177 Lie Algebra
A set of vectors with a mapping that combines two vectors into a third vector that is used to calculate the change of a vector value.
definition [d] (Lie Algebra) From Hall Definition 16.10: a Lie algebra over a field \(F\) is a vector space \(\mathfrak{g}\) over \(F\), together with a bracket map \([\cdot,\cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}\) having the following properties:
- \([\cdot,\cdot]\) is bilinear.
- \([Y,X] = -[X,Y]\) for all \(X,Y \in \mathfrak{g}\).
- \([X,X] = 0\) for all \(X \in \mathfrak{g}\).
- For all \(X,Y,Z \in \mathfrak{g}\) the Jacobi identity holds:
- \([X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0\) .
where
- \(\mathfrak{g}\) is the Lie algebra.
- \(F\) is the field of scalars.
- \([\cdot,\cdot]\) is the Lie bracket.
definition [d] (Lie Algebra) From Hassani Definition 29.2.1: a finite-dimensional vector space \(V\) over \(\mathbb{R}\) or \(\mathbb{C}\) is called a Lie algebra over \(\mathbb{R}\) or \(\mathbb{C}\) if there is a binary operation, called Lie multiplication, \([\cdot,\cdot]: V \times V \rightarrow V\) on \(V\), satisfying
- \([X,Y] = -[Y,X]\) for all \(X,Y \in V\) (antisymmetry).
- \([\alpha X + \beta Y, Z] = \alpha[X,Z] + \beta[Y,Z]\) for \(\alpha,\beta\) scalars (linearity).
- \([X,[Y,Z]] + [Z,[X,Y]] + [Y,[Z,X]] = 0\) (Jacobi identity).
where
- \(V\) is the Lie algebra.
- \([\cdot,\cdot]\) is Lie multiplication.
177.1 Elementary Example
177.1.1 Simple
A Lie algebra is a vector space with a bracket \([\ ,\ ]\). On three elements the bracket closes in the set.
\[ \mathfrak{g} = \{ X,\ Y,\ Z \} \]
\[ [X,Y] = Z,\quad [Y,Z] = X,\quad [Z,X] = Y,\quad [X,X] = 0 \]
where
- \(\mathfrak{g}\) is the Lie algebra.
- \([X,Y]\) is the Lie bracket of \(X\) and \(Y\).
177.1.2 General
Matrix commutators give a concrete bracket: \([A,B] = AB - BA\) on \(2 \times 2\) matrices.
\[ A = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\quad B = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \]
\[ [A,B] = AB - BA = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]
where
- \(A, B\) are matrices in the Lie algebra.
- \(AB - BA\) is the commutator bracket.
177.2 References
- Hall, B. C. Quantum Theory for Mathematicians. Springer. — Definition 16.10 (Lie algebra).
- Hassani, S. Mathematical Physics, 2nd ed. Springer. — Definition 29.2.1 (Lie algebra).
- Alternating Function
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