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The Heisenberg group

LieGroups.HeisenbergGroup — Type
HeisenbergGroup{T} <: AbstractDecoratorManifold{ℝ}

Heisenberg group HeisenbergGroup(n) is the group of $(n+2)×(n+2)$ matrices [BP08]

\[\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}\]

where $I_n$ is the $n×n$ unit matrix, $\mathbf{a}$ is a row vector of length $n$, $\mathbf{b}$ is a column vector of length $n$ and $c$ is a real number. The group operation is matrix multiplication.

The left-invariant metric on the manifold is used.

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Base.exp — Method
exp(G::HeisenbergGroup, g, X)

Exponential map on the HeisenbergGroup G with the left-invariant metric. The expression reads

\[\exp_{\begin{bmatrix} 1 & \mathbf{a}_p & c_p \\ \mathbf{0} & I_n & \mathbf{b}_p \\ 0 & \mathbf{0} & 1 \end{bmatrix}}\left(\begin{bmatrix} 0 & \mathbf{a}_X & c_X \\ \mathbf{0} & 0_n & \mathbf{b}_X \\ 0 & \mathbf{0} & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & \mathbf{a}_p + \mathbf{a}_X & c_p + c_X + \mathbf{a}_X⋅\mathbf{b}_X/2 + \mathbf{a}_p⋅\mathbf{b}_X \\ \mathbf{0} & I_n & \mathbf{b}_p + \mathbf{b}_X \\ 0 & \mathbf{0} & 1 \end{bmatrix}\]

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Base.exp — Method
exp(G::HeisenbergGroup, ::Identity{MatrixMultiplicationGroupOperation}, X)

Lie group exponential for the HeisenbergGroup G of the vector X. The formula reads

\[\exp\left(\begin{bmatrix} 0 & \mathbf{a} & c \\ \mathbf{0} & 0_n & \mathbf{b} \\ 0 & \mathbf{0} & 0 \end{bmatrix}\right) = \begin{bmatrix} 1 & \mathbf{a} & c + \mathbf{a}⋅\mathbf{b}/2 \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}\]

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Base.log — Method
log(G::HeisenbergGroup, g, h)

Compute the logarithmic map on the HeisenbergGroup group. The formula reads

\[\log_{\begin{bmatrix} 1 & \mathbf{a}_p & c_p \\ \mathbf{0} & I_n & \mathbf{b}_p \\ 0 & \mathbf{0} & 1 \end{bmatrix}}\left(\begin{bmatrix} 1 & \mathbf{a}_q & c_q \\ \mathbf{0} & I_n & \mathbf{b}_q \\ 0 & \mathbf{0} & 1 \end{bmatrix}\right) = \begin{bmatrix} 0 & \mathbf{a}_q - \mathbf{a}_p & c_q - c_p + \mathbf{a}_p⋅\mathbf{b}_p - \mathbf{a}_q⋅\mathbf{b}_q - (\mathbf{a}_q - \mathbf{a}_p)⋅(\mathbf{b}_q - \mathbf{b}_p) / 2 \\ \mathbf{0} & 0_n & \mathbf{b}_q - \mathbf{b}_p \\ 0 & \mathbf{0} & 0 \end{bmatrix}\]

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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Base.log — Method
log(G::HeisenbergGroup, ::Identity{MatrixMultiplicationGroupOperation}, g)

Lie group logarithm for the HeisenbergGroup G of the point g. The formula reads

\[\log\left(\begin{bmatrix} 1 & \mathbf{a} & c \\ \mathbf{0} & I_n & \mathbf{b} \\ 0 & \mathbf{0} & 1 \end{bmatrix}\right) = \begin{bmatrix} 0 & \mathbf{a} & c - \mathbf{a}⋅\mathbf{b}/2 \\ \mathbf{0} & 0_n & \mathbf{b} \\ 0 & \mathbf{0} & 0 \end{bmatrix}\]

where $I_n$ is the $n×n$ identity matrix, $0_n$ is the $n×n$ zero matrix and $\mathbf{a}⋅\mathbf{b}$ is dot product of vectors.

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