Lie groups have symmetries. We can consider only the infintesimal symmetries.
For example the unit circle has many symmetries, but we can consider only those which rotate infintesimally.
Take a continous group, such as \(U(1)\). Its Lie algebra is all matrices such that their exponential is in the Lie group.
\(\mathfrak{u}(1)=\{X\in \mathbb {C}^{1\times 1}|e^{tX}\in U(1) \forall t\in \mathbb{R}\}\)
This is satisfied by the matrices where \(M=-M^*\). Note that this means the diagonals are all \(0\).
Because of \(t\).
Consider two members of the Lie algebra: \(A\) and \(B\). The commutator is:
\(A\).
The corresponding Lie group member is:
\(e^{t(A+B)}=e^{tA}e^{tB}\)
While the Lie group multiplication may not commute, the corresponding addition of the Lie algebra does.
We can define the Lie bracket from the ring commutator.
We use the Lie bracket, rather than multiplication, as the operator over a field homomorphism.
\([A, B]\)
This generates another element in the algebra.
This satisifies:
Bilinearity: \([xA+yB,C]=x[A,C]+y[B,C]\)
Alternativity: \([A,A]=0\)
Jacobi identity: \([A, [B,C]]+[C,[A,B]]+[B,[C,A]]=0\)
Anticommutivity: \([A, B]=-[B,A]\)
One option for the Lie bracket is the ring commutor. So that:
\([A,B]=AB-BA\)
We can measure commutation of Lie groups using:
\(ABA^{-1}B^{-1}\)
If the group commutes then:
\(ABA^{-1}B^{-1}=BA^{-1}B^{-1}=I\)
This corresponds to \([A,B]=AB-BA\) in the underlying lie algebra, if we expand.
\(A=e^{ta}\)
\(B=e^{tb}\)
\(ABA^{-1}B^{-1}=e\)
The Lie algebra of \(O(n)\) is defined as:
\(\mathfrak{o}(n)=\{X\in \mathbb {R}^{n\times n}|e^{tX}\in O(n) \forall t\in \mathbb{R}\}\)
This is satisfied by the skew-symmetric matrices where \(M=-M^T\). Note that this means the diagonals are all \(0\).
The Lie algebra of \(U(n)\) is defined as:
\(\mathfrak{u}(n)=\{X\in \mathbb {C}^{n\times n}|e^{tX}\in U(n) \forall t\in \mathbb{R}\}\)
This is satisfied by the skew-Hermitian matrices where \(M=-M^*\). Note that this means the diagonals are all \(0\) or pure imaginary.
The Lie algebra of \(SO(n)\) is defined as:
\(\mathfrak{so}(n)=\{X\in \mathbb {R}^{n\times n}|e^{tX}\in SO(n) \forall t\in \mathbb{R}\}\)
This is satisfied by the skew-symmetric matrices where \(M=-M^T\). Note that this means the diagonals are all \(0\).
The Lie algebra of \(SU(n)\) is defined as:
\(\mathfrak{su}(n)=\{X\in \mathbb {C}^{n\times n}|e^{tX}\in SU(n) \forall t\in \mathbb{R}\}\)
This is satisfied by the skew-Hermitian matrices where \(M=-M^*\) and the trace is \(0\). Note that this means the diagonals are all \(0\) or pure imaginary.