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Symplectic group

In mathematics, the symplectic group is a classical Lie group consisting of all linear transformations of a finite-dimensional vector space that preserve a nondegenerate antisymmetric bilinear form, known as a symplectic form, defined on an even-dimensional space over a field such as the real or complex numbers.^[1] This form ensures the group acts as the automorphism group of a symplectic vector space, maintaining key geometric structures like volume and orientation.^[2] The standard notation for the real symplectic group is \mathrm{Sp}(2n, \mathbb{R}), where n is a positive integer and the dimension of the space is $2n; elements are $2n \times 2n matrices M satisfying M^T J M = J, with J the standard symplectic matrix (a block-diagonal form with identity and negative-identity blocks).^[1] These groups form a family of non-compact Lie groups with dimension n(2n + 1), and their Lie algebras \mathfrak{sp}(2n, \mathbb{R}) consist of matrices X such that X^T J + J X = 0.^[3] For finite fields \mathbb{F}_q, the order of \mathrm{Sp}(2n, q) is given by q^{n^2} \prod_{i=1}^n (q^{2i} - 1), highlighting their role in finite geometry and group theory.^[2] Symplectic groups are fundamental in both pure mathematics and theoretical physics; the term "symplectic" was coined by Hermann Weyl in 1939 to describe these groups, replacing earlier confusing nomenclature like "Abelian linear group."^[4] In physics, they underpin Hamiltonian mechanics by preserving the Poisson bracket structure on phase space and the canonical commutation relations in quantum mechanics via the metaplectic representation, a double cover of the group.^[3] They also appear in optics for modeling ray transformations and in symplectic geometry, where they extend to infinite-dimensional settings and study properties like simplicity (projective versions are simple for most cases except small dimensions over small fields).^[2]

Fundamentals

Definition

In mathematics, a symplectic vector space is a finite-dimensional vector space V over a field F (of characteristic not 2) equipped with a symplectic form \omega: V \times V \to F, which is a bilinear map satisfying two key properties: it is alternating, meaning \omega(v, v) = 0 for all v \in V, and non-degenerate, meaning that if \omega(u, v) = 0 for all v \in V, then u = 0.^[5] Such a form induces a pairing that pairs V with its dual in a perfect manner, and non-degeneracy ensures the space has even dimension $2n for some positive integer n, as skew-symmetric matrices (representing the form in a basis) are invertible only in even dimensions over fields of characteristic not 2. The symplectic group \mathrm{Sp}(2n, F) is defined as the group of all linear automorphisms of V that preserve the symplectic form \omega, i.e., the set of invertible linear maps g: V \to V such that \omega(g(u), g(v)) = \omega(u, v) for all u, v \in V.^[5] This group acts as the automorphism group of the symplectic vector space and consists precisely of those transformations that leave the geometric structure defined by \omega invariant. In a standard basis adapted to \omega, elements of \mathrm{Sp}(2n, F) correspond to $2n \times 2n matrices M over F satisfying M^\top J M = J, where J is the block-diagonal matrix

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

with I_n the n \times n identity matrix; this condition ensures preservation of the form represented by J. As a subgroup of the general linear group \mathrm{GL}(2n, F), \mathrm{Sp}(2n, F) inherits the group operation of matrix multiplication and inversion, and its elements have determinant 1, i.e., \det(M) = 1 for all M \in \mathrm{Sp}(2n, F), a consequence of the preservation condition implying \det(M)^2 = 1 and the group's inclusion in the special linear group.^[5] This determinant property underscores the volume-preserving nature of symplectic transformations, though the full ramifications of the group's structure are explored further in specialized contexts.

Historical development

The origins of the symplectic group trace back to the early 19th century in the context of classical mechanics, where foundational concepts emerged from efforts to describe the dynamics of physical systems. In 1808, Joseph-Louis Lagrange introduced a structure on the manifold of planetary motions that preserved certain variational principles, laying the groundwork for what would later be recognized as the symplectic form through his "Lagrange parentheses," which are the components of the canonical symplectic 2-form. This was extended in 1809 when Siméon Denis Poisson developed the Poisson bracket in his treatise on celestial mechanics, providing a composition law on functions that encodes the symplectic structure and facilitates the discovery of integrals of motion. These developments connected to the phase space formalism later formalized by William Rowan Hamilton in the 1830s, where the symplectic form arises naturally from the Poisson brackets governing Hamiltonian flows. By the late 19th century, the algebraic aspects of these structures began to crystallize through group-theoretic investigations. Sophus Lie, in his 1869 work on line geometry and subsequent studies in the 1870s–1880s, identified groups preserving certain bilinear forms, including what are now known as symplectic groups, arising from contact transformations and line complexes originally studied by Julius Plücker in the 1860s. Wilhelm Killing's classification of simple Lie algebras from 1888 to 1890 placed these groups within the broader Cartan-Killing scheme as the type C_n series, recognizing their role among the classical simple Lie groups. Élie Cartan refined this classification in the early 20th century, particularly through his 1894 thesis and 1913 work on Lie groups, while also developing the theory of exterior differential forms in 1899–1901, which provided tools for handling the antisymmetric bilinear forms central to symplectic structures. The explicit matrix formulation and nomenclature of symplectic groups solidified in the 1930s. Hermann Weyl's 1939 monograph on classical groups systematically treated the symplectic series as one of the four main types (alongside unitary, orthogonal, and linear), introducing the term "symplectic" derived from the Greek symplektikos, meaning "plaited together" or "interwoven," to replace earlier confusing designations like "complex group" or "Abelian linear group" used by Leonard Eugene Dickson. This work highlighted their preservation of a non-degenerate antisymmetric bilinear form, building on Élie Cartan's Lie theory contributions from the 1920s, including his 1926–1928 lectures on differential forms and their applications to mechanics. In the mid-20th century, these groups were firmly established as simple Lie groups of type C_n in the Cartan-Killing classification, with Cartan's 1935–1937 Paris lectures further integrating them into the theory of continuous groups. Post-1950 developments extended symplectic groups into differential geometry, with infinitesimal versions emerging in studies of symplectic manifolds during the 1950s, such as Jean-Marie Souriau's 1953 work on Lagrangian submanifolds. A notable post-1960s advancement was the metaplectic representation, introduced by André Weil in 1964, which provides an infinite-dimensional unitary representation of the double cover of the symplectic group, linking it to theta functions and modular forms. These extensions built on the 1930s matrix realizations, emphasizing the groups' role in preserving symplectic forms across geometric and analytic contexts.

Symplectic groups over fields

General case over fields F

The symplectic group over a field F of characteristic not equal to 2 is constructed on a vector space V of even dimension $2n equipped with a non-degenerate alternating bilinear form \omega: V \times V \to F, meaning \omega is bilinear, \omega(v, v) = 0 for all v \in V, \omega(u, v) = -\omega(v, u) for all u, v \in V, and the radical \{v \in V \mid \omega(v, w) = 0 \ \forall w \in V\} is trivial.^[2] The group \mathrm{Sp}(2n, F) consists of all linear automorphisms of V that preserve \omega, i.e., \omega(Au, Av) = \omega(u, v) for all u, v \in V and A \in \mathrm{Sp}(2n, F).^[2] Such a symplectic vector space (V, \omega) admits a symplectic basis \{e_1, \dots, e_n, f_1, \dots, f_n\} where \omega(e_i, f_j) = \delta_{ij}, \omega(e_i, e_j) = \omega(f_i, f_j) = 0 for all i, j.^[6] In this basis, the Gram matrix of \omega takes the standard skew-symmetric form J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, and \mathrm{Sp}(2n, F) is realized as the subgroup of \mathrm{GL}(2n, F) consisting of matrices A satisfying A^T J A = J.^[2] When F is a finite field \mathbb{F}_q with q elements and characteristic not 2, the order of \mathrm{Sp}(2n, q) is q^{n^2} \prod_{i=1}^n (q^{2i} - 1).^[7] The group \mathrm{Sp}(2n, F) is generated by symplectic transvections, which are unipotent elements of the form T_v(w) = w + \omega(v, w) u for fixed nonzero u, v \in V with \omega(u, v) = 0.^[2] For n \geq 2, the projective symplectic group \mathrm{PSp}(2n, F) = \mathrm{Sp}(2n, F) / Z(\mathrm{Sp}(2n, F)) is simple, except in the cases \mathrm{PSp}(2, q) \cong \mathrm{PSL}(2, q) (which is simple for q \geq 4) and \mathrm{PSp}(4, 2) \cong A_6, \mathrm{PSp}(4, 3) \cong \mathrm{PSp}(4, 3)' (a simple group of order 25920).^[6] The center of \mathrm{Sp}(2n, F) is \{\pm I_{2n}\} when characteristic not 2.^[2] In characteristic 2, the notion of alternating bilinear form coincides with symmetric bilinear forms satisfying \omega(v, v) = 0 for all v \in V, as the antisymmetry condition becomes symmetry.^[8] The symplectic group \mathrm{Sp}(2n, F) is defined analogously as the automorphism group preserving a non-degenerate such form on a $2n-dimensional space V, with non-degeneracy ensuring the radical is zero; symplectic bases exist under these conditions, though the standard matrix J simplifies to \begin{pmatrix} 0 & I_n \\ I_n & 0 \end{pmatrix} since -1 = 1.^[6] In this case, the center is trivial (just the identity), and the group exhibits modified behavior, such as all involutions being totally degenerate, with exceptional isomorphisms or automorphisms possible for small n (e.g., n=2 over perfect fields of characteristic 2).^[6] For finite fields of characteristic 2, the order formula remains q^{n^2} \prod_{i=1}^n (q^{2i} - 1), and \mathrm{PSp}(2n, q) is simple for n \geq 3, with exceptions for small dimensions like \mathrm{PSp}(4, 2).^[7] All non-degenerate symplectic spaces of fixed even dimension $2n over a given field F are isomorphic, regardless of characteristic, ensuring that \mathrm{Sp}(2n, F) is uniquely determined up to isomorphism by n and F.^[6]

Complex case Sp(2n, C)

The symplectic group \mathrm{Sp}(2n, \mathbb{C}) consists of all $2n \times 2n complex matrices A such that A^T J A = J, where J is the standard skew-symmetric matrix with blocks of the identity, preserving a nondegenerate alternating bilinear form on \mathbb{C}^{2n}.^[9] As a complex Lie group, \mathrm{Sp}(2n, \mathbb{C}) is semisimple of type C_n in Cartan's classification, with Lie algebra \mathfrak{sp}(2n, \mathbb{C}) comprising matrices X satisfying X^T J + J X = 0.^[10] The dimension of \mathrm{Sp}(2n, \mathbb{C}), as a complex manifold, is n(2n + 1), matching that of its Lie algebra.^[11] \mathrm{Sp}(2n, \mathbb{C}) forms an algebraic group over \mathbb{C}, defined by polynomial equations in the matrix entries, and it is simply connected, serving as the universal cover of its quotients.^[12] Its maximal compact subgroup is \mathrm{Sp}(n), the compact real form also known as the quaternionic unitary group, which embeds densely and preserves the same symplectic structure in a compatible real sense.^[13] This compact subgroup has the same real dimension n(2n + 1) and plays a key role in the Cartan decomposition of the group. The fundamental representation of \mathrm{Sp}(2n, \mathbb{C}) acts on the standard module \mathbb{C}^{2n} by matrix multiplication, preserving the symplectic form, and this representation is irreducible for n \geq 1.^[14] \mathrm{Sp}(2n, \mathbb{C}) sits as a closed subgroup of \mathrm{SL}(2n, \mathbb{C}), consisting precisely of those special linear transformations that preserve the symplectic form, though this embedding is not normal.^[9] For the case n=1, \mathrm{Sp}(2, \mathbb{C}) is isomorphic to \mathrm{SL}(2, \mathbb{C}), as the symplectic condition reduces to the special linear condition in dimension 2.^[10]

Real case Sp(2n, R)

The real symplectic group \mathrm{Sp}(2n, \mathbb{R}) consists of all $2n \times 2n real matrices M satisfying M^\top J M = J, where J is the standard skew-symmetric symplectic matrix with blocks of the form \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, and \det M = 1.^[15] As a real Lie group, it is non-compact for all n \geq 1 and has dimension n(2n+1).^[15] Topologically, \mathrm{Sp}(2n, \mathbb{R}) is connected and path-connected, with a single connected component for all n \geq 1.^[16] Its fundamental group is isomorphic to \mathbb{Z}, matching that of its maximal compact subgroup.^[17] The maximal compact subgroup of \mathrm{Sp}(2n, \mathbb{R}) is the unitary group U(n).^[15] This leads to the Iwasawa decomposition \mathrm{Sp}(2n, \mathbb{R}) = K A N, where K = U(n), A is the vector subgroup of diagonal matrices with entries e^{t_i} and e^{-t_i} for t_i \in \mathbb{R}, and N is the unipotent subgroup of upper triangular matrices with ones on the diagonal.^[15] The quotient \mathrm{Sp}(2n, \mathbb{R}) / U(n) is contractible, making \mathrm{Sp}(2n, \mathbb{R}) homotopy equivalent to U(n).^[17] For n=1, \mathrm{Sp}(2, \mathbb{R}) is isomorphic to the special linear group \mathrm{SL}(2, \mathbb{R}).^[18] In this case, and more generally for elements in \mathrm{Sp}(2n, \mathbb{R}), conjugacy classes of semisimple elements are classified as elliptic (with eigenvalues on the unit circle, corresponding to compact rotations), hyperbolic (with real eigenvalues \lambda, 1/\lambda > 0, \lambda \neq 1), or parabolic (with eigenvalue 1 and nontrivial Jordan blocks).^[19]

Lie algebra and structure

Lie algebra sp(2n, F)

The Lie algebra \mathfrak{sp}(2n, F) of the symplectic group \mathrm{Sp}(2n, F) over a field F consists of all $2n \times 2n matrices X \in \mathfrak{gl}(2n, F) satisfying the condition X^T J + J X = 0, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} is the standard skew-symmetric matrix with I_n the n \times n identity.^[20] This defining relation captures the infinitesimal preservation of the symplectic form associated to J. The dimension of \mathfrak{sp}(2n, F) is n(2n + 1), computed from the block structure of such matrices, where the upper-left block contributes n^2 parameters (with the lower-right block determined as the negative transpose of the upper-left), the upper-right block n(n+1)/2 (symmetric), and the lower-left block another n(n+1)/2 (symmetric).^[20] Over fields F of characteristic not equal to 2 or 3, \mathfrak{sp}(2n, F) is semisimple, with a non-degenerate Killing form B(X, Y) = \mathrm{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y), which is the standard invariant bilinear form on the algebra.^[20] It is in fact simple, possessing no nontrivial ideals, and admits a root space decomposition relative to a Cartan subalgebra \mathfrak{h} of dimension n (the rank of the algebra), consisting of block-diagonal matrices \mathrm{diag}(a_1, \dots, a_n, -a_1, \dots, -a_n) with a_i \in F.^[20] The root system is of type C_n, irreducible and reduced, with simple roots \alpha_i = \epsilon_i - \epsilon_{i+1} for $1 \leq i < n and \alpha_n = 2\epsilon_n, where \{\epsilon_1, \dots, \epsilon_n\} is the standard basis for the dual space \mathfrak{h}^*; the full set of roots comprises short roots \pm 2\epsilon_i and long roots \pm \epsilon_i \pm \epsilon_j for i < j.^[20] A Chevalley basis for \mathfrak{sp}(2n, F) can be constructed using root vectors e_\alpha and f_\alpha = -e_{-\alpha} for each positive root \alpha, together with coroots h_i corresponding to the simple roots \alpha_i, satisfying the Serre relations [h_i, e_{\alpha_j}] = A_{ij} e_{\alpha_j} (where A is the Cartan matrix of type C_n) and similar for the f_{\alpha_j}, with [e_{\alpha_i}, f_{\alpha_i}] = h_i.^[20] The structure constants are integers independent of F, allowing this basis to generate the algebra over \mathbb{Z}. For simply connected Lie groups with this Lie algebra, such as the complex symplectic group \mathrm{Sp}(2n, \mathbb{C}), the exponential map \exp: \mathfrak{sp}(2n, \mathbb{C}) \to \mathrm{Sp}(2n, \mathbb{C}) provides a local diffeomorphism at the identity, parametrizing elements near the origin via the Lie algebra.^[20]

Infinitesimal generators

The infinitesimal generators of the symplectic group \mathrm{Sp}(2n, F) are the elements of its Lie algebra \mathfrak{sp}(2n, F), which consist of $2n \times 2n matrices X over the field F satisfying the condition X^T J + J X = 0, 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.^[14] This condition arises by considering a smooth curve M(t) in \mathrm{Sp}(2n, F) with M(0) = I and differentiating the defining relation M(t)^T J M(t) = J at t = 0, yielding X = M'(0) as the tangent vector at the identity.^[21] In block form, with respect to the decomposition \mathbb{F}^{2n} = \mathbb{F}^n \oplus \mathbb{F}^n, the elements of \mathfrak{sp}(2n, F) take the structure

X = \begin{pmatrix} A & B \\ C & -A^T \end{pmatrix},

where A \in \mathfrak{gl}(n, F), B = B^T, and C = C^T are n \times n symmetric matrices.^[14] Over the real numbers, these matrices are known as Hamiltonian matrices, as they generate flows preserving the symplectic form.^[22] A basis for \mathfrak{sp}(2n, F) can be constructed from symplectic transvections, which are nilpotent matrices corresponding to root vectors for the non-compact directions, and rotations, which span the compact subalgebra.^[23] In the real case \mathfrak{sp}(2n, \mathbb{R}), the maximal compact subalgebra is isomorphic to \mathfrak{u}(n), consisting of generators of the form \begin{pmatrix} A & B \\ -B & A \end{pmatrix} where A is skew-Hermitian and B is symmetric, embedding \mathbb{R}^{2n} as \mathbb{C}^n with the standard complex structure.^[24] For the low-dimensional case n=1, the Lie algebra \mathfrak{sp}(2, \mathbb{R}) \cong \mathfrak{sl}(2, \mathbb{R}) is three-dimensional and generated by the basis

H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},

satisfying the relations [H, X] = 2X, [H, Y] = -2Y, and [X, Y] = H.^[14]

Representations and examples

Matrix realizations

The symplectic group \mathrm{Sp}(2n, F) over a field F (of characteristic not 2) admits a standard realization as the subgroup of the general linear group \mathrm{GL}(2n, F) consisting of all $2n \times 2n matrices M that preserve the standard symplectic form, defined by the non-degenerate skew-symmetric matrix J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, where I_n is the n \times n identity matrix.^[6] Specifically, M \in \mathrm{Sp}(2n, F) if and only if M^\top J M = J.^[6] This embedding captures the group's action on the vector space F^{2n} equipped with the standard alternating bilinear form \omega(u, v) = u^\top J v.^[15] A distinguished generating set for \mathrm{Sp}(2n, F) consists of the elementary symplectic matrices, which are transvections of the form s_{ij}(a) = I_{2n} + a e_{ij} for a \in F, where the indices i, j are chosen such that the addition preserves the symplectic structure (typically, i and j lie in complementary positions with respect to the block structure of J).^[25] These include shear-type matrices, such as block-upper-triangular forms like \begin{pmatrix} I_n & B \\ 0 & I_n \end{pmatrix} or \begin{pmatrix} I_n & 0 \\ C & I_n \end{pmatrix} with B, C arbitrary n \times n matrices over F, which generate the full group over Euclidean domains or fields.^[25] Over such fields, the elementary symplectic subgroup equals the full symplectic group, providing a constructive way to express any element as a product of these generators.^[6] Any non-degenerate symplectic form on a $2n-dimensional vector space over F is equivalent to the standard form via a change of basis; that is, there exists P \in \mathrm{GL}(2n, F) such that P^\top J P = \tilde{J}, where \tilde{J} represents the given form, ensuring all realizations are conjugate within \mathrm{GL}(2n, F).^[6] This equivalence implies that the matrix realization is canonical up to basis choice. To verify computationally whether a given $2n \times 2n matrix M over F belongs to \mathrm{Sp}(2n, F), one computes the matrix product M^\top J M and checks equality with J; this requires O(n^3) arithmetic operations via standard matrix multiplication algorithms.^[15] In numerical settings over \mathbb{R} or \mathbb{C}, a tolerance threshold is applied to account for floating-point errors, such as \|M^\top J M - J\| < \epsilon for a small \epsilon > 0.^[26] The preservation of a symplectic form by M is intrinsically tied to its relation with quadratic forms: since the symplectic form is alternating (\omega(v, v) = 0 for all v), its preservation ensures that M maps isotropic subspaces to isotropic subspaces, distinguishing it from preservers of non-degenerate quadratic forms (as in orthogonal groups), though in characteristic 2 the distinction blurs as alternating forms derive from quadratic residues.^[6]

Low-dimensional examples

The symplectic group in the lowest dimension, \mathrm{Sp}(2, \mathbb{R}), consists of $2 \times 2 real matrices of the form

\begin{pmatrix} a & b \\ c & d \end{pmatrix}

with ad - bc = 1. This group is isomorphic to the special linear group \mathrm{SL}(2, \mathbb{R}), as the condition of preserving the standard symplectic form on \mathbb{R}^2 coincides exactly with the determinant-one condition for $2 \times 2 matrices.^[18]^[17] These matrices represent area-preserving linear transformations of the plane, which maintain the oriented area of parallelograms under the action on position-momentum coordinates in phase space.^[27] Over the complex numbers, \mathrm{Sp}(2, \mathbb{C}) is likewise isomorphic to \mathrm{SL}(2, \mathbb{C}), where the symplectic preservation condition again reduces to the determinant being unity.^[28]^[29] The group acts on the complex plane via Möbius transformations z \mapsto \frac{az + b}{cz + d}, which preserve the symplectic form inherited from the real case when viewing \mathbb{C} as \mathbb{R}^2.^[30] For the next dimension, n=2, the group \mathrm{Sp}(4, \mathbb{R}) comprises $4 \times 4 real matrices that preserve the standard symplectic form on \mathbb{R}^4, often interpreted as the phase space for two degrees of freedom. An explicit example is the symplectic rotation matrix corresponding to independent rotations in each pair of conjugate coordinates (position and momentum), given by the block-diagonal form

\begin{pmatrix} \cos \theta & \sin \theta & 0 & 0 \\ -\sin \theta & \cos \theta & 0 & 0 \\ 0 & 0 & \cos \phi & \sin \phi \\ 0 & 0 & -\sin \phi & \cos \phi \end{pmatrix},

which generates rotations in the phase space while conserving the symplectic structure.^[31] This illustrates how elements of \mathrm{Sp}(4, \mathbb{R}) can describe coupled or uncoupled Hamiltonian evolutions in multi-particle systems. Notably, \mathrm{Sp}(4, \mathbb{R}) admits exceptional isomorphisms, being locally isomorphic to the Lorentz group \mathrm{SO}(2,3), which highlights its role in connecting symplectic geometry to indefinite orthogonal transformations in five-dimensional spacetime.^[32]^[33] The action of \mathrm{Sp}(2, \mathbb{R}) on \mathbb{R}^2 can be visualized as the linear realization of Hamiltonian flows generated by quadratic functions on the phase space, where group elements correspond to time evolutions under such Hamiltonians, preserving volumes and symplectic areas along integral curves.^[34]^[35]

Subgroups and relations

Important subgroups

The maximal compact subgroup of the symplectic group \mathrm{Sp}(2n, \mathbb{R}) is the unitary group \mathrm{U}(n), which consists of complex matrices preserving both the Hermitian form and the symplectic structure when viewing \mathbb{R}^{2n} as \mathbb{C}^n.^[18] Similarly, for the complex symplectic group \mathrm{Sp}(2n, \mathbb{C}), the maximal compact subgroup is \mathrm{Sp}(n), defined as the intersection \mathrm{U}(2n, \mathbb{C}) \cap \mathrm{Sp}(2n, \mathbb{C}), which is compact and simply connected.^[36] Parabolic subgroups of symplectic groups are stabilizers of isotropic flags in the underlying vector space, generalizing the block upper triangular structure while preserving the symplectic form.^[37] The Borel subgroup, a minimal parabolic, corresponds to the stabilizer of a complete isotropic flag and consists of upper triangular matrices in a suitable basis adapted to the symplectic form.^[38] Among the maximal parabolic subgroups, the Lagrangian subgroups are the stabilizers of maximal isotropic subspaces of dimension n, which play a central role in the geometry of the Lagrangian Grassmannian.^[39] Principal \mathrm{SL}(2) embeddings in \mathrm{Sp}(2n, F) refer to irreducible homomorphisms \iota: \mathrm{SL}(2, F) \to \mathrm{Sp}(2n, F) such that the fundamental $2n-dimensional representation of \mathrm{Sp}(2n, F) restricts to the irreducible representation of \mathrm{SL}(2, F) of dimension $2n.^[40] These embeddings are unique up to conjugation and are used in the study of representations and branching rules for symplectic groups.^[40] Over finite fields \mathbb{F}_q with q odd, the symplectic group \mathrm{Sp}(2n, \mathbb{F}_q) admits finite subgroups classified up to conjugation, including irreducible representations of smaller classical groups and extraspecial groups analogous to Gaussian integer structures in the complex case.^[41] For instance, subgroups generated by elements of prime order p \geq 5 that satisfy specific intersection conditions characterize the symplectic groups themselves among finite groups.^[42]

Connections to other classical groups

The Lie algebra \mathfrak{sp}(2n, \mathbb{C}) of type C_n is the Langlands dual to the orthogonal Lie algebra \mathfrak{so}(2n+1, \mathbb{C}) of type B_n, where the duality interchanges the root datum such that roots of one become coroots of the other.^[43] This correspondence arises in the classification of simple Lie algebras and plays a key role in the endoscopic classification of representations for classical groups.^[44] The special linear group \mathrm{SL}(n, F) embeds into the symplectic group \mathrm{Sp}(2n, F) over a field F (of characteristic not 2) via the action on the space V \oplus V^*, where V is the standard n-dimensional representation of \mathrm{SL}(n, F) and V^* is its dual, equipped with the natural evaluation pairing as the symplectic form. The explicit embedding sends g \in \mathrm{SL}(n, F) to the block-diagonal matrix \begin{pmatrix} g & 0 \\ 0 & (g^t)^{-1} \end{pmatrix}, which preserves the symplectic form because \det g = 1. Over the complex numbers, the unitary group \mathrm{U}(n) embeds into the compact symplectic group \mathrm{Sp}(n) (also denoted \mathrm{USp}(2n)), which is the intersection \mathrm{Sp}(2n, \mathbb{C}) \cap \mathrm{U}(2n), via a similar block construction involving conjugate transposes.^[44] Over finite fields \mathbb{F}_q with q odd, the symplectic group \mathrm{Sp}(2n, q) is a Chevalley group of type C_n, fitting into the uniform framework of simple groups of Lie type alongside the linear \mathrm{[SL](/page/SL)}(n, q), unitary \mathrm{[SU](/page/SU)}(n, q), and orthogonal groups.^[44] These groups share a BN-pair structure, which governs their parabolic subgroups and relates to the geometry of associated Tits buildings, unifying the incidence structures across classical types.^[44] In low dimensions, specific isomorphisms highlight these relations, such as \mathrm{[PSp](/page/PSP)}(4, q) \cong \mathrm{PO}(5, q) for the projective versions over \mathbb{F}_q.^[44] In infinite dimensions, symplectic groups extend to analogs within loop groups and affine Kac-Moody algebras of type C_n^{(1)}, where the finite-dimensional \mathrm{Sp}(2n) embeds as constant loops, facilitating studies in conformal field theory and integrable systems.^[45] The projective symplectic group \mathrm{PSp}(2n, F) = \mathrm{Sp}(2n, F)/Z(\mathrm{Sp}(2n, F)) is the quotient by the center \{\pm I\} (for \mathrm{char} F \neq 2), yielding a simple group that parallels the projective special linear and orthogonal groups in their simplicity properties for n \geq 2.^[44]

Applications

Symplectic geometry

In symplectic geometry, a symplectomorphism is a diffeomorphism \phi: (M, \omega) \to (M, \omega) between symplectic manifolds that preserves the symplectic form, satisfying \phi^* \omega = \omega.^[46] These maps form the symplectomorphism group \mathrm{Symp}(M, \omega), which generalizes the linear symplectic group \mathrm{Sp}(2n, \mathbb{R}) acting on \mathbb{R}^{2n} with the standard symplectic form \omega_0 = \sum_{i=1}^n dq_i \wedge dp_i.^[39] In the linear case, elements of \mathrm{Sp}(2n, \mathbb{R}) are precisely the linear symplectomorphisms of (\mathbb{R}^{2n}, \omega_0), preserving the symplectic structure globally.^[47] The Darboux theorem establishes that every symplectic manifold (M, \omega) of dimension $2nadmits local symplectic coordinates(q_1, \dots, q_n, p_1, \dots, p_n)around any point, such that\omega = \omega_0in this chart, implying that symplectic manifolds have no local invariants beyond their dimension. This local normal form underscores the role of symplectic groups in modeling transformations that maintain the geometric structure, as any two such coordinate neighborhoods are symplectomorphic via an element of\mathrm{Sp}(2n, \mathbb{R})$.^[48] The Hamiltonian subgroup \mathrm{Ham}(M, \omega) \subset \mathrm{Symp}(M, \omega) consists of time-1 maps of flows generated by Hamiltonian vector fields X_H, defined by i_{X_H} \omega = -dH for smooth functions H: M \to \mathbb{R}.^[39] On (\mathbb{R}^{2n}, \omega_0), the subgroup generated by these Hamiltonian flows is dense in \mathrm{Symp}(\mathbb{R}^{2n}, \omega_0) with respect to the compact-open topology, reflecting the flexibility of Hamiltonian perturbations in approximating arbitrary symplectomorphisms.^[47] Moser's theorem provides a stability result: on a compact manifold M, if two symplectic forms \omega_0 and \omega_1 lie in the same de Rham cohomology class [\omega_0] = [\omega_1] \in H^2(M, \mathbb{R}), then there exists a diffeomorphism \phi isotopic to the identity such that \phi^* \omega_1 = \omega_0.^[49] This isotopy highlights the rigidity within fixed cohomology classes, with the proof relying on a homotopy connecting the forms via a time-dependent vector field.^[48] Symplectic groups extend to contact geometry through symplectization, where a contact manifold (Y, \xi) lifts to a symplectic manifold (Y \times \mathbb{R}, d(e^t \alpha)) with contact form \alpha defining \xi = \ker \alpha, and symplectomorphisms preserving the contact structure descend from actions of \mathrm{Sp}(2n, \mathbb{R}) on the transverse directions.^[46] In modern symplectic topology, the symplectic group framework underpins results like Gromov's nonsqueezing theorem, which states that no symplectomorphism of the standard symplectic \mathbb{R}^{2n} maps the ball B^{2n}(R) of radius R into the cylinder Z^{2n}(r) = B^2(r) \times \mathbb{R}^{2n-2} if r < R, demonstrating global rigidity phenomena beyond local Darboux normal forms.

Classical mechanics

In classical mechanics, the phase space of a system with n degrees of freedom is the cotangent bundle T^*Q of the configuration space Q, diffeomorphic to \mathbb{R}^{2n} with coordinates (q_1, \dots, q_n, p_1, \dots, p_n), where q_i are generalized positions and p_i are conjugate momenta. This space carries a natural symplectic structure defined by the canonical 2-form \omega = \sum_{i=1}^n \mathrm{d}q_i \wedge \mathrm{d}p_i, which encodes the geometric properties essential for Hamiltonian dynamics. The real symplectic group \mathrm{Sp}(2n, \mathbb{R}) acts linearly on \mathbb{R}^{2n} by preserving \omega, meaning transformations M satisfy M^\top J M = J, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} is the standard symplectic matrix; this action maintains the pairing between positions and momenta, ensuring the symplectic form remains invariant under group elements.^[50] Canonical transformations are diffeomorphisms \phi: T^*Q \to T^*Q that preserve the symplectic form, i.e., \phi^* \omega = \omega, and thus belong to the symplectomorphism group containing \mathrm{Sp}(2n, \mathbb{R}) as its linear component. These transformations map Hamilton's equations \dot{z} = J \nabla H(z) (with z = (q, p)) to an equivalent form under new coordinates, preserving the dynamical structure. In particular, the time evolution under a Hamiltonian H is generated by the flow of the Hamiltonian vector field X_H, satisfying \iota_{X_H} \omega = -\mathrm{d}H, which integrates to a one-parameter subgroup of symplectomorphisms; for linear systems, these flows lie in \mathrm{Sp}(2n, \mathbb{R}).^[51] The Poisson bracket provides the Lie algebra structure on smooth functions over phase space, defined canonically as

\{f, g\} = \sum_{i=1}^n \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right) = \omega(X_f, X_g),

where X_f is the Hamiltonian vector field of f. Elements of \mathrm{Sp}(2n, \mathbb{R}), being linear symplectomorphisms, preserve this bracket: for \phi \in \mathrm{Sp}(2n, \mathbb{R}), \{\phi^* f, \phi^* g\} = \phi^* \{f, g\}, which ensures that the fundamental commutation relations \{q_i, p_j\} = \delta_{ij}, \{q_i, q_j\} = \{p_i, p_j\} = 0 remain unchanged, underpinning the consistency of canonical quantization in the classical limit.^[50] Liouville's theorem asserts that Hamiltonian flows preserve the Liouville volume measure \mu = \frac{\omega^n}{n!} on phase space, implying that the Jacobian determinant of the flow map is unity and incompressible evolution occurs. For linear Hamiltonian systems, flows in \mathrm{Sp}(2n, \mathbb{R}) directly enforce this volume preservation, as the group preserves the Pfaffian \mathrm{Pf}(\omega) = 1, equivalent to \det(M) = 1 for M \in \mathrm{Sp}(2n, \mathbb{R}); this result extends to nonlinear cases via local linearization and is foundational for ergodic theory in mechanics, where it guarantees long-term statistical stability without referencing ergodicity explicitly.^[50] Noether's theorem links continuous symmetries of the Hamiltonian to conserved quantities: if the dynamics is invariant under the infinitesimal action of a one-parameter subgroup of \mathrm{Sp}(2n, \mathbb{R}), generated by a Hamiltonian vector field X_\xi with \mathcal{L}_{X_\xi} H = 0, then the momentum map component J_\xi = \{H, \xi\} (or more generally the contraction \iota_{X_\xi} \omega) is conserved along trajectories. For finite-dimensional Lie group actions preserving the symplectic structure, the full momentum map J: T^*Q \to \mathfrak{g}^* yields a set of conserved quantities in involution under the Poisson bracket when the action is Hamiltonian, facilitating reduction of the phase space and integrability; this applies, for example, to rotational symmetries in central force problems, where the angular momentum components arise from the \mathrm{SO}(3) \subset \mathrm{Sp}(2n, \mathbb{R}) action.^[51]

Quantum mechanics

In quantum mechanics, the symplectic group \mathrm{Sp}(2n, \mathbb{R}) plays a central role through its double cover, the metaplectic group \mathrm{Mp}(2n, \mathbb{R}), which provides unitary representations essential for quantizing classical phase spaces. The metaplectic group is a two-to-one covering of \mathrm{Sp}(2n, \mathbb{R}), ensuring that the representation is projective on the symplectic group but lifts to a true representation on the cover; this structure arises naturally in the quantization of Hamiltonian systems where the classical Poisson bracket becomes the quantum commutator. Seminal work by André Weil established the metaplectic representation, which acts unitarily on the Hilbert space L^2(\mathbb{R}^n) via the Schrödinger representation, intertwining differential operators with multiplication operators under symplectic transformations.^[52] This representation preserves the canonical commutation relations and underpins the transition from classical symplectic geometry to quantum operator algebras. The Heisenberg group, a nilpotent Lie group that is a central extension of the phase space \mathbb{R}^{2n} by \mathbb{R} (or U(1) in the compact case), is intimately linked to \mathrm{Sp}(2n, \mathbb{R}) through automorphisms induced by the symplectic group. Specifically, \mathrm{Sp}(2n, \mathbb{R}) acts as the group of symplectic automorphisms on the Heisenberg group, preserving its central extension structure. The Stone-von Neumann theorem asserts the uniqueness (up to unitary equivalence) of the irreducible unitary representation of the Heisenberg group on L^2(\mathbb{R}^n), where position and momentum operators satisfy the canonical commutation relations [Q_j, P_k] = i \hbar \delta_{jk}. This theorem, originally proved by Marshall Stone and John von Neumann, guarantees that the Schrödinger representation is the fundamental one, and the metaplectic representation of \mathrm{Sp}(2n, \mathbb{R}) acts irreducibly on this space, facilitating the quantization of symplectic flows. Weyl quantization provides a concrete method to associate self-adjoint operators on L^2(\mathbb{R}^n) with classical symbols on the phase space \mathbb{R}^{2n}, using the symplectic Fourier transform to ensure covariance under the symplectic group action. For a symbol a(z) where z = (x, \xi) \in \mathbb{R}^{2n}, the Weyl operator is defined as

\mathrm{Op}^W(a) \psi(x) = \frac{1}{(2\pi \hbar)^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} e^{i (x-y) \cdot \eta / \hbar} a\left( \frac{x+y}{2}, \eta \right) \psi(y) \, dy \, d\eta,

^[53] or equivalently via the symplectic Fourier transform \hat{a}(z) = \int e^{-i \sigma(z, z')} a(z') \, dz' / (2\pi \hbar)^n, where \sigma is the symplectic form. This procedure, originating from Hermann Weyl's foundational work on group representations in quantum mechanics, transforms classical observables into operators while maintaining the symplectic invariance essential for consistent quantization. The resulting calculus is central to deformation quantization and pseudodifferential operator theory. Coherent states in quantum mechanics can be constructed using the orbit method under the action of \mathrm{Sp}(2n, \mathbb{R}) on coadjoint orbits of the Heisenberg group, providing overcomplete bases that resolve the identity and minimize uncertainty in phase space. These states, generalizing the harmonic oscillator coherent states, are obtained as orbits under the metaplectic representation, where a reference state (like the ground state) is displaced by symplectic transformations; for instance, the Perelomov coherent states are defined as |\gamma\rangle = U(\gamma) |0\rangle, with U(\gamma) in the unitary representation induced by \mathrm{Sp}(2n, \mathbb{R}). This geometric approach, developed by Alexandre Perelomov, highlights the Kähler structure of the orbits and their role in semiclassical approximations. In modern quantum optics, the symplectic group governs the transformation of Gaussian states, particularly squeezed states, which reduce noise in one quadrature at the expense of the other, enabling applications in precision measurements and quantum information processing. Squeezed states arise as metaplectic representations of \mathrm{Sp}(2n, \mathbb{R}) acting on the vacuum, with the squeezing operator S(z) = \exp\left( \frac{z^*}{2} a^2 - \frac{z}{2} (a^\dagger)^2 \right) corresponding to elements of the group that preserve the symplectic form on the covariance matrix. This framework, explored in detail by Han and Kim, unifies coherent and squeezed states as irreducible representations and has been pivotal in experimental realizations since the 1980s, surpassing classical limits in interferometry and gravitational wave detection.^[54]

References

[1]
Symplectic Group -- from Wolfram MathWorld
For every even dimension , the symplectic group is the group of matrices which preserve a nondegenerate antisymmetric bilinear form. , i.e., a symplectic form.
[2]
[PDF] Symplectic groups - MIT Mathematics
Mar 18, 2005 · The goal is to work out the structure of certain subgroups of Sp(V ), and to use.
[3]
[PDF] arXiv:quant-ph/9509002v3 24 Nov 1995
The symplectic groups form one of the three major families of classical semisimple Lie groups, the other two being the real orthogonal family and the ...
[4]
The Symplectic Piece - Ideas | Institute for Advanced Study
The term “symplectic” was introduced in 1939 by Hermann Weyl, a Faculty member at IAS, as a verbum pro verbo for “complex” in his important work on the ...
[5]
[PDF] Lie Group, Lie Algebra and their Representations
Jun 8, 2010 · non-degenerate alternating bilinear form, 2 norm, 7 octonions O, 49 ... Symplectic group Sp(2n), 1, 2 tangent space Tx, 4 tensor ...
[6]
[PDF] SYMPLECTIC GROUPS - School of Mathematics
Jan 2, 2011 · This book, 'Symplectic groups', is based on lectures given at the University of Notre Dame, 1974-1975.<|control11|><|separator|>
[7]
Order formulas for symplectic groups - Groupprops
Aug 30, 2012 · This article describes order formulas for the symplectic group of finite degree over a finite field and its variants.
[8]
first cohomology groups of some linear groups - over fields of ...
If the characteristic of K is 2, the form B determined by Q is alternate. From this point, we will consider only fields of characteristic 2, unless we ...
[9]
[PDF] Lie Groups: Fall, 2022 Lecture VI Structure of Simple Lie Algebras
Nov 28, 2022 · The Killing form of a (finite dimensional, complex) Lie algebra L is non-degenerate if and only if L is semi-simple. The Killing form is a real ...<|separator|>
[10]
[PDF] Complex Semisimple Lie Algebras - Pierre Clare
group in 2n + 1 variables, S0(2n + 1). For n ~ 3, en = sv(2n) is the Lie algebra of the symplectic group in 2n variables, Sp(2n). For n ~ 4, Dn = so(2n) is ...Missing: C_n | Show results with:C_n
[11]
[PDF] MAT 445/1196 - Complex symplectic Lie algebras Let n be an ...
Note that the dimension of sp2n(C) is n(2n + 1). The set of diagonal matrices h in g = sp2n(C) is an abelian subalgebra of g. The elements Hi = Ei,i − En+i,n+i ...
[12]
[PDF] Fundamental Representations of Classical Lie Algebras
(i) Use a similar argument to show that the groups. SLn+1(C) and Sp2n(C) are simply connected for n ≥ 1 (consider their action on nonzero vectors in the vector ...
[13]
[PDF] Lior Silberman's Math 535, Problem Set 5: Compact Lie Groups
(b) Show that the normalizer of the torus acts irreducibly on C2n. (c) Show that Sp(n) is a maximal compact subgroup of Sp2n(C). Roots and root spaces.
[14]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
group, which is usually denoted simply Sp(n), is a “compact form” of the group Sp(2n, C) in the sense we will describe later (see Exercise 3.16). To avoid ...Missing: C_n | Show results with:C_n
[15]
[PDF] notes on symplectic topology - UChicago Math
Mar 5, 2025 · If V is any finite dimensional real vector space, then there is a natural symplectic form ω on V ⊕ V ∗ that restricts to 0 on V and V ∗, and for ...<|control11|><|separator|>
[16]
[PDF] Lecture 2: Symplectic Linear Algebra, Part 2 Brian Tran 1 The ...
(1) The unitary group U(n) is a maximal compact subgroup of Sp(2n). (2) The inclusion ι : U(n) ,→ Sp(2n) is a homotopy equivalence. In particular, Sp(2n).
[17]
[PDF] On the universal covering group of the real symplectic group
The symplectic group is semisimple, so there are no homomorphisms Sp(2n,R) → S1. However, U(n) is a maximal compact subgroup and so the same fundamental group.
[18]
[PDF] symplectic groups, their parametrization and cover
A.4.3 The subgroup U(N) C Sp(2N,*). The maximal compact subgroup of Sp(2N,R) is U(N). This fact may be seen knowing that the maximal compact subgroup of GL(2N ...
[19]
[PDF] Conic Conjugacy Classes of Real 2 × 2 Matrices
In the case of real 2 × 2 matrices, these equivalence classes can be further classified as elliptic, hyperbolic, parabolic, or scalar.
[20]
[PDF] 18.745: lie groups and lie algebras, i - MIT Mathematics
Let g be the symplectic Lie algebra sp2n(k). Thus g consists of square matrices A of size 2n such that. AJ + JAT = 0 where J = 0. 1. −1 0. , with blocks being ...
[21]
[PDF] 1 Lie groups
The condition to lie in Sp(2n) is ATΩA = Ω. Since. A(t)TΩA(t) = I + (ATΩ+ΩA)t + O(t2) ,. (31) we conclude that every matrix in the Lie algebra sp(2n) must ...
[22]
[PDF] An Introduction to Lie Groups and Symplectic Geometry
Jul 23, 2018 · This course introduces Lie groups and symplectic geometry, focusing on basic concepts and examples, for students familiar with differential ...
[23]
[PDF] The Symplectic Lie Algebra sp(2N)
Thus, the unitary symplectic algebra has the same dimension and rank as sp(2N,R) and it can be thought of as a complexification of that algebra. (10) Show that ...
[24]
[PDF] 2n}(\mathbb{R}), root systems and Cartan involution.
Feb 6, 2019 · In sp2n(R), the maximum compact subalgebra is isomorphic to u(n). In sp2n(R), take X = A B. C D. , the condition. AT. CT. BT. DT. 0. I. −I 0. =.
[25]
[PDF] Equality of elementary symplectic group and symplectic group - arXiv
Dec 14, 2024 · The subgroup of Sp2n(R) generated by the elementary symplectic matrices is called the elementary symplectic group, and is denoted by ESp2n ... E2n ...
[26]
[PDF] SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
For E = R2n with the standard symplectic basis and the standard symplectic structure, J is given by a matrix in block form, J = 0 I. −I 0 . Writing. A = a b.
[27]
Sub-Riemannian geodesics on SL(2,ℝ)
... [2]. This paper deals with a sub-Riemannian structure on SL(2) = SL(2, R), the Lie group of area preserving transformations on R2. For this structure, we ...
[28]
[PDF] Exercise Sheet 5 | metaphor
Nov 22, 2018 · Solution: It follows immediately by the definition that Sp(2, C) = SL(2, C) and from this it follows the isomorphism between the associated Lie ...
[29]
[PDF] Lie algebras, lecture notes - Fachbereich Mathematik
There are also the following isomorphisms: so(3,C) ≃ sp(2,C) ≃ sl(2,C), sp ... isomorphic if there exists an isomorphism of vector spaces f : V. // V ...
[30]
[PDF] Geometry of Möbius Transformations - Sites
May 10, 2012 · Let the group SL2(R) act on C by means of linear- fractional ... the phase space whose derived form preserves the symplectic form on the.
[31]
[PDF] two-field quantum systems and the symplectic group Sp(4,R) - HAL
Jun 20, 2022 · M(θ, d, ϕ) = R(θ)Z(d)R(ϕ),. (3.32) where Z = U ZU† and R = U RU† are the squeezing matrix and the rotation matrix in the helicity basis. This.
[32]
[PDF] arXiv:2307.06690v2 [math-ph] 16 Oct 2024
Oct 16, 2024 · We fully exploit the symmetry group isomorphic to Sp(4, R), that is, the two-fold covering of SO0(2, 3) (Sp(4, R) ∼ SO0(2, 3) × Z2), recognized ...
[33]
[PDF] 15.3 More about Orthogonal groups - UC Berkeley math
Exercise 219 Which of the three groups SO5(R), SO4,1(R), SO3,2(R) is Sp4(R) locally isomorphic to? There are several ways to distinguish symplectic groups ...
[34]
[PDF] Examples of Singular Reduction - Cornell Math Department
The moment map for the action of. Sp(T∗R2,ω) ≃ Sp(2,R) on T∗R2 ≃ R4 identifies sp(2,R) with the Poisson algebra of quadratic polynomials. The polynomials ...<|control11|><|separator|>
[35]
[PDF] EXERCISES IN SYMPLECTIC GEOMETRY 1 - UC Berkeley math
4 of Springer's EMS series. 1. Page 2. 2 action of SL(2,R) = Sp(2,R). Indicate in your sketch the region of sign- definite quadratic hamiltonians. 12. Show ...
[36]
The symplectic group Sp2n(F) - Math in Melbourne
A maximal compact subgroup of Sp 2 n ( ℂ ) is. Sp ( n ) = U 2 n ( ℂ ) ∩ Sp 2 n ( ℂ ) . The group Sp ( n ) is a compact, connected, and simply connected real Lie ...
[37]
[PDF] Maximal parabolic subgroups for the symplectic group
Mar 21, 2005 · The goal is to work out the structure of what are called maximal parabolic sub- groups of Sp(V ), and to look at the corresponding geometry.
[38]
[PDF] arXiv:math/0604323v1 [math.AG] 13 Apr 2006
Let B be the subgroup of Sp2n+1 of upper triangular matrices in the odd symplectic basis {e0,e1,...,e2n}. Then B is a Borel subgroup. Proof. B is solvable since ...<|control11|><|separator|>
[39]
[PDF] Symplectic Geometry (Fall 2024)
Symplectic geometry, originating in physics, is a mathematical framework for classical mechanics and geometrical optics, defined by a 2-form on a manifold.
[40]
[PDF] On Minuscule Representations and the Principal SL2
The dual group is ˆG = Sp2ℓ, the representation V is the standard representation, and its restriction to the principal SL2 is the irreducible representation SN ...
[41]
Classification of subgroups of symplectic groups over finite fields ...
May 6, 2014 · In this note we give a self-contained proof of the following classification (up to conjugation) of subgroups of the general symplectic group of ...
[42]
A characterization of the unitary and symplectic groups over finite ...
Let G be a finite group generated by a conjugacy class D of subgroups of prime order p ^ 5, such that for any choice of distinct A and B in D, the subgroup ...
[43]
[PDF] The Endoscopic Classification of Representations
Type Bn: G = SO(2n + 1) is split, and CG = Sp(2n,C) = LG. Type Cn: G = Sp(2n) is split, and CG = SO(2n + 1,C) = LG.Missing: C_n B_n
[44]
[PDF] The Geometry of the Classical Groups
further types of classical groups: symplectic, unitary and orthogonal. These groups correspond to three types of bilinear form which can be defined on a.
[45]
[PDF] Infinite loop spaces associated to affine Kac-Moody groups
Abstract. The main purpose of this paper is to construct infinite loop spaces from affine Kac-Moody groups, It is well known that to each infi-.
[46]
symplectomorphism in nLab
Jun 28, 2023 · Symplectomorphisms are the homomorphisms of symplectic manifolds. In the context of mechanics where symplectic manifolds model phase spaces.
[47]
[PDF] lecture 4: symplectomorphisms
Since any function (modulo constants) gives a family of Hamiltonian symplec- tomorphisms, we see that the group of symplectomorphisms is huge. 2.
[48]
[PDF] Part III - Symplectic Geometry (Theorems with proof) - Dexter Chua
Note that Darboux's theorem says there is always an embedding of B2n(r) into any symplectic manifold as long as r is small enough. 19.
[49]
https://doi.org/10.1090/S0002-9947-1965-0182927-5
[50]
[PDF] C:\Downloaded_files\Arnold V I Mathematical Methods Of Classical ...
In this book we construct the mathematical apparatus of classical mechanics from the very beginning; thus, the reader is not assumed to have any previous ...
[51]
[PDF] Introduction to Mechanics and Symmetry - Fisica NET
Marsden is Professor of Control and Dynamical Systems at Caltech. He got his B.Sc. at Toronto in 1965 and his Ph.D. from Princeton University in.
[52]
Sur certains groupes d'opérateurs unitaires - Project Euclid
1964 Sur certains groupes d'opérateurs unitaires. André Weil. Author Affiliations +. André Weil1 1Institute for Advanced Study. DOWNLOAD PDF + SAVE TO MY ...
[53]
Squeezed States as Representations of Symplectic Groups - arXiv
Mar 11, 1998 · Access Paper: View a PDF of the paper titled Squeezed States as Representations of Symplectic Groups, by D. Han and Y. S. Kim. View PDF · TeX ...Missing: review | Show results with:review