Introduction
Lie groups and Lie algebras form the backbone of modern physics and mathematics, particularly in areas such as quantum mechanics, particle physics, and differential geometry. A Lie group is a continuous group with a smooth structure, where elements can be parameterized by continuous variables. These groups encapsulate symmetries of physical systems and are often expressed in terms of their corresponding Lie algebras, which provide a linearized description of the group.
This article outlines the key properties of groups and Lie groups, focusing on their algebraic structure. We explore concepts such as generators, structure constants, roots, weights, and representations, with explicit examples from \( \text{SU}(2) \) and \( \text{SU}(3) \). \( \text{SU}(2) \) is foundational in describing spin and angular momentum in quantum mechanics, while \( \text{SU}(3) \) is central to the theory of strong interactions in quantum chromodynamics.
Properties of a Group
A group is a set of elements \( G \) that satisfy the following properties:
- Closure under multiplication: For every \( A, B \in G \), \( C = AB \in G \).
- Associativity: For every \( A, B, C \in G \), \( A(BC) = (AB)C \).
- Existence of an identity element: There exists an \( I \in G \) such that for all \( A \in G \), \( IA = AI = A \).
- Existence of an inverse element: For every \( A \in G \), there exists \( A^{-1} \in G \) such that \( AA^{-1} = I \).
Continuous Groups and Lie Algebras
For a continuous group (Lie group), elements are specified by parameters \( p_{i} = (p_{1}, p_{2}, \dots, p_{n}) \) and generators \( \Sigma_{i} = (\Sigma_{1}, \Sigma_{2}, \dots, \Sigma_{n}) \), where \( n \) is the number of group generators. Any group element can be expressed as:
\[
R = e^{i p_{i} \Sigma_{i}}.
\]
The commutator of two group generators can be written as a linear combination of the generators:
\[
[\Sigma_{i}, \Sigma_{j}] = \sum_{k=1}^{n} f_{ij}^{k} \Sigma_{k},
\]
where the constants \( f_{ij}^{k} \) are the structure constants of the group. This forms the Lie algebra of the group.
The adjoint representation has a dimension equal to the number of group generators. The characters (traces of matrices) simplify determining group representations. The rank of a group corresponds to the number of independent quantum numbers. For example, \( \text{SU}(2) \) has rank 1, and \( \text{SU}(3) \) has rank 2.
Casimir Operators and Cartan Subalgebra
The Casimir operator commutes with all the generators of the algebra, e.g., \( [J^{2}, J_{i}] = 0 \), and has the same eigenvectors as the generators. The generators that commute with each other, \( H_{i} \), form the Cartan subalgebra, while the remaining generators, \( E_{j} \), are ladder operators. For ladder operators:
\[
[H_{i}, E_{j}] = r^{i} E_{j},
\]
where \( r^{i} \) are the roots. The number of fundamental roots equals the rank of the group.
The weights are the eigenvalues of the Cartan subalgebra operators. Any weight is a linear combination of fundamental weights, whose number equals the number of fundamental roots. The maximum weight is associated with the group representation. For \( \text{SU}(2) \), the weights correspond to eigenvalues of \( S_{z} \), which are \( \frac{n}{2} \) for integer \( n \).
The fundamental weights can be determined using the relation:
\[
2\frac{\mu_{i} \cdot \alpha_{j}}{\alpha_{j}^{2}} = \delta_{ij},
\]
where \( \alpha_{j} \) are the fundamental roots, \( \mu_{i} \) are the fundamental weights, and \( i, j = 1, \dots, N \), with \( N \) being the number of Cartan subalgebra generators.
\( \text{SU}(2) \): Roots and Weights for the 2-Dimensional Representation
The Cartan subalgebra generator is:
\[
H_{1} = \frac{S_{z}}{2}.
\]
The ladder operators are:
\[
E_{\pm} = \frac{S_{x} \pm iS_{y}}{2}.
\]
This yields:
\[
[H_{1}, E_{\pm}] = \pm E_{\pm}.
\]
The roots are \( r_{1} = 1 \) and \( r_{2} = -1 \). The fundamental root is \( \alpha = 1 \), and the fundamental weight is:
\[
2\frac{\lambda \cdot \alpha}{\alpha^{2}} = 1 \quad \Rightarrow \quad \lambda = \frac{1}{2}.
\]
\( \text{SU}(3) \): Roots and Weights for the 3-Dimensional Representation
The Cartan subalgebra generators are:
\[
H_{1} = \frac{\lambda_{3}}{2}, \quad H_{2} = \frac{\lambda_{8}}{2}.
\]
The ladder operators are:
\[
E_{1\pm} = \frac{\lambda_{1} \pm i\lambda_{2}}{2}, \quad
E_{2\pm} = \frac{\lambda_{4} \pm i\lambda_{5}}{2}, \quad
E_{3\pm} = \frac{\lambda_{6} \pm i\lambda_{7}}{2}.
\]
Commutation relations yield:
\[
[H_{1}, E_{1\pm}] = \pm E_{1\pm}, \quad [H_{2}, E_{1\pm}] = 0,
\]
\[
[H_{1}, E_{2\pm}] = \pm \frac{1}{2} E_{2\pm}, \quad [H_{2}, E_{2\pm}] = \pm \frac{\sqrt{3}}{2} E_{2\pm},
\]
\[
[H_{1}, E_{3\pm}] = \mp \frac{1}{2} E_{3\pm}, \quad [H_{2}, E_{3\pm}] = \pm \frac{\sqrt{3}}{2} E_{3\pm}.
\]
The roots of \( \text{SU}(3) \) are:
\[
r_{1} = (1, 0), \quad r_{2} = (-1, 0), \quad r_{3} = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right),
\]
\[
r_{4} = \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right), \quad
r_{5} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right), \quad
r_{6} = \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right).
\]
The fundamental roots are:
\[
\alpha_{1} = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right), \quad
\alpha_{2} = \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right).
\]
The fundamental weights are derived as follows:
For \( \lambda_{1} \):
\[
2\frac{\lambda_{1} \cdot \alpha_{1}}{\alpha_{1}^{2}} = 1 \quad \Rightarrow \quad \lambda_{1x} + \lambda_{1y} \sqrt{3} = 1,
\]
\[
2\frac{\lambda_{1} \cdot \alpha_{2}}{\alpha_{2}^{2}} = 0 \quad \Rightarrow \quad \lambda_{1x} – \lambda_{1y} \sqrt{3} = 0,
\]
\[
\Rightarrow \lambda_{1} = \left(\frac{1}{2}, \frac{1}{2\sqrt{3}}\right).
\]
For \( \lambda_{2} \):
\[
2\frac{\lambda_{2} \cdot \alpha_{1}}{\alpha_{1}^{2}} = 0 \quad \Rightarrow \quad \lambda_{2x} + \lambda_{2y} \sqrt{3} = 0,
\]
\[
2\frac{\lambda_{2} \cdot \alpha_{2}}{\alpha_{2}^{2}} = 1 \quad \Rightarrow \quad \lambda_{2x} – \lambda_{2y} \sqrt{3} = 1,
\]
\[
\Rightarrow \lambda_{2} = \left(\frac{1}{2}, -\frac{1}{2\sqrt{3}}\right).
\]
Conclusion
The study of Lie groups and Lie algebras, particularly \( \text{SU}(2) \) and \( \text{SU}(3) \), provides a fundamental framework for understanding symmetry in physics and mathematics. These concepts allow for elegant descriptions of complex systems, from quantum spin to particle interactions.