Lie Theory

\(\mathrm{SO}(3)\) has two elements in its fundamental group \(\pi\).

• Class I: which wraps around, with odd number of crossing
• Class II: which does not wrap, with even number of crossing

\(\mathrm{SU}(2)\) is, however, simply connected, and it double covers \(\mathrm{SO}(3)\). Spinor is the thing that is transformed by the \(\mathrm{SU}(2)\).

Lie bracket let us calculate the group multiplication of Lie group within Lie algebra, and the properties of Lie bracket match the properties of group multiplication.

Lie bracket is defined to satisfy the Jacobi identity: \[ [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y]] = 0. \]

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

\(\mathfrak{sl}(2)\) is three dimensional with the basis vectors satisfying:

\[ [H, E] = 2E,\quad [H, F] = -2F,\quad [E, F] = H. \]

Two real dimensional representation is \[ \textcolor{RoyalBlue}{H} \equiv \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} \quad \textcolor{OrangeRed}{E} \equiv \begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix} \quad \textcolor{ForestGreen}{F} \equiv \begin{pmatrix} 0 & 0 \\ 1 & 0\end{pmatrix}. \]

The representation of sl(2) on five dimensional vector space acting on the vector space with five basis vectors:

In matrix form, \[ H= \begin{pmatrix} -4&0&0&0&0\\ 0&-2&0&0&0\\ 0&0&0&0&0\\ 0&0&0&2&0\\ 0&0&0&0&4 \end{pmatrix},\quad E= \begin{pmatrix} 0&0&0&0&0\\ 1&0&0&0&0\\ 0&2&0&0&0\\ 0&0&3&0&0\\ 0&0&0&4&0 \end{pmatrix},\quad F= \begin{pmatrix} 0&4&0&0&0\\ 0&0&3&0&0\\ 0&0&0&2&0\\ 0&0&0&0&1\\ 0&0&0&0&0 \end{pmatrix}. \]

Since sl(2,C) is the complexification of sl(2,R), the representations of sl(2,R) can be used for sl(2,C) with complex coefficients.

\(\mathfrak{sl}(2,\mathbb{C})\) can also be represented by the direct sum of two representations of su(2) related by complex conjugation, due to the peculiar relation:

where the subscript \(\mathbb{C}\) means the complexification, and the inclusion \(\iota\) is such that the representations of left and right chiral spinor are related by complex conjugation.

In physics, each representation is labeled by the spins of the two su(2) representations. e.g. (1/2, 1/2).

\(\mathfrak{sl}(3, \mathbb{R})\) is a eight dimension with basis vectors \[ H_1, H_2, \textcolor{OrangeRed}{E_1}, \textcolor{ForestGreen}{E_2}, F_1, F_2, [F_1, F_2], [E_1, E_2] \] and whose adjoint representation looks like:

where arrows mean adjoint action defined to be \[ \mathrm{ad}_X\colon \mathfrak{g} \to \mathfrak{g} \colon Y \mapsto [X,Y]. \]

For any complex dimension \(d = 2j + 1\), the three su(2) generators in the standard basis are:

\begin{aligned} T_1 &=-\frac{i}{2} \begin{bmatrix} 0&c_1&&&\\ c_1&0&c_2&&\\ &c_2&0&\ddots&\\ &&\ddots&\ddots&c_{d-1}\\ &&&c_{d-1}&0 \end{bmatrix}, \\ T_2 &=\frac12 \begin{bmatrix} 0&-c_1&&&\\ c_1&0&-c_2&&\\ &c_2&0&\ddots&\\ &&\ddots&\ddots&-c_{d-1}\\ &&&c_{d-1}&0 \end{bmatrix}, \\ T_3 &=-\frac{i}{2}\mqty[\dmat{d-1,d-3,\ldots,1-d}] \end{aligned}

where \(c_k=\sqrt{k(d-k)}\).

They satisfy: \[ [T_a,T_b]=\epsilon_{abc}T_c,\qquad T_a^\dagger=-T_a. \]

The Hermitian angular-momentum matrices are directly related to this: \(J_a=iT_a\).

For example, d=1 \[ T_1=T_2=T_3=0. \]

d=2 (or spin-1/2): \[ T_1=-\frac{i}{2} \begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad T_2=\frac12 \begin{pmatrix}0&-1\\1&0\end{pmatrix},\qquad T_3=-\frac{i}{2} \begin{pmatrix}1&0\\0&-1\end{pmatrix}, \] or equivalently using Pauli matrices, \[ T_a = - \frac{i}{2}\sigma_a. \]

d=3 (or spin-1): \[ T_1=-\frac{i}{\sqrt2} \begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix},\qquad T_2=\frac1{\sqrt2} \begin{pmatrix}0&-1&0\\1&0&-1\\0&1&0\end{pmatrix},\qquad T_3=-i \begin{pmatrix}1&0&0\\0&0&0\\0&0&-1\end{pmatrix}. \]

SU(2) has unique irreducible representation for each complex dimension derived from the representations of su(2).

SU(3) has two distinct irreducible representation on three complex dimension related by complex conjugation: \(\mathbf{3}\) and \(\bar{\mathbf{3}}\).

Dimension	Irrep	Label
1	trivial (singlet) \(\mathbf{1}\)	(0,0)
2	none	—
3	\(\mathbf{3}\)	(1,0)
	\(\bar{\mathbf{3}}\)	(0,1)
4	none	—
5	none	—
6	\(\mathbf{6}\)	(2,0)
	\(\bar{\mathbf{6}}\)	(0,2)
8	\(\mathbf{8}\)	(1,1)
10	\(\mathbf{10}\)	(3,0)
	\(\bar{\mathbf{10}}\)	(0,3)
15	\(\mathbf{15}\)	(2,1)
	\(\bar{\mathbf{15}}\)	(1,2)
	\(\mathbf{15}'\)	(4,0)
	\(\bar{\mathbf{15}}'\)	(0,4)
27	\(\mathbf{27}\)	(2,2)

A Lie algebra \(\mathfrak{g}\) can be decomposed into \[ \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_{\alpha} \] where \(\mathfrak{h}\subset \mathfrak{g}\) is Cartan subalgebra, and \(\mathfrak{g}_{\alpha}\) is defined to be \[ \mathfrak{g}_\alpha := \{X\in \mathfrak{g} \colon \forall H \in \mathfrak{h}, [H,X] = \alpha(H)X\}. \] with nonzero linear functionals \(\alpha\colon \mathfrak{h} \to \mathbb{C}\) known as the roots.

For semisimple Lie algebra the vectors corresponds to the adjoint representations of Lie algebra elements, since \([\mathfrak{g}_{\alpha}, \mathfrak{g}_{\beta}] = \mathfrak{g}_{\alpha + \beta}\).

• \(\Delta_+\)

Roots that are on the upper half, including the ones on the positive half of the axes.

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

Dynkin diagram corresponds to a root system

Finite Dynkin Diagrams

There are only following finite dimensional complex simple Lie algebras:

• Four infinite families \(A_n,\ B_n,\ C_n,\ D_n\)
• Five exceptional Lie algebras \(E_6, E_7, E_8, F_4, G_2\).

\(n\) is the rank which is the number of nodes in the Dynkin diagram, and also the number of simple roots.

• \(A_n\) is \(\mathfrak{sl}(n+1)\)
• \(B_n\) is \(\mathfrak{so}(2n+1)\)
• \(C_n\) is \(\mathfrak{sp}(2n)\) symplectic Lie algebra
• \(D_n\) is \(\mathfrak{so}(2n)\)

For low dimension, several importanct isomorphism occurs:

\begin{align*} \mathfrak{su}(2) &\cong \mathfrak{so}(3) \\ \mathfrak{su}(2) \oplus \mathfrak{su}(2) &\cong \mathfrak{so}(4) \\ \mathfrak{su}(4) &\cong \mathfrak{so}(6). \end{align*}

One Dynkin type can have multiple real forms. For example \(A_2 = \mathfrak{sl}(3,\mathbb{C})\) has three: \[ \mathfrak{su}(3), \mathfrak{su}(2,1), \mathfrak{sl}(3,\mathbb{R}). \] In particular, complexification of \(\mathfrak{su}(n)\) always has following isomorphism \[ \mathfrak{su}(n)_{\mathbb{C}} \cong \mathfrak{sl}(n, \mathbb{C}). \]

The canonical complexification of Lie algebra \(\mathfrak{g}\) is: \[ \mathfrak{g}_{\mathbb{C}} := \mathfrak{g}\otimes_{\mathbb{R}} \mathbb{C} = \mathfrak{g} \oplus i\mathfrak{g}. \]

Universal complexification of a Lie group \(G\) is a complex Lie group \(G_{\mathbb{C}}\), such that for arbitrary continuous homomorphism \(f\) into a complex Lie group \(H\) there exists \(F\) such that the following diagram commutes

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: \[ Z(X,Y) = X+Y + \frac{1}{2}[X,Y] + \frac{1}{12}([X,[X,Y]] + [Y,[Y,X]]) -\frac{1}{24}[Y,[X,[X,Y]]] -\cdots \]