SO(5)

In mathematics, particularly in the field of Lie groups, SO(5) refers to the special orthogonal group of degree 5. It is the group of rotations in 5-dimensional Euclidean space, preserving orientation. More formally, SO(5) is the group of 5x5 orthogonal matrices with determinant 1.

The "S" in SO(5) denotes "special," indicating that the determinant of the matrices in the group must be +1. The "O" stands for "orthogonal," meaning that the matrix multiplied by its transpose equals the identity matrix (A^TA = I). The "(5)" signifies that the matrices are 5x5.

SO(5) is a compact, connected, non-abelian Lie group. Its Lie algebra is denoted by so(5) or $\backslash mathfrak\{so\}(5)$, and consists of 5x5 skew-symmetric matrices. The dimension of SO(5) is 10.

SO(5) finds applications in various areas of physics and mathematics, including theoretical physics, particle physics, and differential geometry. It is related to other Lie groups and Lie algebras, such as Spin(5) (its double cover) and Sp(2) (which is isomorphic to Spin(5)). Its representation theory is well-studied and plays a role in understanding the symmetries of physical systems.