A,B]=AB-BA\).⁵
- Notable example of this is the \([\log\mathbf{k}, \log\mathbf{j}] = \log\mathbf{i}\). Rotating \(\log\mathbf{k}\) about \(\mathbf{j}\) axis, the rate of change at each point is \(\log\mathbf{i}\).
Conjugation diagram⁶ excalidraw:./conjugation.excalidraw.svg

Distribution Rule
- \[ [X,[Y,Z]] = [[X, Y], Z] + [Y, [X, Z]] \]
  - This is the generalization of the adjoint operation.⁷
- \[ [[X,Y],Z] = [X, [Y, Z]] - [Y, [X, Z]] \]

Four infinite families \(A_n,\ B_n,\ C_n,\ D_n\)
Five exceptional Lie algebras \(E_6, E_7, E_8, F_4, G_2\).
Theorem: There is no more.
\(n\) is the rank which is the number of nodes in the Dynkin diagram, and also the number of simple roots.
Heavy use of matrix calculus

Some of the Lie algebra can be represented in a finite way, which in turn form a simple set of vectors that represents the structure of the Lie algebra.

\[ \textcolor{RoyalBlue}{H} \equiv \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} \quad \textcolor{OrangeRed}{E} \equiv \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix} \quad \textcolor{ForestGreen}{F} \equiv \begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix} \]
acting on the vector space with five basis vectors:
- excalidraw:./sl2.excalidraw
Notice that they satisfies: \[ [H, E] = 2E,\quad [H, F] = -2F,\quad [E, F] = H. \]

\[ H_1, H_2, \textcolor{OrangeRed}{E_1}, \textcolor{ForestGreen}{E_2}, F_1, F_2 \]
Lie algebra can also act on themselves
- excalidraw:./sl3.excalidraw

The vectors that corresponds to the basis elements and their Lie brackets within a Lie algebra.
The Lie bracket corresponds to the vector addition of roots.

\(i\quad j\)	\(d_i\)	\(a_{ij}\)	\(\langle \alpha_i, \alpha_j\rangle = d_ia_{ij}\)
\(i = j\)	---	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.

The value of \(Z\) that solves \(e^Xe^Y = e^Z\) with \(X, Y\) in the Lie algebra.
It is not necessarily convergent.
Baker, Campbell, Hausdorff only stated the qualitative form of the formula, that it can be written in terms of possibly infinitely nested commutators yielding another element in the Lie algebra if it is convergent.

First explicit formula
\[ \log(\exp X\exp Y) = \sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sum_{\begin{smallmatrix}r_1+s_1 >0\\[-.5em] \vdots\\[.3em] r_n+s_n >0\end{smallmatrix}} \frac{[X^{r_1}Y^{s_1}X^{r_2}Y^{s_2}\cdots X^{r_n}Y^{s_n}]}{\left(\sum_{j=1}^n(r_j+s_j)\right)\cdot \prod_{i=1}^nr_i!s_i!} \]
- where the bracket is the notation for the Lie bracket nested on the right iterated by the exponent, and \(r_i, s_i\) are assumed to be nonnegative, with the understanding that \([X] := X\).
The first few terms are well-known to be:
- \[ Z(X,Y) = X+Y + \frac{1}{2}[X,Y] + \frac{1}{12}([X,[X,Y]] + [Y,[Y,X]]) -\frac{1}{24}[Y,[X,[X,Y]]] -\cdots \]