PhyslibAlpha.ClassicalMechanics.MomentMapCohomology

Given a symplectic form $\sigma$ on a real vector space $E$, the multiplication of two elements $a$ and $b$ in the affine symplectic group $\text{Sp}(E, \sigma) \ltimes E$ is defined by the composition of their affine transformations. If $a$ is represented by the pair $(B_a, C_a)$ and $b$ by $(B_b, C_b)$, where $B$ is a symplectic linear transformation and $C$ is a translation vector, then their product is $a \cdot b = (B_a \circ B_b, B_a(C_b) + C_a)$.

The identity element of the affine symplectic group $\text{ASp}(E, \sigma)$ is the pair $(I, 0)$, where $I$ is the identity linear map on the symplectic vector space $E$ and $0$ is the zero vector in $E$. This element represents the identity transformation $x \mapsto x$ on $E$.

For an element $a = (B, C)$ in the affine symplectic group $\text{Sp}(E, \sigma) \ltimes E$ acting on a symplectic vector space $(E, \sigma)$, where $B$ is the linear symplectic transformation and $C \in E$ is the translation vector, the inverse $a^{-1}$ is defined by the linear part $B^{-1}$ and the translation vector $-B^{-1}C$.

For any two elements $a$ and $b$ in the affine symplectic group $\text{ASp}(E, \sigma)$, the linear part of their product $a \cdot b$ is equal to the composition of the linear part of $a$ with the linear part of $b$. That is, $(a \cdot b).\text{linear} = a.\text{linear} \circ b.\text{linear}$.

For any two elements $a$ and $b$ in the affine symplectic group $\text{Sp}(E, \sigma) \ltimes E$ of a symplectic vector space $(E, \sigma)$, the translation component of their product $a \cdot b$ is given by $B_a(C_b) + C_a$, where $B_a$ is the linear symplectic transformation part of $a$, and $C_a, C_b$ are the translation vectors of $a$ and $b$ respectively.

In the affine symplectic group $\text{ASp}(E, \sigma)$ of a symplectic vector space $(E, \sigma)$, the linear component of the identity element $1$ is the identity linear transformation $\text{id}_E$ on $E$.

In the affine symplectic group $\text{ASp}(E, \sigma)$ of a symplectic vector space $(E, \sigma)$, the translation component of the identity element $1$ is the zero vector $0 \in E$.

For any element $a$ in the affine symplectic group $\text{Sp}(E, \sigma) \ltimes E$ of a symplectic vector space $(E, \sigma)$, the linear component of the inverse element $a^{-1}$ is equal to the inverse of the linear component of $a$.

Let $(E, \sigma)$ be a symplectic vector space. For any element $a$ in the affine symplectic group $\text{ASp}(E, \sigma)$, let $B$ denote its linear symplectic part and $C$ its translation part. The translation part of the group inverse $a^{-1}$ is given by $-B^{-1}(C)$, where $B^{-1}$ is the inverse of the linear symplectic transformation.

Let $\sigma$ be a symplectic form on a vector space $E$. For any two elements $a$ and $b$ in the affine symplectic group $\text{ASp}(E, \sigma)$, if their linear symplectic parts are equal and their translation parts are equal, then $a = b$.

The affine symplectic group $\text{ASp}(E, \sigma)$ (also written as $\text{Sp}(E, \sigma) \ltimes E$) of a symplectic vector space $(E, \sigma)$ forms a group. Elements are pairs $(B, C)$ where $B$ is a symplectic linear transformation and $C \in E$ is a translation vector. The group structure is defined by: - **Multiplication**: $(B_a, C_a) \cdot (B_b, C_b) = (B_a \circ B_b, B_a(C_b) + C_a)$ - **Identity**: $1 = (I, 0)$, where $I$ is the identity linear map. - **Inverse**: $(B, C)^{-1} = (B^{-1}, -B^{-1}C)$.

The affine symplectic group $\text{ASp}(E, \sigma)$ acts on the symplectic vector space $E$ via affine transformations. For an element $a \in \text{ASp}(E, \sigma)$ with linear symplectic part $B$ and translation part $C$, the action on a vector $x \in E$ is defined by: $$a \cdot x = B(x) + C$$ This mapping satisfies the axioms of a group action: the identity element acts as the identity transformation, and the action of a product of group elements is the composition of their individual actions.

Let $(E, \sigma)$ be a real symplectic vector space. For any element $a$ of the affine symplectic group $\text{ASp}(E, \sigma)$ and any vector $x \in E$, the group action $a \cdot x$ is defined as the sum of the linear part of $a$ applied to $x$ and the translation part of $a$: $$a \cdot x = a_{\text{linear}}(x) + a_{\text{translation}}$$ where $a_{\text{linear}}$ is the linear symplectic transformation and $a_{\text{translation}} \in E$ is the translation vector.

Given a bilinear form $\sigma$ on a real vector space $E$, the affine symplectic algebra is the $\mathbb{R}$-submodule of $\text{End}(E) \times E$ consisting of pairs $(A, b)$ such that the linear map $A$ is infinitesimally symplectic with respect to $\sigma$. That is, for all $v, w \in E$, the condition $\sigma(A v, w) + \sigma(v, A w) = 0$ is satisfied.

Let $E$ be a real vector space and $\sigma$ a bilinear form on $E$. A pair $Z = (A, b)$, where $A$ is a linear endomorphism of $E$ and $b \in E$, belongs to the affine symplectic algebra of $\sigma$ if and only if for all $v, w \in E$, $\sigma(A v, w) + \sigma(v, A w) = 0$.

Given a real vector space $E$ with a bilinear form $\sigma$, let $Z = (A, b)$ be an element of the affine symplectic algebra (where $A$ is an infinitesimally symplectic linear map and $b \in E$). The affine vector field associated with $Z$ at a point $x \in E$ is defined by the formula $Z_E(x) = Ax + b$.

For an element $a = (B, C)$ of the affine symplectic group associated with a bilinear form $\sigma$ (representing the transformation $x \mapsto Bx + C$ where $B$ is a symplectic linear map and $C \in E$) and an element $Z = (A, b)$ of the affine symplectic algebra (representing the affine vector field $Z_E(x) = Ax + b$), the adjoint action $\text{Ad}_a(Z)$ is the linear map defined by: $$\text{Ad}_a(A, b) = (B A B^{-1}, B b - B A B^{-1} C)$$ This map satisfies the condition that the resulting pair is also an element of the affine symplectic algebra.

The adjoint action of the identity element $1$ of the affine symplectic group $\text{ASp}(E, \sigma)$ on its associated Lie algebra is the identity linear map.

For any elements $a$ and $b$ of the affine symplectic group associated with a bilinear form $\sigma$, the adjoint action of their product $a \cdot b$ on the affine symplectic algebra is equal to the composition of the adjoint action of $a$ with the adjoint action of $b$. That is, $\text{Ad}_{a \cdot b} = \text{Ad}_a \circ \text{Ad}_b$.

Let $(E, \sigma)$ be a real vector space equipped with a bilinear form $\sigma$. Let $a \in \text{ASp}(E, \sigma)$ be an element of the affine symplectic group with linear part $B$ (so that the action on $E$ is given by $a \cdot x = Bx + C$ for some $C \in E$). For any element $Z$ of the affine symplectic algebra and its associated affine vector field $Z_E(x)$, the adjoint action $\text{Ad}_a Z$ is characterized by the identity: $$(\text{Ad}_a Z)_E(a \cdot x) = B (Z_E(x))$$ for all $x \in E$.

Let $E$ be a real vector space and $\sigma$ a bilinear form on $E$. Let $Z$ and $Z'$ be elements of the affine symplectic algebra associated with $\sigma$ (consisting of pairs $(A, b)$ where $A$ is an infinitesimally symplectic linear map and $b \in E$). If for all $x \in E$, the associated affine vector fields satisfy $Z_E(x) = Z'_E(x)$, where $Z_E(x) = Ax + b$, then $Z = Z'$.

Given a real vector space $E$ with a bilinear form $\sigma$, let $G$ be the affine symplectic group (the group of transformations $x \mapsto Bx + C$ where $B$ is symplectic) and let $\mathfrak{g}^*$ be the dual of the affine symplectic algebra. This definition establishes the scalar multiplication (action) of an element $a \in G$ on a linear functional $\mu \in \mathfrak{g}^*$ defined by the rule $a \cdot \mu = \mu \circ \text{Ad}_{a^{-1}}$, where $\text{Ad}_{a^{-1}}$ is the adjoint action of the inverse of $a$ on the algebra.

Let $E$ be a real vector space equipped with a bilinear form $\sigma$. Let $G$ be the affine symplectic group (consisting of transformations $x \mapsto Bx + C$ where $B$ is a symplectic linear map) and let $\mathfrak{g}^*$ be the dual of the affine symplectic algebra $\mathfrak{g}$. For any group element $a \in G$, linear functional $\mu \in \mathfrak{g}^*$, and algebra element $Z \in \mathfrak{g}$, the coadjoint action of $a$ on $\mu$ evaluated at $Z$ satisfies: $$(a \cdot \mu)(Z) = \mu(\text{Ad}_{a^{-1}} Z)$$ where $\text{Ad}_{a^{-1}}$ denotes the adjoint action of the inverse element $a^{-1} \in G$ on the algebra $\mathfrak{g}$.

This instance defines the multiplicative group action of the affine symplectic group $\text{ASp}(E, \sigma)$ on the dual of the affine symplectic algebra $\mathfrak{g}^*$. For an element $a \in \text{ASp}(E, \sigma)$ and a linear functional $\mu \in \mathfrak{g}^*$, the action is given by the coadjoint action $a \cdot \mu = \mu \circ \text{Ad}_{a^{-1}}$, where $\text{Ad}_{a^{-1}}$ is the adjoint action of the inverse of $a$. This definition satisfies the group action axioms: $1 \cdot \mu = \mu$ and $(a \cdot b) \cdot \mu = a \cdot (b \cdot \mu)$ for all $a, b \in \text{ASp}(E, \sigma)$ and $\mu \in \mathfrak{g}^*$.

This instance defines the distributive multiplicative action (the coadjoint action) of the affine symplectic group $\text{ASp}(E, \sigma)$ on the dual of the affine symplectic algebra $\mathfrak{g}^*$. For any group element $a \in \text{ASp}(E, \sigma)$ and linear functionals $\mu, \nu \in \mathfrak{g}^*$, the action $a \cdot \mu = \mu \circ \text{Ad}_{a^{-1}}$ is distributive over addition and preserves the zero element, such that $a \cdot (\mu + \nu) = a \cdot \mu + a \cdot \nu$ and $a \cdot 0 = 0$.

Given a bilinear form $\sigma$ on a real vector space $E$, the moment map $\mu$ at a point $x \in E$ is a linear functional on the affine symplectic algebra $\mathfrak{g}$ of $\sigma$. For any element $(A, b) \in \mathfrak{g}$, where $A$ is an infinitesimally symplectic linear map and $b \in E$ is a translation vector, the moment is defined as: $$\mu(x)(A, b) = -\frac{1}{2} \sigma(Ax, x) - \sigma(b, x)$$

Let $E$ be a real vector space equipped with a bilinear form $\sigma$. For any point $x \in E$ and any element $Z = (A, b)$ in the affine symplectic algebra $\mathfrak{g}$ of $\sigma$ (where $A$ is an infinitesimally symplectic linear map and $b$ is a translation vector), the moment map $\mu$ at $x$ evaluated at $Z$ is given by: $$\mu(x)(Z) = -\frac{1}{2} \sigma(Ax, x) - \sigma(b, x)$$

Given a bilinear form $\sigma$ on a real vector space $E$, the moment map $\mu$ of the affine symplectic action evaluated at the origin $0 \in E$ is the zero linear functional. That is, $\mu(0) = 0$.

Let $(E, \sigma)$ be a real symplectic vector space. Let $G = \text{ASp}(E, \sigma)$ be the affine symplectic group, where an element $a \in G$ acts on $E$ via $a \cdot x = Bx + C$ (with $B$ a symplectic linear map and $C \in E$ a translation vector). Let $\mathfrak{g}^*$ be the dual of the affine symplectic algebra, equipped with the coadjoint action $a \cdot \xi = \xi \circ \text{Ad}_{a^{-1}}$. Let $\mu : E \to \mathfrak{g}^*$ be the moment map defined by $\mu(x)(A, b) = -\frac{1}{2} \sigma(Ax, x) - \sigma(b, x)$. For any point $x \in E$ and any group element $a \in G$, the difference between the moment of the transformed point and the coadjoint-transformed moment is independent of $x$ and given by: $$\mu(a \cdot x) - a \cdot \mu(x) = \mu(C)$$ where $C$ is the translation part of $a$.

Let $(E, \sigma)$ be a real symplectic vector space. Let $G$ be the affine symplectic group of $(E, \sigma)$, consisting of transformations $a: x \mapsto B_a x + C_a$ where $B_a$ is a symplectic linear map and $C_a \in E$ is a translation vector. Let $\mathfrak{g}^*$ be the dual of the affine symplectic algebra, equipped with the coadjoint action $a \cdot \xi = \xi \circ \operatorname{Ad}_{a^{-1}}$. The map $\theta : G \to \mathfrak{g}^*$ defined by $\theta(a) = \mu(C_a)$, where $\mu(C_a)$ is the moment map evaluated at the translation vector $C_a$, is a 1-cocycle. Specifically, for all $a, b \in G$, it satisfies the identity: $$\theta(a \cdot b) = \theta(a) + a \cdot \theta(b)$$

For a finite-dimensional real normed vector space $E$, a bilinear form $\sigma$ on $E$, and a linear operator $A: E \to E$, this definition represents the continuous linear map $E \to \mathcal{L}(E, \mathbb{R})$ that maps each element $y \in E$ to the continuous linear functional $v \mapsto \sigma(Ay, v)$. This map is utilized to compute the Fréchet derivative of quadratic forms associated with the moment map.

Let $E$ be a finite-dimensional real normed vector space, $\sigma$ a bilinear form on $E$, and $A: E \to E$ a linear operator. For any vectors $y, v \in E$, the value of the continuous linear map $\text{sigmaLinear } \sigma A$ (which maps $y$ to the functional $v \mapsto \sigma(Ay, v)$) satisfies $$(\text{sigmaLinear } \sigma A)(y)(v) = \sigma(Ay, v).$$

Let $E$ be a finite-dimensional real normed vector space. Let $\sigma$ be a bilinear form on $E$ and $A: E \to E$ be a linear operator. For any $x \in E$, the Fréchet derivative of the map $y \mapsto \sigma(Ay, y)$ at the point $x$ is the continuous linear map mapping $v$ to $\sigma(Ax, v) + \sigma(Av, x)$.

Let $E$ be a finite-dimensional real normed vector space equipped with a symplectic form $\sigma$. Let $Z$ be an element of the affine symplectic algebra of $\sigma$ and $Z_E(x)$ be its associated affine vector field at $x \in E$. The Fréchet derivative of the map $y \mapsto \mu(y)(Z)$ at the point $x$ is the continuous linear map $v \mapsto -\sigma(Z_E(x), v)$, where $\mu$ is the moment map of the affine symplectic action. This identity is often expressed as $\nabla \mu(x)(Z) = -\sigma(Z_E(x), \cdot)$.

Let $E$ be a finite-dimensional real normed vector space equipped with a symplectic form $\sigma$. Let $\mathfrak{g}$ be the affine symplectic algebra consisting of pairs $(A, b)$, where $A$ is an infinitesimally symplectic linear map and $b \in E$ is a translation vector. Let $\gamma : \mathbb{R} \to \text{ASp}(E, \sigma)$ be a curve in the affine symplectic group such that $\gamma(0)$ is the identity element and the derivative of the translation component $C(t)$ of $\gamma(t)$ at $t=0$ is $b' \in E$. Then for any $Z = (A, b) \in \mathfrak{g}$, the derivative of the map $t \mapsto \mu(C(t))(Z)$ at $t=0$ is equal to $\sigma(b', b)$, where $\mu$ is the moment map of the affine symplectic action.

Let $E$ be a non-zero real vector space equipped with a symplectic form $\sigma$. Let $G$ be the affine symplectic group (the group of transformations $x \mapsto Bx + C$ where $B$ is a symplectic linear map) and let $\mathfrak{g}^*$ be the dual of the affine symplectic algebra, which is a $G$-module under the coadjoint action. Let $\theta : G \to \mathfrak{g}^*$ be the map defined by $\theta(a) = \mu(C)$, where $C$ is the translation component of the affine transformation $a$ and $\mu(C)$ is the moment map at point $C$. Then $\theta$ is not a 1-coboundary. That is, there does not exist an element $\mu_0 \in \mathfrak{g}^*$ such that for all $a \in G$, $\theta(a) = a \cdot \mu_0 - \mu_0$.

Let $\sigma$ be the standard symplectic form on the plane $\mathbb{R}^2$, defined by $\sigma(v, w) = v_1 w_2 - v_2 w_1$. Let $G$ be the affine symplectic group of the plane, consisting of transformations $a: x \mapsto Bx + C$, where $B$ is a symplectic linear map and $C \in \mathbb{R}^2$ is the translation vector. Let $\mu: \mathbb{R}^2 \to \mathfrak{g}^*$ be the moment map associated with the affine action, which maps a vector $x$ to a linear functional on the affine symplectic algebra $\mathfrak{g}$. The mapping $\theta: G \to \mathfrak{g}^*$, defined by $\theta(a) = \mu(C)$, is a 1-cocycle for the coadjoint action of $G$ on $\mathfrak{g}^*$. This theorem states that $\theta$ is not a 1-coboundary, which implies that the affine symplectic group of the symplectic plane has a non-zero cohomology class.