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Symplectic vector space

A symplectic vector space is a pair (V, \omega), where V is a finite-dimensional vector space over the real numbers \mathbb{R} and \omega: V \times V \to \mathbb{R} is a non-degenerate skew-symmetric bilinear form, called the symplectic form.^[1] This form satisfies \omega(u, v) = -\omega(v, u) for all u, v \in V, and non-degeneracy means that if \omega(u, v) = 0 for all v \in V, then u = 0.^[2] Consequently, the dimension of V must be even, say \dim V = 2n for some positive integer n, as the symplectic form induces a non-degenerate pairing that pairs the space with itself in a canonical way.^[1] Symplectic vector spaces form the algebraic cornerstone of symplectic geometry, providing the linear model for phase spaces in classical Hamiltonian mechanics, where positions and momenta are coordinated via the symplectic form to preserve the structure of dynamical systems.^[3] Key structural features include the existence of a Darboux basis \{e_1, \dots, e_n, f_1, \dots, f_n\} such that \omega(e_i, e_j) = \omega(f_i, f_j) = 0 and \omega(e_i, f_j) = \delta_{ij}, which standardizes the form to the canonical matrix J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}.^[2] All symplectic vector spaces of the same dimension are symplectomorphic, meaning there exists a linear isomorphism preserving the symplectic form, underscoring their uniformity.^[1] Subspaces of a symplectic vector space V are classified by their interaction with \omega: an isotropic subspace S satisfies S \subseteq S^\omega = \{v \in V \mid \omega(v, s) = 0 \ \forall s \in S\}; a Lagrangian subspace is maximal isotropic with \dim S = n; a coisotropic subspace has S^\omega \subseteq S; and a symplectic subspace restricts \omega to a non-degenerate form on itself.^[1] The group of linear symplectomorphisms, denoted Sp(2n, \mathbb{R}), consists of invertible maps A: V \to V such that \omega(Au, Av) = \omega(u, v) for all u, v \in V, and it plays a central role in preserving the symplectic structure under transformations.^[2] These elements extend to broader contexts, such as cotangent bundles in mechanics, where the canonical symplectic form facilitates the formulation of Hamiltonian vector fields and Poisson brackets.^[3]

Definition and Properties

Symplectic Form

A symplectic vector space is a finite-dimensional vector space V over the real numbers \mathbb{R} equipped with a symplectic form \omega: V \times V \to \mathbb{R}, which is a bilinear map satisfying two key properties: skew-symmetry, meaning \omega(u, v) = -\omega(v, u) for all u, v \in V, and non-degeneracy, meaning that if \omega(u, v) = 0 for all v \in V, then u = 0.^[1] This non-degeneracy condition ensures that \omega induces a natural isomorphism between V and its dual space V^*, establishing a perfect pairing.^[1] The skew-symmetry of \omega over the real field implies that the dimension of V must be even; if \dim V were odd, the form would necessarily be degenerate, as the determinant of the associated skew-symmetric matrix would vanish.^[1] Thus, \dim V = 2n for some positive integer n, where n represents the number of degrees of freedom in the underlying physical interpretation from classical mechanics.^[1] In a suitable basis of V \cong \mathbb{R}^{2n}, the symplectic form admits a standard matrix representation: \omega(u, v) = u^T J v, where J is the block-diagonal skew-symmetric matrix

J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix},

with I_n denoting the n \times n identity matrix.^[1] This representation highlights the form's structure, with the superdiagonal blocks consisting of +1 entries (via I_n) and the subdiagonal blocks of -1 entries (via -I_n). The concept of the symplectic vector space formalizes the linear structure underlying phase spaces in Hamiltonian mechanics, serving as the infinitesimal model for symplectic manifolds.^[4] The term "symplectic" itself was coined by Hermann Weyl in 1939, as a direct Greek calque of "complex" to describe the associated linear group, replacing earlier nomenclature like "Abelian linear group."^[4]

Key Properties

A symplectic vector space (V, \omega) is equipped with a bilinear form \omega: V \times V \to \mathbb{R} that satisfies the alternating property, meaning \omega(v, v) = 0 for all v \in V. This condition implies skew-symmetry, \omega(u, v) = -\omega(v, u) for all u, v \in V, distinguishing the symplectic form from symmetric or other bilinear forms.^[1]^[5] Non-degeneracy requires that the linear map v \mapsto \omega(v, \cdot) is an isomorphism from V to its dual space V^*, ensuring that if \omega(v, w) = 0 for all w \in V, then v = 0. This property implies that V must be even-dimensional, say \dim V = 2n, and the symplectic form induces a natural orientation on V via the Liouville volume form \frac{\omega^n}{n!}.^[1]^[5] Every symplectic vector space admits a symplectic basis \{e_1, \dots, e_n, f_1, \dots, f_n\} such that \omega(e_i, f_j) = \delta_{ij} and \omega(e_i, e_j) = \omega(f_i, f_j) = 0 for all i, j. This basis canonicalizes the form, allowing representation in a standard block-diagonal structure.^[1]^[5] The symplectic form induces a natural volume element on V, given by the n-fold wedge product

\frac{\omega^n}{n!},

which is non-vanishing and defines a volume up to sign, reflecting the oriented structure of the space.^[1]^[5] The symplectic structure is compatible with linear automorphisms, preserved precisely by symplectomorphisms—linear maps \phi: V \to V satisfying \omega(\phi(u), \phi(v)) = \omega(u, v) for all u, v \in V. This preservation ensures that the intrinsic properties of \omega remain invariant under such transformations.^[1]^[5]

Canonical Realization

Standard Symplectic Space

The standard symplectic vector space is constructed on the Euclidean space V = \mathbb{R}^{2n} equipped with the symplectic form \omega_0: V \times V \to \mathbb{R} defined by

\omega_0(x, y) = \sum_{i=1}^n (x_i y_{n+i} - x_{n+i} y_i)

for x = (x_1, \dots, x_n, x_{n+1}, \dots, x_{2n}) and y = (y_1, \dots, y_n, y_{n+1}, \dots, y_{2n}).^[1] This bilinear form can equivalently be expressed in matrix notation as \omega_0(x, y) = x^T J y, where

J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}

and I_n is the n \times n identity matrix.^[6] The non-degeneracy of \omega_0 ensures that V forms a symplectic vector space, providing a prototypical model for the algebraic structure.^[7] In coordinates adapted to this structure, a symplectic basis for V consists of position-momentum pairs \{q_1, \dots, q_n, p_1, \dots, p_n\}, where the first n basis vectors correspond to positions and the latter n to momenta. With respect to this basis, the symplectic form takes the explicit values \omega_0(q_i, q_j) = 0, \omega_0(p_i, p_j) = 0, and \omega_0(q_i, p_j) = \delta_{ij}.^[6] This basis reflects the canonical pairing in the construction, facilitating computations in symplectic linear algebra.^[1] Adapting differential notation to the linear setting, the form \omega_0 can be written as \omega_0 = \sum_{i=1}^n dq_i \wedge dp_i, where dq_i and dp_i are the dual basis elements corresponding to the symplectic basis.^[6] This expression highlights the wedge product structure, analogous to the volume-preserving aspect in higher-dimensional contexts, though here it remains a constant bilinear form on the vector space.^[7] Every finite-dimensional symplectic vector space (E, \omega) of dimension $2n is symplectomorphic to the standard one (\mathbb{R}^{2n}, \omega_0) via a linear isomorphism that preserves the symplectic form.^[6] This universality underscores the standard space's role as a canonical representative, allowing abstract properties to be studied concretely in coordinates.^[1] For n=1, the standard symplectic space is \mathbb{R}^2 with \omega_0((x_1, x_2), (y_1, y_2)) = x_1 y_2 - x_2 y_1, which coincides with the signed area form on the plane.^[1] This case models the phase space of a single classical particle, where coordinates (q, p) represent position and momentum, and \omega_0 encodes the Poisson bracket structure fundamental to Hamiltonian mechanics.^[6]

Darboux Theorem

The linear Darboux theorem asserts that for any symplectic vector space (V, \omega) over \mathbb{R} of finite dimension $2n, there exists a basis \{e_1, \dots, e_n, f_1, \dots, f_n\}, called a symplectic basis, such that \omega(e_i, e_j) = 0 = \omega(f_i, f_j) for all i, j and \omega(e_i, f_j) = \delta_{ij} for the Kronecker delta \delta_{ij}. This basis induces a linear symplectomorphism from (V, \omega) to the standard symplectic space (\mathbb{R}^{2n}, \omega_0), where \omega_0 = \sum_{i=1}^n dx_i \wedge dy_i.^[8] The proof proceeds by induction on n. For the base case n=1, select any nonzero e_1 \in V; non-degeneracy ensures there exists f_1 \in V with \omega(e_1, f_1) \neq 0, which can be normalized to \omega(e_1, f_1) = 1. For the inductive step, assume the result holds for dimension $2(n-1). Let [W](/page/W) be the span of \{e_1, \dots, e_{n-1}, f_1, \dots, f_{n-1}\}, a symplectic subspace of dimension $2(n-1). The symplectic complement W^\omega is a symplectic subspace of dimension 2. Choose a nonzero e_n \in W^\omega; then select f_n \in W^\omega such that \omega(e_n, f_n) = 1, which is possible by non-degeneracy of \omega on W^\omega. This ensures \omega(e_n, e_i) = 0 = \omega(e_n, f_i), \omega(f_n, e_i) = 0 = \omega(f_n, f_i) for i < n, and linear independence. This extends the basis to dimension $2n.^[8]^[5] A key consequence is that all symplectic vector spaces of the same even dimension $2n are symplectomorphic via linear maps, implying no symplectic invariants exist beyond the dimension itself.^[8] The theorem's linear case originated in the late 19th century, rooted in Henri Poincaré's foundational work on invariant integrals and area-preserving transformations in the 1880s, with Gaston Darboux formalizing the canonical form for linear structures in 1882 as part of solving Pfaff's problem on differential forms; this predates Élie Cartan's extensions to manifolds in the 1920s using exterior calculus.^[9] The Darboux theorem also underpins the Williamson normal form for quadratic Hamiltonians in linear symplectic algebra: for a quadratic form H(x) = \frac{1}{2} \langle x, A x \rangle on (V, \omega) with symmetric A, there exists a symplectic basis in which H decomposes into n_+ positive squares, n_- negative squares, and n_0 hyperbolic terms \frac{1}{2}(q_i^2 - p_i^2), where n_+ + n_- + 2n_0 = 2n and the signature is determined by the inertia indices.

Symplectic Transformations

Symplectic Maps

In a symplectic vector space (V, \omega), a linear map \phi: V \to V is called a symplectic map if it preserves the symplectic form, that is, \omega(\phi(u), \phi(v)) = \omega(u, v) for all u, v \in V.^[1] This preservation ensures that \phi acts as an automorphism of the bilinear form \omega, maintaining the symplectic structure of the space.^[10] An equivalent matrix formulation arises when choosing a Darboux basis for V, in which \omega is represented by the block matrix J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, where I_n is the n \times n identity matrix. In this basis, the matrix M representing \phi satisfies M^T J M = J.^[1] This condition directly implies that \phi preserves \omega as a skew-symmetric bilinear map.^[11] Symplectic maps possess several key properties. They are invertible, as the preservation of the non-degenerate form \omega implies that \phi is injective, and hence bijective on finite-dimensional spaces.^[1] Additionally, \det \phi = 1, which means symplectic maps preserve the Liouville volume form \frac{\omega^n}{n!} and are thus volume-preserving transformations.^[1] This determinant condition also ensures that they are orientation-preserving.^[11] Representative examples illustrate these maps in standard settings. In the phase space \mathbb{R}^2 with \omega = dq \wedge dp, rotations given by the matrix

\begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}

preserve \omega and thus qualify as symplectic.^[8] Shear maps in canonical coordinates, such as (q, p) \mapsto (q + a p, p) for a constant a \in \mathbb{R}, also preserve \omega and represent simple canonical transformations.^[8] Symplectic maps play a fundamental role in Hamiltonian dynamics, where the time-t flow of a quadratic Hamiltonian, linearized at an equilibrium, yields a linear symplectic transformation on the tangent space.^[11] Such flows maintain the symplectic structure, reflecting the conservation laws inherent in Hamiltonian systems.

Symplectic Group

The symplectic group \mathrm{Sp}(2n, \mathbb{R}) consists of all $2n \times 2n real matrices A \in \mathrm{GL}(2n, \mathbb{R}) satisfying A^T J A = J, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} is the standard symplectic matrix, with I_n the n \times n identity matrix. This condition ensures that elements of the group preserve the standard symplectic form \omega_0 on \mathbb{R}^{2n}, making \mathrm{Sp}(2n, \mathbb{R}) the automorphism group of the symplectic vector space (\mathbb{R}^{2n}, \omega_0). Symplectic maps, as linear transformations preserving the symplectic form, are precisely the elements of this group.^[1] As a Lie group, \mathrm{Sp}(2n, \mathbb{R}) has dimension n(2n+1), reflecting the number of independent parameters needed to specify matrices satisfying the defining relation after accounting for symmetries.^[12] It is non-compact, but admits compact subgroups such as the unitary symplectic group \mathrm{USp}(2n) = \mathrm{Sp}(2n, \mathbb{C}) \cap \mathrm{U}(2n), which serves as the compact real form of the complex symplectic Lie algebra \mathfrak{sp}(2n, \mathbb{C}).^[13] The group is generated by transvections, which act as shears along symplectic hyperplanes, and rotations within the canonical 2×2 blocks of the symplectic basis.^[14] In terms of representations, the fundamental representation of \mathrm{Sp}(2n, \mathbb{R}) is its defining action on \mathbb{R}^{2n}, which is irreducible and preserves the symplectic structure. Every element has determinant 1, so \mathrm{Sp}(2n, \mathbb{R}) embeds as a subgroup of \mathrm{SL}(2n, \mathbb{R}).^[15] The name "symplectic group" was coined by Hermann Weyl in 1939, drawing from the Greek root for "interwoven" to distinguish it from complex linear groups while highlighting its role among the classical groups alongside orthogonal and unitary groups.

Subspace Classifications

Isotropic Subspaces

In a symplectic vector space (V, \omega) of finite dimension $2n over \mathbb{R}, a subspace W \subseteq V is isotropic if \omega(u, v) = 0 for all u, v \in W. Equivalently, W \subseteq W^\perp, where the symplectic orthogonal complement is defined as W^\perp = \{ v \in V \mid \omega(u, v) = 0 \ \forall \, u \in W \}.^[16]^[1] The dimension of an isotropic subspace satisfies \dim W \leq n. This bound arises because \dim W + \dim W^\perp = 2n for any subspace W, and the inclusion W \subseteq W^\perp implies \dim W \leq \dim W^\perp, so \dim W \leq n.^[1]^[16] A subspace W is coisotropic if W^\perp \subseteq W, which is the dual condition to being isotropic and yields the complementary dimension bound \dim W \geq n. The notions of isotropic and coisotropic subspaces are interchanged under the orthogonal complement operation, as the complement of an isotropic subspace of dimension k \leq n is coisotropic of dimension $2n - k \geq n.^[16]^[1] In the standard symplectic vector space (\mathbb{R}^{2n}, \omega_0) where \omega_0 = \sum_{i=1}^n dx_i \wedge dy_i, the subspace of position coordinates spanned by \{ \partial/\partial x_1, \dots, \partial/\partial x_k \} for k \leq n is isotropic, as the symplectic form vanishes identically on it. Similarly, the momentum subspace spanned by \{ \partial/\partial y_1, \dots, \partial/\partial y_k \} for k \leq n is isotropic.^[7]^[1] The symplectic orthogonal complement operation is an involution, satisfying (W^\perp)^\perp = W for any subspace W \subseteq V, a consequence of the non-degeneracy of \omega. For a general subspace W, the space V decomposes as a direct sum V = W \oplus W^\perp if and only if W is a symplectic subspace (i.e., the restriction of \omega to W is non-degenerate, or equivalently W \cap W^\perp = \{0\}).^[1]^[16]

Lagrangian Subspaces

In a symplectic vector space (V, \omega) of even dimension $2nover\mathbb{R}, a **Lagrangian subspace** L \subseteq Vis defined as a subspace satisfyingL = L^\perp, where L^\perp = { v \in V \mid \omega(v, w) = 0 \ \forall w \in L }denotes the symplectic orthogonal complement ofL.[1] This condition implies that \omegavanishes identically onL \times L, making Lisotropic, while the equalityL = L^\perp$ ensures maximality.^[7] Equivalently, a Lagrangian subspace is characterized as an isotropic subspace of dimension exactly n, the midpoint of \dim V.^[1] Such subspaces exist in every symplectic vector space and play a central role in symplectic reductions and decompositions, as they represent the largest possible null sets for the symplectic form.^[7] Representative examples include the position subspace \{p = 0\} \subseteq \mathbb{R}^{2n} equipped with the standard symplectic form \omega = \sum_{i=1}^n dq_i \wedge dp_i, which has dimension n and satisfies \omega|_{L \times L} = 0.^[7] In the linear model of a cotangent bundle T^* \mathbb{R}^n \cong \mathbb{R}^{2n}, the zero section (corresponding to the position subspace) and the graph of a closed linear 1-form (i.e., a linear functional

\alpha: \mathbb{R}^n \to \mathbb{R}^n^*

such that d\alpha = 0, which is exact) are Lagrangian. Two Lagrangian subspaces L, M \subseteq V are said to be transverse if L \cap M = \{0\}, which is equivalent to L + M = V given their dimensions.^[7] In this case, V decomposes as a direct sum V = L \oplus M, providing a symplectic basis adapted to the pair (e.g., extending bases of L and M yields a Darboux basis for V).^[7] The Maslov index provides an invariant for paths of Lagrangian subspaces in the Lagrangian Grassmannian \Lambda(n), the space of n-dimensional subspaces of \mathbb{R}^{2n}. For a smooth path \Lambda: [a,b] \to \Lambda(n) relative to a fixed reference Lagrangian V, the linear Maslov index \mu(\Lambda, V) is the signed count of crossings with the codimension-1 strata of the Maslov cycle \Sigma(V), where \Sigma(V) consists of Lagrangians intersecting V nontrivially; regular crossings contribute \pm 1 based on the sign of the crossing form.^[17] This index, originally due to Maslov and interpreted geometrically by Arnold, detects topological changes along the path, such as dimension jumps in intersections with V.^[17]

Induced Structures

Volume Form

In a symplectic vector space (V, \omega) of dimension $2n, the symplectic form \omegainduces a natural volume form through itsn-th exterior power. Specifically, the volume element is given by \mathrm{vol} = \frac{\omega^n}{n!}

, which is a non-vanishing &#36;2n

-form on V.^[5] This construction arises because \omega is a closed, non-degenerate 2-form, and raising it to the top power yields a top-degree differential form that serves as the Liouville volume form, up to normalization by the factorial to align with standard conventions in coordinates.^[5] The non-degeneracy of \omega ensures that \omega^n \neq 0 everywhere on V, implying that \mathrm{vol} is indeed a volume form without zeros.^[5] In a Darboux basis adapted to \omega, this volume form corresponds to the standard Lebesgue measure on the underlying real vector space, providing a canonical way to integrate over V.^[5] For instance, on the standard symplectic space \mathbb{R}^{2n} with coordinates (q_1, \dots, q_n, p_1, \dots, p_n) and \omega = \sum_{i=1}^n \mathrm{d}q_i \wedge \mathrm{d}p_i, the induced volume form is \mathrm{vol} = \mathrm{d}q_1 \wedge \cdots \wedge \mathrm{d}q_n \wedge \mathrm{d}p_1 \wedge \cdots \wedge \mathrm{d}p_n, representing the phase space volume in classical mechanics.^[5] Symplectic linear maps, which preserve \omega, also preserve this volume form, as the pullback satisfies \phi^* \omega = \omega, hence \phi^* (\omega^n / n!) = \omega^n / n!.^[5] This volume preservation is the linear analog of Liouville's theorem, ensuring that the Lebesgue measure induced by \mathrm{vol} is invariant under the action of the symplectic group.^[5] Moreover, if A is a matrix representing a symplectic transformation, then \det(A) = 1, which directly follows from the preservation of \omega and confirms the volume-preserving property.^[5] The volume form \omega^n further defines a canonical orientation on V, compatible with the symplectic structure. In the standard realization, this orientation is specified by requiring that the Pfaffian \mathrm{Pf}(J) > 0, where J is the skew-symmetric matrix associated to \omega in an adapted basis; since \mathrm{Pf}(J)^2 = \det(J) = 1, the positive sign selects the standard orientation aligning with \omega^n.^[18] This ensures that \mathrm{vol} is positively oriented, providing a consistent choice across all symplectomorphic spaces.^[18]

Compatibility with Complex Structures

A compatible complex structure on a symplectic vector space (V, \omega) is a linear endomorphism J: V \to V satisfying J^2 = -\mathrm{Id}_V and \omega(Ju, Jv) = \omega(u, v) for all u, v \in V, ensuring J preserves the symplectic form.^[7] Additionally, J is called \omega-tamed if \omega(u, Ju) > 0 for all nonzero u \in V, which induces a positive definite inner product g(u, v) = \omega(u, Jv).^[19] This construction establishes an analogy between symplectic and complex structures: just as a complex structure equips V with multiplication by i, the pair (\omega, J) transforms \omega into the imaginary part of a Hermitian form h(u, v) = g(u, v) - i \omega(u, v), where h(Ju, v) = i h(u, v).^[20] The triple (V, \omega, J) with g positive definite forms a linear Kähler structure, analogous to Kähler manifolds but restricted to vector spaces.^[19] In coordinates on the standard symplectic space \mathbb{R}^{2n} with the canonical form \omega_0 = \sum_{k=1}^n dx_k [\wedge](/page/Wedge) dy_k, a compatible J acts as J(\partial_{x_k}) = \partial_{y_k} and J(\partial_{y_k}) = -\partial_{x_k}, identifying V \cong \mathbb{C}^n where \omega_0 corresponds to the imaginary part of the standard Hermitian inner product \langle z, w \rangle = \sum z_k \overline{w_k}.^[7] Compatible almost complex structures always exist on any symplectic vector space and can be chosen such that the induced metric g is diagonalized in a suitable basis; on vector spaces, such J defines a true complex structure without additional integrability conditions.^[20] Not every symplectic vector space is inherently Kähler, as this requires selecting a compatible J and verifying the positive definiteness of g; however, such structures exist on any even-dimensional symplectic space.^[19] This compatibility underscores linear parallels to Hodge theory, where symplectic forms interplay with complex structures to decompose spaces, though the emphasis here remains on finite-dimensional vector spaces without differential topology.^[7]

Associated Groups

Heisenberg Group

The Heisenberg group H_{2n+1} associated with a symplectic vector space (V, \omega) of dimension $2n over \mathbb{R} is the nilpotent Lie group with underlying manifold V \times \mathbb{R} and multiplication given by (x, t) \cdot (y, s) = (x + y, t + s + \frac{1}{2} \omega(x, y)) for x, y \in V and t, s \in \mathbb{R}.^[21] This structure makes H_{2n+1} a central extension $0 \to \mathbb{R} \to H_{2n+1} \to V \to 0, where the center is \{ (0, t) \mid t \in \mathbb{R} \} and the quotient group V is abelianized by the symplectic form \omega, which serves as the 2-cocycle in the extension.^[21] The group is simply connected and 2-step nilpotent, with the derived subgroup contained in the center.^[21] Irreducible unitary representations of H_{2n+1} are realized via the Schrödinger representation on the Hilbert space L^2(\mathbb{R}^n), where elements act as \pi(q, p, t) \psi(r) = e^{i t} e^{i p \cdot (r + q/2)} \psi(r + q) in adapted coordinates x = (q, p) with q, p \in \mathbb{R}^n, ensuring unitarity through the Plancherel theorem.^[21] Equivalent representations are related by the symplectic Fourier transform, which interchanges position and momentum variables while preserving the unitary structure.^[21] The Stone–von Neumann theorem in its linear form asserts that, up to unitary equivalence, there is a unique irreducible unitary representation of H_{2n+1} on a separable Hilbert space, corresponding to the Schrödinger model; this uniqueness holds for the canonical central character \chi(t) = e^{i t}.^[22] For n=1, the 3-dimensional Heisenberg group H_3 admits a faithful matrix representation as the group of upper-triangular $3 \times 3 matrices

\begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix},

with a, b, c \in \mathbb{R}, under matrix multiplication, where the symplectic form on \mathbb{R}^2 is \omega((a,b), (a',b')) = a b' - b a'.^[21] This realization arises from exponentiating the Lie algebra basis elements X = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, Y = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}, Z = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} satisfying [X, Y] = Z, [X, Z] = [Y, Z] = 0.^[21] In quantum mechanics, the Heisenberg group exponentiates the central extension of the symplectic Lie algebra \mathfrak{sp}(2, \mathbb{R}), capturing the canonical commutation relations [\hat{q}, \hat{p}] = i for position \hat{q} and momentum \hat{p} operators in the Schrödinger representation, thereby unifying algebraic and geometric aspects of phase space quantization.^[21]

Oscillator Group

The oscillator group associated to a symplectic vector space (V, \omega) of finite dimension $2n over \mathbb{R} is defined as the semi-direct product H(V) \rtimes \mathrm{Sp}(V, \omega), where H(V) is the Heisenberg group over V and \mathrm{Sp}(V, \omega) is the symplectic group preserving \omega. The Heisenberg group H(V) consists of elements (v, t) \in V \times \mathbb{R} with the group law (v, t) \cdot (v', t') = (v + v', t + t' + \frac{1}{2} \omega(v, v')), and \mathrm{Sp}(V, \omega) acts on H(V) by automorphisms via g \cdot (v, t) = (g v, t) for g \in \mathrm{Sp}(V, \omega), preserving the central \mathbb{R}-factor since \omega(gv, gv) = \omega(v, v).^[23] This structure captures the symmetries combining translations in phase space with linear canonical transformations, central to the dynamics of the classical harmonic oscillator on V.^[24] The Lie algebra of the oscillator group is the semi-direct product \mathfrak{h}(V) \rtimes \mathfrak{sp}(V, \omega), where \mathfrak{h}(V) is the Heisenberg Lie algebra with basis elements from V and a central generator Z satisfying [X, Y] = \omega(X, Y) Z for X, Y \in V, and \mathfrak{sp}(V, \omega) acts by derivations preserving this bracket. In the standard realization on \mathbb{R}^{2n} with the canonical symplectic form \omega = \sum_{i=1}^n dq_i \wedge dp_i, the oscillator group extends the Galilei group or Euclidean motions in higher dimensions, appearing in models of non-relativistic quantum mechanics. For n=1, it reduces to the 4-dimensional oscillator group generated by position Q, momentum P, Hamiltonian H, and central element E, with relations [Q, P] = E and [H, Q] = P, [H, P] = -Q.^[24]^[25] Irreducible unitary representations of the oscillator group are induced from characters of the Heisenberg subgroup, yielding the Segal–Shale–Weil (or oscillator) representation, which is projective on \mathrm{Sp}(V, \omega) and realized on L^2(\mathbb{R}^n) via the Schrödinger model with creation and annihilation operators a^\dagger, a satisfying [a, a^\dagger] = 1. This representation lifts to a true representation of the metaplectic group, the double cover of \mathrm{Sp}(V, \omega), and plays a key role in quantization, where the prequantum line bundle over the phase space V corresponds to the central extension. Seminal work established these representations for the infinite-dimensional case, with finite-dimensional analogs in oscillator models over finite fields or discrete phase spaces.^[26]^[27] The oscillator group thus bridges symplectic geometry with harmonic analysis, providing a framework for Stone–von Neumann type theorems on uniqueness of representations up to unitary equivalence.^[23]

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[PDF] SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
A subspace F ⊆ E of a symplectic vector space is called. (a) isotropic if F ⊆ Fω,. (b) co-isotropic if Fω ⊆ F. (c) Lagrangian if F = Fω,. (d) symplectic if F ∩ ...
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[PDF] symplectic geometry: lecture 1
Note that being symplectomorphic is an equivalence relation on vector spaces of finite dimension. The group of symplectomorphisms of (V,ω) is de- noted Sp(V ).
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[PDF] Lectures on Symplectic Geometry - UC Berkeley math
Apr 24, 2024 · In our first result we state a linear version of Darboux's theorem and some elementary facts about symplectic vector spaces. Darboux's ...
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[PDF] Early History of Symplectic Geometry
Currently, symplectic geometry refers to the study of symplectic manifolds. A symplectic manifold is an even dimensional manifold endowed with a closed.
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[PDF] Symplectic Geometry - Mathematics
4.6 Some History . ... In the sequel (E, σ) will be a symplectic vector space, i.e. E is a finite-dimensional vector space.
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[PDF] Symplectic Maps
Symplectic maps arise from Hamiltonian dynamics, because these preserve the loop action. Thus, for example, the time t map of any Hamiltonian flow is symplectic ...
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[PDF] Understanding reductive group representations
Sp(2n, R) = ng ∈ GL(2n, R) ω(g·v1, g·v2) = ω(v1, v2) (vi ∈ R2n)o. The symplectic group is a Lie group of dimension 2n2 + n. It is a great example of a reductive ...
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[PDF] Mathematical Methods in Physics - 231B – Group Theory –
Feb 26, 2019 · 2.4 The symplectic group Sp(2n) and Lie algebra sp(2n) . ... The group USp(2n) = Sp(2n;C)∩U(2n) is the maximal compact subgroup of Sp(2n;C) ...
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[PDF] Generators of the Symplectic Group
May 11, 2005 · The best way to picture a transvection is as a shear - you fix some hyperplane, and take the orthogonal vector to that hyperplane and apply a ...Missing: 2n+ | Show results with:2n+
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[PDF] 4 Symplectic groups
Proposition 4.3 (a) Sp 2n F is a subgroup of SL 2n F . (b) PSp 2n F 0 / Sp 2n F 1! I" . Proof (a) If P 2 Sp 2n F , then Pf A Pf PAP det P Pf A , so det P. 1.
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[PDF] notes on symplectic topology - UChicago Math
Mar 5, 2025 · 1.1. Symplectic vector spaces. Definition 1.1 (Symplectic vector space). A symplectic vector space is a real, finite di- mensional vector space ...
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[PDF] The Maslov index for paths
Maslov's famous index for a loop of Lagrangian subspaces was interpreted by Arnold [1] as an intersection number with an algebraic variety known as.
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[PDF] An Introduction to Lie Groups and Symplectic Geometry - CIMAT
Feb 19, 2003 · The Pfaffian. Let V be a vector space of dimension 2n. Fix a basis b ... (iii) Let (V,Ω) be a symplectic vector space and let G = Sp(V ...
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[PDF] 1. Kähler manifolds - UChicago Math
Nov 20, 2013 · Linear algebra. We describe three kinds of structure on a (real) vector space: a positive inner product, a complex structure, and a symplectic ...
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[PDF] lecture 1: linear symplectic geometry
Symplectic subspaces are of course important. However, in symplectic vector spaces there are many other types of vector subspaces that are even more important.
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[PDF] The Heisenberg group and its representations
This group is a central extension ... The subgroup of GL(V ) preserving S and taking symplectic bases to sym- plectic bases is by definition the symplectic group ...
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[PDF] A Selective History of the Stone-von Neumann Theorem - UMD MATH
Here Mp(2m, R), often called the metaplectic group, is the double cover of the symplectic group. Proof. If we combine the results of section 4 and subsection ...
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[PDF] The symplectic group and the oscillator representation
Since the definition of the Heisenberg Lie algebra and Lie group only depend on the antisymmetric bilinear form S on V = R2n, the group Sp(2n, R) of linear.
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The Representations of the Oscillator Group* - Project Euclid
One may define the elements h', pf , q', e' of G' (for the oscillator group) by their scalar products in a natural way, e.g.. (hf, hy = 1, (hf, p} = 0 etc. The ...
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[PDF] Representations of the Canonical group, (the semi-direct product of ...
The Oscillator group has the semi-direct product structure G = OsH1, 3L ... of the Oscillator group as well as the Weyl-Heisenberg group and so the stabilizer ...
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on the role of the heisenberg group in harmonic analysis
Reciprocally, the Plancherel Theorem serves to prove and strengthen a celebrated fact about the Heisenberg group, the Stone-von. Neumann Theorem, which lies at ...<|control11|><|separator|>
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The moment map on symplectic vector space and oscillator ... - arXiv
Aug 28, 2014 · The aim of this paper is to show that the canonical quantization of the moment maps on symplectic vector spaces naturally gives rise to the ...