Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
A torus in is an abelian subgroup which is connected (and compact). Note that any compact connected abelian Lie group must be a torus , hence the name.
A maximal torus is a subgroup which is maximal with these properties.
The Lie algebra of a maximal torus of is a Cartan subalgebra of the Lie algebra of .
Suppose throughout that the compact Lie group is connected.
Given a choice of maximal torus , then each element is conjugate to an element of its maximal torus, i.e. there exists such that .
(e.g. Johansen, theorem 2.7.3)
For the unitary group , by example , prop. is the spectral theorem that unitary matrices may be diagonalized.
Any two choices of maximal tori are conjugate.
(e.g. Johansen, corollary 2.7.5)
Any two elements of a maximal torus are conjugate in precisely if they are conjugate via the Weyl group.
(e.g. Johansen, prop 2.7.13)
The conjugacy classes of are in bijection to .
Under the assumptions on , the exponential map is surjective.
The exponential map is surjective; similarly, if is a Cartan subalgebra of , then the exponential map maps homomorphically onto a maximal torus. Also the exponential map preserves conjugation by any element of , i.e., . Then surjectivity follows from Proposition .
The maximal torus of a unitary group is the subgroup of diagonal matrices with unitary entries.
The maximal torus of the special unitary group is again (one dimension lower) included as the subgroup of diagonal matrices whose first diagonal entries are arbitrary elements of and whose last diagonal entry is the inverse of their product.
The maximal torus of the special orthogonal group and is included block-diagonal (with, in the odd-dimensional case, the remaining block entry being the unit).
e.g.
Last revised on April 15, 2024 at 17:19:10. See the history of this page for a list of all contributions to it.