Matrix Lie group

A matrix Lie group is a subgroup of a general linear group that is also a Lie group for its matrix topology. Near the identity, the matrix exponential and matrix logarithm give inverse coordinate charts whenever they are restricted to sufficiently small neighborhoods in the group and its tangent space.

The indefinite special unitary group preserves a Hermitian form of signature $(p,q)$ and has determinant one: $M^{†}JM=J$, where $J=\mathrm{diag}(I_p,-I_q)$. With both $p,q$ positive it is noncompact, unlike the group preserving a positive-definite form. Its Lie algebra satisfies $X^{†}J+JX=0$ and $\mathrm{Tr}X=0$.

Matrices $\left(\begin{array}{cc}\alpha & \beta \\ {\beta }^{\ast }& {\alpha }^{\ast }\end{array}\right)$ with the displayed constraint form a real three-dimensional noncompact Lie group. A real Lie algebra basis is $m_1={\sigma }_x/2$, $m_2={\sigma }_y/2$, $m_3=i{\sigma }_z/2$, with $[m_1,m_2]=m_3$, $[m_2,m_3]=-m_1$, and $[m_3,m_1]=-m_2$.

Pair creation and annihilation in two independent oscillator modes obey $[K_{-},K_{+}]=2K_3$ and $[K_3,K_{\pm }]=\pm K_{\pm }$. The Hermitian components $K_1=(K_{+}+K_{-})/2$ and $K_2=(K_{+}-K_{-})/(2i)$ satisfy $[K_1,K_2]=-i{K}_3$. Multiplying all three Hermitian components by $i$ gives the real Lie bracket signs of SU(1,1). Both occupations change equally, so their difference is conserved. This realizes the algebra used in two-mode squeezed states.

The identities $K_{+}{K}_{-}=N_a{N}_b$ and $K_{-}{K}_{+}=(N_a+1)(N_b+1)$ give the displayed quadratic Casimir operator. It commutes with all generators because the number difference does. A fixed number-difference sector therefore has a fixed Casimir value even though pair creation changes its total occupation.

For a real quadratic form $Q(v)=v^{T}Av$ with $A$ a symmetric matrix, its stabilizer is $G_Q=\{g\in {\mathrm{GL}}_n(\mathbb{R}):g^{T}Ag=A\}$. Its Lie algebra consists exactly of $X$ satisfying $X^{T}A+AX=0$. Differentiation proves necessity; the matrix exponential proves sufficiency because $e^{t{X}^T}A{e}^{tX}=A$. The result holds even when the quadratic form is degenerate.

For a Matrix Lie group $G$ with Lie algebra $\mathfrak{g}$, its exponential map is the restriction of the matrix exponential, $X\mapsto e^X$. It maps $\mathfrak{g}$ into $G$ and is a local diffeomorphism at zero, but it need not be surjective globally.

A real matrix with at least one simple negative eigenvalue cannot be a matrix exponential of a real matrix: a logarithm would commute with it and preserve each one-dimensional real negative-eigenvalue space, where exponentiation can only produce a positive eigenvalue. In particular $\mathrm{diag}(-2,-1/2)$ belongs to the connected special linear group ${\mathrm{SL}}_2(\mathbb{R})$ but has no real logarithm.

For a matrix Lie group $G$ with Lie algebra $\mathfrak{g}=T_{I}G$, choose neighborhoods on which $\exp :\mathfrak{g}\to G$ and $\log :G\to \mathfrak{g}$ are inverse. Left translation gives a chart near $g\in G$ byThe Baker--Campbell--Hausdorff formula expresses multiplication smoothly in these coordinates.