Physlib.Relativity.Tensors.RealTensor.Vector.Representation

For a given natural number $d$ representing spatial dimensions, the representation of the Lorentz group $\mathcal{L}$ on the space of Lorentz vectors $\text{Vector}_d$ is defined as the group homomorphism from $\mathcal{L}$ to the group of linear automorphisms of $\text{Vector}_d$. For any Lorentz transformation $\Lambda \in \mathcal{L}$ and vector $v \in \text{Vector}_d$, the action of the representation is given by the matrix-vector multiplication $\Lambda v$, utilizing the standard basis of $\text{Vector}_d$ indexed by $\text{Fin } 1 \oplus \text{Fin } d$.

For any spatial dimension $d \in \mathbb{N}$, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ and $v$ be a Lorentz vector in $\text{Vector}_d$. The action of the representation of the Lorentz group on $v$, denoted by $\text{rep} \Lambda v$, is equal to the matrix-vector multiplication $\Lambda v$.

For a given spatial dimension $d$, let $\Lambda$ be an element of the Lorentz group $\mathcal{L}$ and $v$ be a Lorentz vector in $\text{Vector}_d$. For any spacetime index $k \in \text{Fin } 1 \oplus \text{Fin } d$, the $k$-th component of the vector resulting from the representation $\rho$ of $\Lambda$ acting on $v$ is equal to the sum over all indices $j$ of the product of the matrix entries $\Lambda_{kj}$ and the vector components $v_j$: $$(\rho(\Lambda)v)_k = \sum_j \Lambda_{kj} v_j$$

For a given number of spatial dimensions $d$, let $\Lambda$ be a Lorentz transformation in the Lorentz group $\mathcal{L}$ and let $\{e_j\}_{j \in \text{Fin } 1 \oplus \text{Fin } d}$ be the standard basis for the space of Lorentz vectors $\text{Vector}_d$. The action of the representation of $\Lambda$ on the $\mu$-th basis vector $e_\mu$ is given by the linear combination $$\text{rep}(\Lambda)(e_\mu) = \sum_j \Lambda_{j\mu} e_j$$ where $\Lambda_{j\mu}$ denotes the $(j, \mu)$-th entry of the matrix representing $\Lambda$.

For a given natural number $d$ and any element $\Lambda$ of the Lorentz group $\mathcal{L}$, the matrix representation of the linear map $\text{rep}(\Lambda)$ with respect to the standard basis of Lorentz vectors is equal to the matrix $\Lambda$.

For any natural number $d$ and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the linear map $\rho(\Lambda): \text{Vector}_d \to \text{Vector}_d$ corresponding to the representation of the Lorentz group on the space of Lorentz vectors is injective.

For a given natural number $d$ and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$ for $d$ spatial dimensions, the linear map $\text{rep}(\Lambda)$ acting on the space of Lorentz vectors $\text{Vector}_d$ is surjective.

For a given natural number $d$ representing spatial dimensions and any Lorentz transformation $\Lambda$ in the Lorentz group $\mathcal{L}$, the linear map $\text{rep}(\Lambda) : \text{Vector}_d \to \text{Vector}_d$ defining the action of $\Lambda$ on the space of Lorentz vectors is a bijection.

For any spatial dimension $d \in \mathbb{N}$, any $n$, and any Lorentz transformation $\Lambda \in \mathcal{L}$, the representation map $\text{rep}(\Lambda) : \text{Vector}_d \to \text{Vector}_d$ is $n$-times continuously differentiable over the real numbers $\mathbb{R}$.

For any natural number $d$ representing spatial dimensions, the representation $\text{rep}$ of the Lorentz group $\mathcal{L}$ on the space of Lorentz vectors $\text{Vector}_d$ is injective.