Maximum torus

In mathematics , a maximal torus of a compact Lie group is a maximally compact, connected , Abelian subgroup . ${\ displaystyle G}$ ${\ displaystyle T}$

It is a - torus , its dimension is by definition the rank of the compact Lie group . ${\ displaystyle r}$ ${\ displaystyle r}$ ${\ displaystyle G}$

The theorem of the maximal torus states that every element is conjugate to an element of . ${\ displaystyle g \ in G}$ ${\ displaystyle T}$

Examples

For the subgroup of diagonal matrices is a maximal torus. ${\ displaystyle G = U (n)}$ ${\ displaystyle T = \ left \ {\ operatorname {diag} (e ^ {i \ theta _ {1}}, \ ldots, e ^ {i \ theta _ {n}}): \ theta _ {1}, \ ldots, \ theta _ {n} \ in \ mathbb {R} \ right \}}$

For the subgroup of all block diagonal matrices with 2 × 2 blocks of is a maximal torus. ${\ displaystyle G = SO (2n)}$ ${\ displaystyle SO (2)}$

properties

The theorem of the maximal torus states that every element is conjugate to an element of . ${\ displaystyle g \ in G}$ ${\ displaystyle T}$

Numerous consequences result from this theorem:

All maximal tori are conjugate to each other.

All maximum tori have the same dimension, the rank of .

{\ displaystyle G}

A maximal torus is a maximal Abelian subgroup.

The maximum tori are the images of maximum Abelian subalgebras under the exponential map .

{\ displaystyle exp: {\ mathfrak {g}} \ to G}

Each element lies in a maximal torus.

{\ displaystyle g \ in G}

The difference between the dimension and the rank of is an even number . ${\ displaystyle G}$

literature

T. Bröcker, T. tom Dieck: Representations of compact Lie groups , Graduate Texts in Mathematics 98 (2nd ed.), Springer, 1995, ISBN 3540136789
JFAdams: Lectures on Lie Groups , University of Chicago Press, 1969, ISBN 0226005305
N. Bourbaki: Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, 1982, ISBN 354034392X
J. Dieudonné: Treatise on analysis 5 (Chapter XXI), Academic Press, 1977, ISBN 012215505X
J. Duistermaat, A. Kolk: Lie groups , Universitext, Springer, 2002, ISBN 3540152938
B. Hall, Brian: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction , Graduate Texts in Mathematics, 222 (2nd ed.), Springer, 2015, ISBN 978-3319134666
S. Helgason: Differential geometry, Lie groups, and symmetric spaces , Academic Press, 1977, ISBN 0821828487

Web links

Maximal Torus (Encyclopedia of Mathematics)
Maximal torus (nLab)
Maximal Tori Theorem (MathWorld)