- 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.

**Readings**

Parsiųstas iš http://www.ms.lt/sodas/Book/DefineClassicalLieGroups

Puslapis paskutinį kartą pakeistas 2019 gegužės 17 d., 18:24