SU(p,q)
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| SU(p,q) | |
|---|---|
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| Name | SU(p,q) |
| Type | Special unitary group of signature (p,q) |
| Dimension | (p+q)^2-1 |
| Rank | min(p,q) |
| Field | Complex numbers |
SU(p,q)
SU(p,q) is the real Lie group of complex (p+q)×(p+q) matrices preserving a nondegenerate Hermitian form of signature (p,q) and having determinant one. It is a noncompact, semisimple group that generalizes special unitary symmetries appearing in models related to Lorentz group, Minkowski space, and certain Hermitian symmetric space structures. SU(p,q) plays a central role in the study of unitary representations, bounded symmetric domain theory, and in constructions related to Erlangen program-style symmetry classifications.
Definition and basic properties
SU(p,q) is defined as the subgroup of GL(p+q,ℂ) preserving a Hermitian form with signature (p,q) and having determinant one. The group is semisimple with no nontrivial compact center when p≠q, and has center isomorphic to a finite cyclic subgroup when p=q. SU(p,q) is a real form of the complex Lie group SL(n,ℂ) for n=p+q and is closely related to other classical groups such as SO(p,q), Sp(2n,ℝ), and U(p,q). SU(p,q) arises in contexts connected to Cartan decomposition, Iwasawa decomposition, and to discrete subgroups related to Arithmetic group theory and Borel–Harish-Chandra theorem applications.
Matrix representation and Lie algebra
Elements of SU(p,q) can be represented as block matrices satisfying A* I_{p,q} A = I_{p,q} and det A = 1, where I_{p,q} is the diagonal matrix with p entries +1 and q entries −1. Its Lie algebra su(p,q) consists of (p+q)×(p+q) complex traceless matrices X with X* I_{p,q} + I_{p,q} X = 0. The algebra su(p,q) is a real form of sl(n,ℂ) and shares structural features with Lie algebras appearing in the Cartan classification of simple Lie algebras, such as types A_{n-1}. Matrix realizations facilitate connections to representation theory constructions like highest-weight modules and to geometric actions on complex projective space and Siegel upper half-space analogues.
Topology and connectedness
As a Lie group, SU(p,q) is connected for all p,q with p+q≥2, and its maximal compact subgroup is isomorphic to S(U(p)×U(q)). The fundamental group of SU(p,q) and its universal cover relate to those of SU(n), U(n), and to central extensions that appear in the study of projective unitary representations such as those encountered in Quantum mechanics and in Borel–Weil–Bott theorem contexts. The noncompactness of SU(p,q) for pq>0 yields nontrivial discrete series representations by results of Harish-Chandra and topological properties relevant to Matsushima's formula and cohomology of locally symmetric spaces like quotients by arithmetic lattices.
Representation theory
The representation theory of SU(p,q) includes unitary highest-weight representations, principal series, complementary series, and discrete series when rank conditions are satisfied. Classification uses methods from Harish-Chandra, Langlands program, and Kac–Moody algebra techniques for related affine constructions. Finite-dimensional irreducible representations correspond to highest weights for the complexification sl(n,ℂ) and are realized by tensor constructions familiar from Young tableau combinatorics and Schur–Weyl duality linking to Symmetric group representations. Unitary dual descriptions exploit Cartan subalgebras and Blattner-type formulae connecting to cohomological induction pioneered by Zuckerman and to dualities such as Howe duality.
Root system and Cartan decomposition
The root system of su(p,q) is that of type A_{n-1} for n=p+q, with a choice of noncompact and compact roots determined by a Cartan involution whose fixed-point algebra is Lie(S(U(p)×U(q))). The Cartan decomposition g = k ⊕ p provides a decomposition into compact part k and noncompact part p; corresponding restricted root systems and Weyl groups govern structure analogous to that in Weyl character formula settings. Parabolic subalgebras tied to flag varieties such as generalized Grassmannians connect to Borel subalgebras and to Bruhat decompositions central to geometry of Flag variety and to the theory of Schubert calculus.
Symmetric space and Hermitian structure
The quotient SU(p,q)/S(U(p)×U(q)) is a noncompact Hermitian symmetric space of the noncompact type, realized as a bounded symmetric domain of type I in the classification of Élie Cartan. This symmetric space admits a complex structure, Kähler metric, and Bergman kernel structures studied in complex analysis on domains and in Kähler–Einstein metrics research. Geodesic and curvature properties tie to results by Helgason, with applications to automorphic forms on domains related to Siegel modular forms and to period domains arising in Hodge theory linked to Griffiths transversality.
Examples and special cases
Notable special cases include SU(1,1), locally isomorphic to SL(2,ℝ) and related to Möbius transformation groups acting on the unit disk; SU(1,2) connected to complex hyperbolic geometry and to Picard modular group arithmetic; and SU(n,0) ≅ SU(n) compact cases linked to classical unitary group representation theory. Equal-rank case SU(p,p) features additional center and accidental isomorphisms in low dimensions, such as relations with Spin groups and with groups appearing in the classification of simple Lie groups catalogued by Dynkin diagram correspondences. SU(p,q) lattices provide examples for rigidity phenomena like Mostow rigidity and Margulis superrigidity in locally symmetric spaces.
Category:Lie groups Category:Classical groups