Physlib.Relativity.Tensors.RealTensor.Vector.Pre.Modules

Let $\text{ContrMod } d$ be the module of contravariant real Lorentz vectors in $1+d$ dimensions. For any two vectors $\psi, \psi' \in \text{ContrMod } d$, if their underlying component values are equal (i.e., $\psi.\text{val} = \psi'.\text{val}$), then the vectors themselves are equal ($\psi = \psi'$).

This definition establishes a canonical equivalence (isomorphism) between the module of contravariant real Lorentz vectors $\text{ContrMod } d$ and the space of functions mapping the index set $\text{Fin } 1 \oplus \text{Fin } d$ to the real numbers $\mathbb{R}$. Effectively, it identifies a contravariant vector with its $(1+d)$ real components $(v^0, v^1, \dots, v^d)$, where the index set is decomposed into one temporal dimension and $d$ spatial dimensions.

The space of contravariant real Lorentz vectors $\text{ContrMod } d$ is equipped with an additive commutative monoid structure. This structure (defining vector addition and the zero vector) is induced by transporting the standard component-wise addition from the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ via the canonical equivalence $\text{ContrMod } d \cong \mathbb{R}^{1+d}$.

The space of contravariant real Lorentz vectors $\text{ContrMod } d$ is equipped with an additive commutative group structure. This structure is defined by transporting the standard component-wise addition, negation, and zero element from the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ via the canonical equivalence $\text{ContrMod } d \cong \mathbb{R}^{1+d}$.

The space of contravariant real Lorentz vectors $\text{ContrMod } d$ is equipped with a module structure over the field of real numbers $\mathbb{R}$. This structure is defined by transporting the standard component-wise scalar multiplication and addition from the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ via the canonical equivalence $\text{ContrMod } d \cong \mathbb{R}^{1+d}$, where the index set represents one temporal and $d$ spatial dimensions.

For any two contravariant real Lorentz vectors $\psi$ and $\psi'$ in the space $\text{ContrMod } d$, the underlying vector representation (the value) of their sum is equal to the sum of their individual underlying vector representations. That is, $$(\psi + \psi').\text{val} = \psi.\text{val} + \psi'.\text{val},$$ where the addition on the left is the addition in $\text{ContrMod } d$ and the addition on the right is the component-wise addition in the underlying function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$.

For any real number $r$ and any contravariant real Lorentz vector $\psi$ in the space $\text{ContrMod } d$, the underlying vector representation of the scalar multiplication $r \cdot \psi$ is equal to the scalar multiplication of $r$ and the underlying vector representation of $\psi$. That is, $$(r \cdot \psi).\text{val} = r \cdot \psi.\text{val},$$ where the scalar multiplication on the left is defined within the $\mathbb{R}$-module structure of $\text{ContrMod } d$, and the scalar multiplication on the right is the component-wise multiplication in the underlying function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$.

This is the linear equivalence (isomorphism of $\mathbb{R}$-modules) between the space of contravariant real Lorentz vectors $\text{ContrMod } d$ and the coordinate space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$. It identifies a contravariant vector with its $(1+d)$ real components $(v^0, v^1, \dots, v^d)$, where the index set is partitioned into one temporal and $d$ spatial dimensions, preserving both vector addition and scalar multiplication over $\mathbb{R}$.

For a contravariant real Lorentz vector $\psi \in \text{ContrMod } d$, this function returns its representation as a coordinate vector in the space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$. It maps the abstract vector $\psi$ to its $1+d$ real components $(v^0, v^1, \dots, v^d)$ using the linear isomorphism \text{toFin1d\mathbb{R}Equiv}.

For a contravariant real Lorentz vector $\psi$ in the module $\text{ContrMod } d$, its representation as a coordinate vector in $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$, denoted by $\psi.\text{toFin1d}\mathbb{R}$, is equal to its underlying value $\psi.\text{val}$.

The standard basis for the space of contravariant real Lorentz vectors $\text{ContrMod } d$, indexed by the set $\text{Fin } 1 \oplus \text{Fin } d$ (representing one temporal and $d$ spatial dimensions). This basis is defined by pulling back the canonical basis of the coordinate space $\mathbb{R}^{1+d}$ via the linear isomorphism \text{toFin1d\mathbb{R}Equiv}, such that the $\mu$-th basis vector corresponds to the tuple $(v^0, v^1, \dots, v^d)$ where $v^\nu = \delta_\mu^\nu$.

For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, let $e_\mu$ denote the $\mu$-th vector of the standard basis for the space of contravariant real Lorentz vectors $\text{ContrMod } d$. Under the canonical linear isomorphism $\Phi: \text{ContrMod } d \cong \mathbb{R}^{1+d}$ (represented by `toFin1dℝEquiv`), the $\mu$-th component of $e_\mu$ is equal to $1$.

For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, let $e_\mu$ denote the $\mu$-th vector of the standard basis for the space of contravariant real Lorentz vectors $\text{ContrMod } d$. The $\mu$-th component of $e_\mu$ is equal to $1$.

In the space of contravariant real Lorentz vectors $\text{ContrMod } d$, let $e_\mu$ be the $\mu$-th vector of the standard basis (denoted by `stdBasis μ`), where $\mu \in \text{Fin } 1 \oplus \text{Fin } d$ represents the spacetime indices (one temporal and $d$ spatial). Let $\phi: \text{ContrMod } d \to ((\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R})$ be the linear isomorphism `toFin1dℝEquiv` that maps a Lorentz vector to its component representation. For any indices $\mu$ and $\nu$ such that $\mu \neq \nu$, the $\nu$-th component of the representation of $e_\mu$ is $0$.

In the space of contravariant real Lorentz vectors $\text{ContrMod } d$, let $e_0$ be the standard basis vector corresponding to the temporal dimension (indexed by $\text{Sum.inl } 0$ in the set $\text{Fin } 1 \oplus \text{Fin } d$). For any spatial index $i \in \text{Fin } d$, the component of $e_0$ at the index $\text{Sum.inr } i$ is $0$.

In the space of contravariant real Lorentz vectors $\text{ContrMod } d$, the standard basis vectors $\{e_\mu\}$ are indexed by $\mu \in \text{Fin } 1 \oplus \text{Fin } d$ (representing one temporal and $d$ spatial dimensions). For any two indices $\mu$ and $\nu$, the $\nu$-th component of the $\mu$-th basis vector is given by the Kronecker delta: $$(e_\mu)_\nu = \begin{cases} 1 & \text{if } \mu = \nu \\ 0 & \text{if } \mu \neq \nu \end{cases}$$

For any contravariant real Lorentz vector $v \in \text{ContrMod } d$, let $\{e_i\}_{i \in \text{Fin } 1 \oplus \text{Fin } d}$ be the standard basis vectors and $v^i$ be the $i$-th coordinate component of $v$ (represented by `toFin1dℝ`). The vector $v$ can be decomposed as the sum of its components multiplied by the corresponding basis vectors: $$v = \sum_i v^i e_i$$

Given a square matrix $M$ of size $(1+d) \times (1+d)$ over the real numbers $\mathbb{R}$ and a contravariant Lorentz vector $v$ in the $(1+d)$-dimensional real vector space $\text{ContrMod } d$, this function computes the resulting vector in $\text{ContrMod } d$. The operation is defined by treating $M$ as a linear operator acting on $v$ with respect to the standard basis of the module.

The notation $M *_{\text{v}} v$ represents the multiplication of a $(1+d) \times (1+d)$ real matrix $M$ and a contravariant Lorentz vector $v$ belonging to the module $\text{ContrMod } d$.

Let $d$ be a natural number representing the spatial dimension. For any square matrix $M$ of size $(1+d) \times (1+d)$ with real entries and any contravariant Lorentz vector $v \in \text{ContrMod } d$, the coordinate representation of the matrix-vector product $M v$ is equal to the standard matrix-vector multiplication of $M$ and the coordinate representation of $v$. That is, $(M v).\text{toFin1d}\mathbb{R} = M \cdot (v.\text{toFin1d}\mathbb{R})$, where $\text{toFin1d}\mathbb{R}$ maps a Lorentz vector to its components in $\mathbb{R}^{1+d}$.

Let $M$ be a real square matrix of size $(1+d) \times (1+d)$ and $v$ be a contravariant Lorentz vector in the $(1+d)$-dimensional real vector space $\text{ContrMod } d$. Let $v.\text{val}$ denote the underlying coordinate vector of $v$. The coordinate vector of the matrix-vector product $M v$ is equal to the standard matrix-vector multiplication of $M$ and the coordinate vector of $v$, that is, $(M v).\text{val} = M \cdot v.\text{val}$.

Let $M$ be a $(1+d) \times (1+d)$ real matrix indexed by $\text{Fin } 1 \oplus \text{Fin } d$, and let $v, w \in \text{ContrMod } d$ be contravariant real Lorentz vectors. The matrix-vector multiplication distributes over vector subtraction such that $M(v - w) = Mv - Mw$.

For any two real matrices $M$ and $N$ of size $(1+d) \times (1+d)$ (indexed by one temporal and $d$ spatial dimensions) and any contravariant real Lorentz vector $v \in \text{ContrMod } d$, the matrix-vector multiplication distributes over matrix subtraction such that $(M - N) v = M v - N v$.

For any $(1+d) \times (1+d)$ real matrix $M$ and any contravariant real Lorentz vectors $v, w$ in the space $\text{ContrMod } d$, the matrix-vector multiplication is distributive over vector addition: $$M(v + w) = Mv + Mw$$ where $Mv$ denotes the action of the matrix $M$ on the vector $v$.

For any contravariant real Lorentz vector $v$ in the $(1+d)$-dimensional space $\text{ContrMod } d$, multiplying $v$ by the $(1+d) \times (1+d)$ identity matrix results in $v$.

For any real square matrices $M$ and $N$ of size $(1+d) \times (1+d)$ and any contravariant real Lorentz vector $v$ in the space $\text{ContrMod } d$, the composition of matrix-vector multiplications satisfies the associativity property $M(Nv) = (MN)v$.

The space of contravariant Lorentz vectors $\text{ContrMod } d$ in $1+d$ dimensions is equipped with the structure of a normed additive commutative group. For a vector $v \in \text{ContrMod } d$, the norm $\|v\|$ is defined as the standard norm of its underlying component vector in $\mathbb{R}^{1+d}$ (typically the $L^\infty$ or supremum norm of its components). This structure is intended for topological purposes and is distinct from the indefinite Minkowski quadratic form (often colloquially called the Minkowski norm). This definition is not registered as a global instance to prevent it from being applied automatically in contexts where it might conflict with other structures.

For a contravariant Lorentz vector $v$ in $1+d$ dimensions (represented by the type `ContrMod d`), this function extracts the spatial components by removing the first (temporal) component. The resulting vector consists of the "tail" of the original vector and is mapped to the Euclidean space $\mathbb{R}^d$ equipped with the $L^2$ norm.

This definition defines the representation $\rho$ of the Lorentz group $\text{LorentzGroup } d$ on the $\mathbb{R}$-module of contravariant real Lorentz vectors $\text{ContrMod } d$. For an element $g$ of the Lorentz group (represented as a matrix), the linear automorphism $\rho(g)$ of $\text{ContrMod } d$ is defined as the linear map corresponding to the matrix $g$ with respect to the standard basis of $\text{ContrMod } d$.

For any Lorentz transformation $g \in \text{LorentzGroup } d$ and any contravariant real Lorentz vector $\psi \in \text{ContrMod } d$, the coordinate representation of the transformed vector $\rho(g)\psi$ is obtained by multiplying the matrix representing $g$ by the coordinate vector of $\psi$. Mathematically, this is expressed as: \[ [\rho(g)\psi] = \Lambda \cdot [\psi] \] where $[\cdot]$ denotes the coordinate representation mapping `toFin1dℝ` into $\mathbb{R}^{1+d}$, $\rho$ is the representation of the Lorentz group on contravariant vectors, and $\Lambda$ is the matrix underlying the group element $g$.

This definition establishes a linear isomorphism between the $\mathbb{R}$-module of contravariant real Lorentz vectors in $3+1$ dimensions, denoted as $\text{ContrMod } 3$, and the space of $2 \times 2$ self-adjoint (Hermitian) complex matrices, $\text{selfAdjoint}(M_2(\mathbb{C}))$. Given a contravariant vector $x$ with components $(x^0, x^1, x^2, x^3)$, the mapping is defined by the linear combination of Pauli matrices: \[ x \mapsto x^0 \sigma_0 - x^1 \sigma_1 - x^2 \sigma_2 - x^3 \sigma_3 \] where $\sigma_0$ is the $2 \times 2$ identity matrix and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices. This map preserves the $\mathbb{R}$-linear structure of both spaces.

For any contravariant real Lorentz vector $x \in \text{ContrMod } 3$ in $3+1$ dimensions, the linear isomorphism $\text{toSelfAdjoint}$ maps $x$ to the $2 \times 2$ self-adjoint (Hermitian) matrix: \[ \text{toSelfAdjoint}(x) = x^0 \sigma_0 - x^1 \sigma_1 - x^2 \sigma_2 - x^3 \sigma_3 \] where $x^0$ is the temporal component of $x$, $x^1, x^2, x^3$ are its spatial components, and $\sigma_0, \sigma_1, \sigma_2, \sigma_3$ are the standard $2 \times 2$ matrices (with $\sigma_0$ being the identity matrix and $\sigma_1, \sigma_2, \sigma_3$ being the Pauli matrices).

For a contravariant real Lorentz vector $x \in \text{ContrMod } 3$ in $3+1$ dimensions, the complex $2 \times 2$ matrix associated with the self-adjoint matrix $\text{toSelfAdjoint}(x)$ is given by the linear combination: \[ (\text{toSelfAdjoint } x)_{ij} = x^0 \sigma_0 - x^1 \sigma_1 - x^2 \sigma_2 - x^3 \sigma_3 \] where $x^0$ is the temporal component of $x$, $x^1, x^2, x^3$ are its spatial components, $\sigma_0$ is the $2 \times 2$ identity matrix, and $\sigma_1, \sigma_2, \sigma_3$ are the standard Pauli matrices.

For any index $i \in \text{Fin } 1 \oplus \text{Fin } 3$, the linear isomorphism $\text{toSelfAdjoint}$ maps the $i$-th standard basis vector $\text{stdBasis}_i$ of the space of contravariant Lorentz vectors $\text{ContrMod } 3$ to the $i$-th element $\text{pauliBasis}'_i$ of the Pauli-related basis for the space of $2 \times 2$ self-adjoint matrices.

For any index $i \in \text{Fin } 1 \oplus \text{Fin } 3$, the inverse of the linear isomorphism $\text{toSelfAdjoint}$ maps the $i$-th Pauli-related basis element $\text{pauliBasis}'_i$ to the $i$-th standard basis vector $\text{stdBasis}_i$ of the space of contravariant Lorentz vectors $\text{ContrMod } 3$. That is, \[ \text{toSelfAdjoint}^{-1}(\text{pauliBasis}'_i) = \text{stdBasis}_i \]

The space of contravariant real Lorentz vectors $\text{ContrMod } d$ is equipped with a topological structure. This topology is the induced topology (the coarsest topology making the map continuous) via the canonical linear equivalence $\text{ContrMod } d \cong \mathbb{R}^{1+d}$, where $\mathbb{R}^{1+d}$ is represented as the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ endowed with the standard product topology. This effectively identifies the topology of the Lorentz vector space with the standard Euclidean topology of its $(1+d)$ components.

The canonical linear isomorphism $\Phi: \text{ContrMod } d \xrightarrow{\cong} \mathbb{R}^{1+d}$, which identifies a contravariant real Lorentz vector with its $1+d$ components in the space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$, is an inducing map. This means the topology on the space of contravariant Lorentz vectors is the initial topology induced by $\Phi$ from the standard product topology on $\mathbb{R}^{1+d}$.

The inverse of the canonical linear isomorphism $\Phi: \text{ContrMod } d \xrightarrow{\cong} \mathbb{R}^{1+d}$, which maps the coordinate space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ to the space of contravariant real Lorentz vectors in $1+d$ dimensions, is an inducing map. This means the topology on the coordinate space $\mathbb{R}^{1+d}$ is the initial topology induced by $\Phi^{-1}$ from the topology on $\text{ContrMod } d$.

For any two covariant Lorentz vectors $\psi, \psi' \in \text{CoMod } d$, if their underlying component representations are equal ($\psi.val = \psi'.val$), then the vectors themselves are equal ($\psi = \psi'$). Here, $\text{CoMod } d$ represents the space of covariant Lorentz vectors in $1+d$ dimensions.

This equivalence provides a bijection between the space of covariant Lorentz vectors $\text{CoMod } d$ and the space of real-valued functions on the index set $\{0\} \oplus \{1, \dots, d\}$. It maps a covariant vector $v$ to its underlying component function $f: (1 \oplus d) \to \mathbb{R}$, and conversely, wraps a function to form a covariant vector. This represents the identification of a $1+d$ dimensional Lorentz vector with its components in the standard basis.

The space of covariant Lorentz vectors $\text{CoMod } d$ is equipped with an additive commutative monoid structure. This structure is defined by transporting the pointwise addition and zero element from the space of real-valued functions $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ to $\text{CoMod } d$ using the canonical equivalence between them.

The space of covariant Lorentz vectors $\text{CoMod } d$ is equipped with an additive commutative group structure. This structure is defined by transporting the pointwise addition, zero element, and additive inverse operations from the space of real-valued functions $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ to $\text{CoMod } d$ via the canonical equivalence between them.

The space of covariant Lorentz vectors $\text{CoMod } d$ is equipped with a module structure over the real numbers $\mathbb{R}$. This structure is defined by transporting the scalar multiplication and additive operations from the space of real-valued functions $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ to $\text{CoMod } d$ via the canonical equivalence between them.

The linear equivalence $\text{CoMod } d \cong_{\mathbb{R}} (\text{Fin } 1 \oplus \text{Fin } d \to \mathbb{R})$ identifies the module of covariant Lorentz vectors of dimension $1+d$ with the space of real-valued functions defined on the index set $\{0\} \cup \{1, \dots, d\}$. This map establishes a vector space isomorphism that associates each covariant vector with its components in the standard basis.

Given a covariant Lorentz vector $\psi \in \text{CoMod } d$ in $1+d$ dimensions, this function returns its representation as a real-valued function on the index set $\text{Fin } 1 \oplus \text{Fin } d$ (representing the time and space components) by applying the linear equivalence \text{toFin1d\mathbb{R}Equiv}.

The standard basis for the space of covariant Lorentz vectors $\text{CoMod } d$ is a basis indexed by the set $\text{Fin } 1 \oplus \text{Fin } d$. This basis is defined by pulling back the standard basis of the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$ through the linear equivalence \text{toFin1d\mathbb{R}Equiv} : \text{CoMod } d \cong_{\mathbb{R}} ((\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}). Essentially, it represents the set of vectors $\{e_\mu\}$ where each $e_\mu$ corresponds to a vector whose components are all zero except for the $\mu$-th component, which is 1.

Let $\text{CoMod } d$ be the module of covariant Lorentz vectors in $1+d$ dimensions, and let $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ be its standard basis. For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the $\mu$-th component of the basis vector $e_\mu$, obtained via the linear equivalence to the function space $(\text{Fin } 1 \oplus \text{Fin } d) \to \mathbb{R}$, is equal to $1$.

Let $\text{CoMod } d$ be the space of covariant Lorentz vectors in $1+d$ dimensions, and let $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ be its standard basis. For any index $\mu \in \text{Fin } 1 \oplus \text{Fin } d$, the $\mu$-th component of the basis vector $e_\mu$ is equal to $1$.

Let $\text{CoMod } d$ be the space of covariant Lorentz vectors and let $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ be its standard basis. For any indices $\mu$ and $\nu$ such that $\mu \neq \nu$, the $\nu$-th component of the basis vector $e_\mu$ (as determined by the linear equivalence to the coordinate space $\text{toFin1dℝEquiv}$) is $0$.

Let $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ be the standard basis for the space of covariant Lorentz vectors $\text{CoMod } d$. For any indices $\mu$ and $\nu$, the $\nu$-th component of the $\mu$-th basis vector $e_\mu$ is given by $$(e_\mu)_\nu = \begin{cases} 1 & \text{if } \mu = \nu \\ 0 & \text{if } \mu \neq \nu \end{cases}$$ In other words, the components of the standard basis vectors are given by the Kronecker delta $\delta_{\mu\nu}$.

In the space of covariant Lorentz vectors $\text{CoMod } d$, any vector $v$ can be uniquely decomposed into the standard basis $\{e_i\}_{i \in \text{Fin } 1 \oplus \text{Fin } d}$ as: $$v = \sum_{i} v_i e_i$$ where $v_i$ is the $i$-th component of the vector $v$ (obtained via the representation function $v.\text{toFin1dℝ}$) and $e_i$ is the $i$-th vector of the standard basis $\text{stdBasis}$.

Given a square matrix $M$ of size $(1+d) \times (1+d)$ over the real numbers $\mathbb{R}$ and a covariant Lorentz vector $v \in \text{CoMod}_d$, this function computes the matrix-vector product $M v$. The operation is defined by applying the linear transformation represented by the matrix $M$ with respect to the standard basis $\{e_\mu\}_{\mu \in \text{Fin } 1 \oplus \text{Fin } d}$ to the vector $v$, resulting in a new covariant Lorentz vector in $\text{CoMod}_d$.

The infix operator $*_{\text{v}}$ denotes the multiplication of a square matrix $M$ of size $(1+d) \times (1+d)$ over $\mathbb{R}$ with a covariant Lorentz vector $v \in \text{CoMod}_d$. The result $M *_{\text{v}} v$ is a new covariant Lorentz vector in $\text{CoMod}_d$.

Let $M$ be a real matrix of size $(1+d) \times (1+d)$ and $v$ be a covariant Lorentz vector in $\text{CoMod}_d$. Let \text{toFin1d\mathbb{R}} be the function that maps a covariant Lorentz vector to its representation as a real-valued function on the index set $\{0, 1, \dots, d\}$. Then the representation of the matrix-vector product $M v$ is equal to the standard matrix-vector product of $M$ and the representation of $v$: (M v).\text{toFin1d\mathbb{R}} = M (v.\text{toFin1d\mathbb{R}})

For any real matrix $M$ of size $(1+d) \times (1+d)$ and any covariant Lorentz vector $v \in \text{CoMod}_d$, the underlying vector components of the matrix-vector product $M v$ are equal to the standard matrix-vector multiplication of $M$ and the components of $v$. This is expressed as $(M v).\text{val} = M (v.\text{val})$, where $\text{val}$ denotes the underlying coordinate vector in $\mathbb{R}^{1+d}$.

The representation $\rho$ of the Lorentz group $\text{O}(1, d)$ on the space of covariant Lorentz vectors $\text{CoMod } d \cong \mathbb{R}^{1+d}$. For an element $g$ of the Lorentz group, let $\Lambda$ be its corresponding matrix. The representation maps $g$ to the linear automorphism of $\text{CoMod } d$ whose matrix with respect to the standard basis is the transpose inverse $(\Lambda^{-1})^\intercal$. This defines the standard transformation law for covariant vectors (covectors) under Lorentz transformations.