PhyslibAlpha.Mathematics.LadderSystem.Basic

For a fixed element $y$ in a Lie algebra $L$ over a field $K$, this is the $K$-linear map from $L$ to itself defined by $x \mapsto [x, y]$, where $[ \cdot, \cdot ]$ denotes the Lie bracket.

Given a Lie algebra $L$ over a field $K$ and a fixed element $x \in L$, `bracketLeft x` is the $K$-linear map from $L$ to itself defined by mapping an element $y \in L$ to the Lie bracket $[x, y]$.

Let $K$ be a field, and let $L$ and $L'$ be Lie algebras over $K$. Let $\{b_i\}_{i \in I}$ be a basis for $L$. If $f: L \to L'$ is a $K$-linear map such that $f([b_i, b_j]) = [f(b_i), f(b_j)]$ for all $i, j \in I$, then for any $x, y \in L$, it holds that $f([x, y]) = [f(x), f(y)]$. Here, $[\cdot, \cdot]$ denotes the Lie bracket.

Let $L$ and $L'$ be Lie algebras over a field $K$, and let $s \subseteq L$ be a subset such that its linear span is the entire space $L$. If a linear map $f: L \to L'$ preserves the Lie bracket for all elements in the spanning set, i.e., $f([x, y]) = [f(x), f(y)]$ for all $x \in s$ and $y \in s$, then $f$ preserves the Lie bracket for all $x, y \in L$.

Let $V$ be a vector space over a field $K$. For any linear operators $x, y, z \in \text{End}_K(V)$, the commutator satisfies the Leibniz rule: $$[x, yz] = [x, y]z + y[x, z]$$ where $[A, B] = AB - BA$ denotes the commutator of two operators.

Let $V$ be a vector space over a field $K$. For any endomorphisms $x, y, z \in \text{End}(V)$, the commutator of the product $xy$ with $z$ satisfies the identity $[xy, z] = x[y, z] + [x, z]y$, where $[A, B] \coloneqq AB - BA$ denotes the Lie bracket (commutator) of endomorphisms.

Given a ladder system on a $K$-vector space $V$ with $d$ modes, the operator $E_{ij} \in \text{End}_K(V)$ is defined as the product of the $i$-th creation operator $a^\dagger_i$ and the $j$-th annihilation operator $a_j$: $$E_{ij} = a^\dagger_i a_j$$ where $i, j \in \{0, \dots, d-1\}$. These operators serve as the $\mathfrak{gl}(d)$ bilinear generators for the representation on $V$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. For any indices $i, j, k, l \in \{0, \dots, d-1\}$, let $E_{ij} = a^\dagger_i a_j$ be the bilinear operators of the system. The commutator (Lie bracket) of these operators satisfies the commutation relations of the general linear Lie algebra $\mathfrak{gl}(d)$: $$[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - \delta_{li} E_{kj}$$ where $\delta$ denotes the Kronecker delta.

Let $E_{ij}$ denote the $d \times d$ matrix unit over a field $K$ with $1$ at the $(i, j)$-th entry and $0$ elsewhere. For any indices $i, j, k, l \in \{0, \dots, d-1\}$, the Lie bracket of these matrices satisfies: $$[E_{ij}, E_{kl}] = \delta_{jk} E_{il} - \delta_{li} E_{kj}$$ where $\delta_{jk}$ is the Kronecker delta.

For a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, this is the $K$-linear map from the space of $d \times d$ matrices $\text{M}_d(K)$ to the endomorphism algebra $\text{End}_K(V)$ that sends each matrix unit $E_{ij}$ (a matrix with $1$ at $(i, j)$ and $0$ elsewhere) to the bilinear operator $E_{ij} = a^\dagger_i a_j$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Let $\text{toGlHomLinear} : \text{M}_d(K) \to \text{End}_K(V)$ be the $K$-linear map that defines the $\mathfrak{gl}(d)$ representation on $V$. For any indices $i, j \in \{0, \dots, d-1\}$, let $e_{ij}$ be the standard basis matrix with $1$ at the $(i, j)$-th entry and $0$ elsewhere. Then, the image of $e_{ij}$ under this linear map is the bilinear operator $E_{ij} = a^\dagger_i a_j$: $$\text{toGlHomLinear}(e_{ij}) = E_{ij}$$

For a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, the $K$-linear map $\rho: \text{M}_d(K) \to \text{End}_K(V)$ (which defines the $\mathfrak{gl}(d)$ representation on $V$) maps the elementary matrix with $1$ at position $(i, j)$ and $0$ elsewhere to the bilinear operator $E_{ij} = a^\dagger_i a_j$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Let $\phi: \text{M}_d(K) \to \text{End}_K(V)$ be the $K$-linear map (defined as `toGlHomLinear`) that sends the matrix units $E_{ij}$ to the bilinear operators $a^\dagger_i a_j$. For any two matrices $x, y \in \text{M}_d(K)$, the map $\phi$ preserves the Lie bracket: $$\phi(\llbracket x, y \rrbracket) = \llbracket \phi(x), \phi(y) \rrbracket,$$ where $\llbracket \cdot, \cdot \rrbracket$ denotes the standard commutator bracket $[A, B] = AB - BA$.

Given a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, this is the Lie algebra homomorphism from the General Linear Lie algebra $\mathfrak{gl}_d(K)$ (represented by $d \times d$ matrices over $K$ under the commutator bracket) to the Lie algebra of endomorphisms $\text{End}_K(V)$. This homomorphism defines the $\mathfrak{gl}_d(K)$-representation on $V$ generated by the ladder operators, mapping the elementary matrix $E_{ij}$ to the bilinear operator $a^\dagger_i a_j$.

For a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, this defines the structure of a Lie ring module on $V$ over the general linear Lie algebra $\mathfrak{gl}_d(K)$ (the space of $d \times d$ matrices over $K$ equipped with the commutator bracket). The representation is induced by the Lie algebra homomorphism $L.\text{toGlHom}: \mathfrak{gl}_d(K) \to \text{End}_K(V)$, which maps the elementary matrix $E_{ij}$ to the bilinear operator $a^\dagger_i a_j$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Then $V$ carries the structure of a Lie module over the Lie algebra $\mathfrak{gl}_d(K)$ (the space of $d \times d$ matrices over $K$ equipped with the commutator bracket). The Lie action is induced by the Lie algebra homomorphism $L.\text{toGlHom}: \mathfrak{gl}_d(K) \to \text{End}_K(V)$, which maps the elementary matrix $E_{ij}$ to the bilinear operator $a^\dagger_i a_j$.

For a ladder system on a $K$-vector space $V$ with $d$ modes, the number operator $N_i \in \text{End}_K(V)$ for the $i$-th mode is defined as the diagonal bilinear operator $E_{ii}$. It corresponds to the product of the $i$-th creation operator $a^\dagger_i$ and the $i$-th annihilation operator $a_i$: $$N_i = E_{ii} = a^\dagger_i a_i$$ where $i \in \{0, \dots, d-1\}$.

In a ladder system on a $K$-vector space $V$ with $d$ modes, for any indices $i, j \in \{0, \dots, d-1\}$, the commutator of the number operator $N_i$ and the annihilation operator $a_j$ is given by: $$[N_i, a_j] = -\delta_{ij} a_i$$ where $\delta_{ij}$ is the Kronecker delta, $N_i$ is the number operator for the $i$-th mode, $a_j$ is the annihilation operator for the $j$-th mode, and the bracket $[ \cdot, \cdot ]$ denotes the Lie bracket (commutator) of endomorphisms in $\text{End}_K(V)$.

For a ladder system on a $K$-vector space $V$ with $d$ modes, let $N_i$ be the number operator for mode $i$ and $a^\dagger_j$ be the creation operator for mode $j$, where $i, j \in \{0, \dots, d-1\}$. The commutator of these operators satisfies: $$[N_i, a^\dagger_j] = \delta_{ij} a^\dagger_i$$ where $\delta_{ij}$ is the Kronecker delta.

For any indices $i, j \in \{0, \dots, d-1\}$, the number operators $N_i$ and $N_j$ of a ladder system commute, meaning their Lie bracket (commutator) in the algebra of endomorphisms is zero: $$[N_i, N_j] = 0$$

For a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, the total number operator $\text{totalN} \in \text{End}_K(V)$ is defined as the sum of the individual number operators $N_i$ for each mode $i \in \{0, \dots, d-1\}$: $$\text{totalN} = \sum_{i=0}^{d-1} N_i$$ This operator represents the total excitation count across every mode.

For a ladder system $L$ on a $K$-vector space $V$ with $d$ modes, let $N_{\text{total}} = \sum_{i=0}^{d-1} N_i$ be the total number operator and $E_{jk} = a^\dagger_j a_k$ be the $\mathfrak{gl}(d)$ generators (bilinears of creation and annihilation operators). For any indices $j, k \in \{0, \dots, d-1\}$, the total number operator commutes with the generators, meaning their Lie bracket in the algebra of endomorphisms is zero: $$[N_{\text{total}}, E_{jk}] = 0$$