USp2m

USp(2m) denotes the compact unitary symplectic group, also known as the unitary symplectic group. It consists of 2m by 2m complex matrices that are both unitary and preserve a fixed nondegenerate skew-symmetric form. With the standard symplectic form J = [0 I; −I 0], USp(2m) is the set of matrices U in GL(2m, C) such that U† U = I and U^T J U = J. Equivalently, it can be viewed as the group of 2m-dimensional quaternionic unitary matrices, i.e., those preserving a quaternionic Hermitian form. It is the compact real form of the complex group Sp(2m, C), and in some literature this group is denoted Sp(m).

Properties of USp(2m) include being a compact, connected Lie group of real dimension m(2m+1) and rank m. Its Lie algebra usp(2m) is simple of type C_m. The center of USp(2m) is {±I}, and the group is simply connected.

Representations and structure: the defining representation is 2m-dimensional and preserves a quaternionic structure, making it of pseudoreal type. This serves as the fundamental representation from which other representations can be constructed via standard Lie-theoretic methods. The maximal torus consists of diagonal matrices within the unitary-symplectic constraints, yielding a rank-m structure.

Relation to other groups: USp(2m) is the compact form of Sp(2m, C) and is related to Sp(n) in the literature, where Sp(n) often denotes the compact symplectic group of degree n. It is a classical example of a simple compact Lie group used throughout representation theory and geometry.