Sp(n)

Sp(n)
Jgmoxness · CC BY-SA 3.0 · source
Name	Sp(n)
Type	Compact Lie group
Dimension	n(2n+1)
Root system	C_n
Related	SU(2n), SO(2n+1), USp(2n)

Sp(n) Sp(n) is the compact symplectic group consisting of unitary quaternionic-linear transformations on an n-dimensional right quaternionic vector space. It appears alongside classical groups such as SU(n), SO(n), and U(n) in the Cartan classification and plays a central role in the theory of compact Lie groups, symmetric spaces, and representation theory. Sp(n) connects to structures studied by figures like Élie Cartan, Hermann Weyl, and Kostant through its root system, highest-weight theory, and applications in gauge theory and string theory.

Definition and constructions

Sp(n) can be defined as the group of 2n×2n complex matrices preserving a nondegenerate skew-Hermitian form and commuting with the standard quaternionic structure, equivalently identified with the group of quaternionic-linear isometries of H^n with the standard Hermitian form. Constructions relate Sp(n) to classical matrix groups: it is the intersection of U(2n) with the complex symplectic group Sp(2n,C) (often denoted C_n in the Cartan list) and sits naturally inside SU(2n) and USp(2n). As a compact, connected, simply connected Lie group it arises in constructions of homogeneous spaces such as the quaternionic projective space HP^n ≅ Sp(n+1)/(Sp(n)·Sp(1)) and in flag varieties studied by Borel and Bruhat.

Topological and Lie group properties

Topologically, Sp(n) is a compact, connected, simply connected simple Lie group of rank n and real dimension n(2n+1). Its maximal tori are conjugate to copies of (S^1)^n and its Weyl group is isomorphic to the semidirect product of the symmetric group S_n with (Z/2Z)^n, reflecting the C_n root system studied by Cartan and Dynkin. Sp(n) admits a bi-invariant Riemannian metric coming from the negative Killing form; this metric makes Sp(n) into a compact Riemannian manifold whose geodesic structure is relevant in works by Morse and Bott. The homotopy groups of Sp(n) connect to classical computations by Hatcher and Serre: for example, π_3(Sp(n)) ≅ Z and stable homotopy phenomena link Sp(n) to the stable homotopy groups of spheres investigated by Adams.

Lie algebra sp(n) and root system

The Lie algebra sp(n) (often denoted 𝔰𝔭(n)) is the set of skew-Hermitian quaternionic n×n matrices; over C it corresponds to the complex simple Lie algebra of type C_n. Its Cartan subalgebras, simple roots, and coroots were classified by Cartan and encoded in the Dynkin diagram C_n. The Killing form on sp(n) is nondegenerate and negative-definite on the compact real form; the positive roots and highest root determine the adjoint representation and the structure of nilpotent orbits studied by Kostant and Bala–Carter. The universal enveloping algebra and Harish-Chandra homomorphism for sp(n) feature in analyses by Harish-Chandra and in the classification of primitive ideals by Joseph.

Representations and harmonic analysis

Finite-dimensional irreducible representations of Sp(n) are classified by highest weights in the dominant integral lattice for type C_n, following highest-weight theory developed by Weyl and Cartan. Characters are computed via the Weyl character formula and branching rules relate Sp(n) representations to those of SU(2n), SO(2n+1), and subgroups such as Sp(n-1) and Sp(1) studied in work by Littlewood and Branching rules (representation theory). The representation ring and tensor product decompositions are central in applications to quantum groups and link invariants investigated by Drinfeld and Reshetikhin–Turaev. Harmonic analysis on Sp(n) and on homogeneous spaces like Sp(n)/Sp(k) uses spherical functions, the Peter–Weyl theorem, and Plancherel measure explored by Harish-Chandra, Gelfand, and Helgason.

Applications in geometry and physics

Sp(n) appears in differential geometry as the holonomy group of quaternionic-Kähler and hyperkähler manifolds studied by Berger and Salamon; groups Sp(n)·Sp(1) classify quaternionic-Kähler holonomy and Sp(n) itself is holonomy for hyperkähler manifolds such as certain gravitational instantons and moduli spaces studied by Hitchin and Kronheimer. In theoretical physics Sp(n) and its representations occur in supersymmetric gauge theories, string theory, and in the classification of global symmetries in models by Witten and Seiberg–Witten, and in the construction of Yang–Mills instantons where ADHM data use quaternionic linear algebra linked to Sp(n). Sp(n) also appears in the study of classical integrable systems, topological quantum field theory, and in the geometry of special holonomy relevant to compactifications investigated by Candelas and Duff.

Category:Lie groups Category:Compact groups Category:Classical groups