Semi-simple lie group

In mathematics , a semi-simple Lie group is a connected Lie group whose Lie algebra is semi-simple .

Equivalent characterizations

A connected Lie group is semi-simple if and only if it satisfies one of the following equivalent conditions:

• the killing form is not degenerate ,
• there are no normal non-trivial resolvable subgroups,
• there are no normal non-trivial Abelian subsets.

Examples

• Special linear groups : , ${\displaystyle SL(n,\mathbb {R} )}$ ${\displaystyle SL(n,\mathbb {C} )}$
• Special orthogonal group ${\displaystyle SO(p,q)}$
• Symplectic group ${\displaystyle Sp(n)}$
• The above examples are simple Lie groups . The direct products of a finite number of simple Lie groups are also semi-simple Lie groups.
• Semi-simple algebraic groups above are semi- simple Lie groups. ${\displaystyle \mathbb {C} }$

Maximum compact subgroup

For a semi-simple Lie group there is a maximal compact subgroup that is unique except for conjugation . For example, SO (n) is a maximally compact subgroup of and SU (n) is a maximally compact subgroup of . ${\displaystyle G}$ ${\displaystyle K}$ ${\displaystyle SL(n,\mathbb {R} )}$ ${\displaystyle SL(n,\mathbb {C} )}$

Symmetrical space

Let be a maximally compact subgroup of the (non-compact) semi-simple Lie group . The quotient is a symmetrical space of non-compact type . ${\displaystyle K}$ ${\displaystyle G}$ ${\displaystyle G/K}$

The dual symmetrical space is denoted by. Its isometric group is a compact Lie group. ${\displaystyle G^{u}/K}$ ${\displaystyle G^{u}}$

literature

• Brian C. Hall: Lie groups, Lie algebras, and representations. An elementary introduction. (= Graduate Texts in Mathematics. 222). Springer-Verlag, New York 2003, ISBN 0-387-40122-9 .