GL(n), SL(n), O(n), SO(n), U(n), SU(n) as geometric objects
Most Lie groups that arise in practice are matrix Lie groups — groups of invertible matrices with smooth dependence on their entries. The “classical” families GL(n), SL(n), O(n), SO(n), U(n), and SU(n) each arise by imposing algebraic constraints on matrices, carving out smooth submanifolds of the space of all matrices.
Each constraint has a geometric meaning: det(A) = 1 confines you to a hypersurface, AᵀA = I forces orthogonality, and A†A = I adds unitarity. Understanding these constraints as geometric conditions is key to visualizing the topology of these groups.
The space of 2×2 real matrices is 4-dimensional. Here we visualize a 2D slice through this space, colored by the determinant. The det = 0 boundary (where the matrix becomes singular) glows hot, while the det = 1 surface (SL(2,ℝ)) appears as a crystalline rose curve. The orthogonality constraint AᵀA = I carves out an even smaller submanifold.
Key insight: Matrix groups are defined by polynomial equations (det = 1, AᵀA = I), and the implicit function theorem guarantees these level sets are smooth manifolds — hence Lie groups.
Each classical matrix group has a distinctive topology: some are connected, some have multiple components; some are compact (bounded and closed), some are not. The fundamental group π₁ tells you about loops that cannot be contracted — SO(3) has π₁ = ℤ/2ℤ (the basis of the belt trick), while SU(2) is simply connected.
Starting from the full general linear group GL(2,ℝ), we can progressively impose constraints: det ≠ 0 (invertibility), AᵀA = I (orthogonality), and det = 1 (special orthogonality). Each constraint reduces the dimension, carving out a smaller submanifold. Watch the layers peel away as constraints are tightened.
The classical matrix groups sit inside each other in a rich hierarchy of subgroup relationships. SO(n) ⊂ O(n) ⊂ GL(n,ℝ), and similarly SU(n) ⊂ U(n) ⊂ GL(n,ℂ). Click any node to learn more about that group, and toggle the dimension to see how the relationships change.