# Book: DefineClassicalLieGroups

Meaningfully define the four classical Lie groups.

• How are {$SL(n,\mathbb{C})$}, {$SO(n,\mathbb{C})$} and {$Sp(n,\mathbb{C})$} related to the complex numbers, real numbers and quaternions and their inner products?
• Why is it that the entries of these groups are complex numbers?
• What is it about even numbered dimensional vector spaces and the related Lie groups that leads to coordinate systems?
• Why do odd numbered vector spaces and the related Lie groups Bn and the symplectic group Cn have Lie algebras with the same Weyl group?

Concepts

Complex classical Lie groups

The complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The complex classical groups are {$SL(n,\mathbb{C})$}, {$SO(n,\mathbb{C})$} and {$Sp(n,\mathbb{C})$}.

• These are matrices whose entries are complex numbers.
• There is either no form, or there is a symmetric form, or there is a skew-symmetric form.
• {$SO(n,\mathbb{C})=\{Q\in GL(n,\mathbb{C}) | Q^TQ=QQ^T=I, \mathrm{det}(Q)=1\}$}
• A Lie group is a transformation group.
• A transformation group is a subgroup of an automorphism group.
• An automorphism group of an object X is the group consisting of automorphisms of X. Loosely speaking, it is the symmetry group.
• An automorphism is an isomorphism from a mathematical object to itself.
• An isomorphism is a morphism that can be reversed by an inverse morphism.
• A morphism is a structure preserving map from one mathematical structure to another of the same type.
• A map is a set function (perhaps structure preserving) or a morphism (in category theory) which respects associativity of composition.
• A classical group is a group that preserves a bilinear or sesquilinear form on finite-dimensional vector spaces over {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$}.

Notes

• If the only types of groups are rotations, then the concept of a nonrotational group is an artifact, and so the definition of Lie groups may be an artifact.
• The type of a structure indicates that structure and thus allows for two such structures to be compared. The type is the basis for comparison.