Orthogonal symmetric Lie algebra

In mathematics, an orthogonal symmetric Lie algebra is a pair ${\displaystyle ({\mathfrak {g}},s)}$ consisting of a real Lie algebra ${\displaystyle {\mathfrak {g}}}$ and an automorphism ${\displaystyle s}$ of ${\displaystyle {\mathfrak {g}}}$ of order ${\displaystyle 2}$ such that the eigenspace ${\displaystyle {\mathfrak {u}}}$ of s corresponding to 1 (i.e., the set ${\displaystyle {\mathfrak {u}}}$ of fixed points) is a compact subalgebra. If "compactness" is omitted, it is called a symmetric Lie algebra. An orthogonal symmetric Lie algebra is said to be effective if ${\displaystyle {\mathfrak {u}}}$ intersects the center of ${\displaystyle {\mathfrak {g}}}$ trivially. In practice, effectiveness is often assumed; we do this in this article as well.

The canonical example is the Lie algebra of a symmetric space, ${\displaystyle s}$ being the differential of a symmetry.

Every orthogonal symmetric Lie algebra decomposes into a direct sum of ideals "of compact type", "of noncompact type" and "of Euclidean type".

References

• S. Helgason, Differential geometry, Lie groups, and symmetric spaces