The Cartan Involution and the Cartan Decomposition

Introduction

Let $\mathrm{G}_0$ be a real semisimple Lie algebra with Killing form $\kappa$. A Cartan involution is an involution $\theta$ of $\mathrm{G}_0$ such that the form

$$ \kappa_\theta(x,y)=-\kappa(x,\theta y) $$

is positive definite. The involution splits the algebra into its eigenspaces,

$$ \mathrm{G}_0=\mathrm{K}\oplus\mathrm{P},\qquad \mathrm{K}=\{x:\theta x=x\},\qquad \mathrm{P}=\{x:\theta x=-x\}, $$

the Cartan decomposition; $\mathrm{K}$ is a subalgebra, $\mathrm{P}$ a module over it, and the pair $(\mathrm{G}_0,\mathrm{K})$ is the algebraic model of a symmetric pair. This article is the - * Theory entry of the category: it reads the Lie algebra with an involution on its elements, defines the Cartan involution and the Cartan decomposition, establishes the bracket relations, identifies $\mathrm{K}$ as the fixed subalgebra and records its role as the Lie algebra of a maximal compact subgroup, a compactness statement that belongs to a later Part and is named only. The involution as an operator on the algebra, the adjoint of the action and the self-adjointness of the Casimir operator belong to the - * Operator Theory group and are deferred; the Killing form is The Killing Form Operator and its structure theory is Structure of Lie Algebras.

The base is $\mathbb{R}$; the algebra $\mathrm{G}_0$ is finite-dimensional real semisimple, $\kappa$ is its Killing form, and $\theta$ is an involution. The article uses the eigenspace decomposition of $\theta$ and the bracket only; the definiteness of $\kappa_\theta$ is stated as the defining condition, and no length, distance or geometric reading is taken from it.

The Cartan Involution

Definition. An involution of a Lie algebra $\mathrm{G}_0$ is an automorphism $\theta$ with $\theta^2=\mathrm{id}$. A Cartan involution is an involution $\theta$ of the real semisimple algebra $\mathrm{G}_0$ such that $\kappa_\theta(x,y)=-\kappa(x,\theta y)$ is positive definite.

Proposition. $\kappa_\theta$ is a symmetric bilinear form, and it is invariant under $\theta$:

$$ \kappa_\theta(\theta x,\theta y)=\kappa_\theta(x,y),\qquad \kappa_\theta(\theta x,y)=\kappa_\theta(x,\theta y). $$

Proof. Symmetry follows from the symmetry of $\kappa$ and $\theta^2=\mathrm{id}$; the invariance under $\theta$ is the same computation. $\square$

Proposition. If $\theta$ is a Cartan involution then so is every conjugate $g\theta g^{-1}$ by an automorphism $g$ that preserves the form up to the appropriate condition, and the Cartan involution is unique up to conjugacy when $\mathrm{G}_0$ is semisimple: any two differ by an inner automorphism.

Proof. The definiteness of $\kappa_\theta$ is preserved by the conjugation when $g$ preserves $\kappa$; the uniqueness up to conjugacy is the standard theorem of the theory of real semisimple Lie algebras, recorded in Real Forms of a Complex Lie Algebra, and is quoted. $\square$

The Cartan Decomposition

Theorem. A Cartan involution $\theta$ decomposes the algebra as the direct sum of its two eigenspaces,

$$ \mathrm{G}_0=\mathrm{K}\oplus\mathrm{P},\qquad \theta=\mathrm{id}\text{ on }\mathrm{K},\qquad \theta=-\mathrm{id}\text{ on }\mathrm{P}, $$

and the eigenspaces satisfy

$$ [\mathrm{K},\mathrm{K}]\subseteq\mathrm{K},\qquad [\mathrm{K},\mathrm{P}]\subseteq\mathrm{P},\qquad [\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}. $$

Proof. An involution of a vector space whose square is the identity is diagonalisable with eigenvalues $\pm1$ in characteristic not two, giving the direct sum; the bracket relations follow from applying $\theta$ to $[x,y]$, which is an automorphism, and reading the sign according to the degrees of $x$ and $y$. $\square$

Corollary. $\mathrm{K}$ is a subalgebra of $\mathrm{G}_0$, $\mathrm{P}$ is a $\mathrm{K}$-module under the adjoint action, and the quotient $\mathrm{G}_0/\mathrm{K}$ is identified with $\mathrm{P}$ as a $\mathrm{K}$-module; the pair $(\mathrm{G}_0,\mathrm{K})$ is a symmetric pair, and the involution is the one of Symmetric Pairs of a Lie Algebra.

Proposition. The two summands are the kernels of the projectors $\tfrac12(\mathrm{id}\pm\theta)$, which are operators of the algebra commuting with the adjoint action; the map $x\mapsto\theta x$ is the operator that multiplies $\mathrm{K}$ by $+1$ and $\mathrm{P}$ by $-1$.

Proof. The projectors are the standard spectral projectors of a diagonalisable involution, and they commute with every automorphism commuting with $\theta$, in particular with the $\operatorname{ad}_x$ for $x$ in $\mathrm{K}$. $\square$

The Brackets and the Symmetric Structure

Theorem. The bracket relations of the Cartan decomposition make $\mathrm{G}_0$ a $\mathbb{Z}/2$-graded Lie algebra with the even part $\mathrm{K}$ and the odd part $\mathrm{P}$, and the involution $\theta$ is the grade involution of that grading.

Proof. The relations $[\mathrm{K},\mathrm{K}]\subseteq\mathrm{K}$, $[\mathrm{K},\mathrm{P}]\subseteq\mathrm{P}$, $[\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}$ are exactly the axioms of a $\mathbb{Z}/2$-graded Lie algebra with even part $\mathrm{K}$, and $\theta$ acts by $+1$ on the even part and $-1$ on the odd part, which is the grade involution. $\square$

Corollary. The symmetric pair $(\mathrm{G}_0,\mathrm{K})$ is the graded Lie algebra of Graded Lie Algebras with an Involution, and the structure of the pair is the structure of the grading.

The Fixed Subalgebra and Maximal Compactness

Definition. The fixed subalgebra of the Cartan involution is $\mathrm{K}=\ker(\theta-\mathrm{id})$, called the maximal compact subalgebra of $\mathrm{G}_0$.

Proposition. $\mathrm{K}$ is the largest subalgebra of $\mathrm{G}_0$ on which $\theta$ acts as the identity, and it is a reductive subalgebra: its Killing form is the restriction of $\kappa$ up to the factor and its centre is contained in the centre of $\mathrm{K}$.

Proof. The fixed set of an automorphism is a subalgebra, and it is the largest on which the automorphism is the identity; the reductive statement is the standard property of the fixed algebra of a Cartan involution, recorded in Real Forms of a Complex Lie Algebra. $\square$

Remark (forward reference). The name compact records that $\mathrm{K}$ is the Lie algebra of a maximal compact subgroup of the adjoint group of $\mathrm{G}_0$; the compactness of that subgroup is a group-theoretic and topological condition, treated with the Lie groups, and is named here only. The Cartan decomposition is the algebraic shadow of the decomposition of a semisimple Lie group into a compact subgroup and a subspace, and the group statement is deferred.

Worked Case: $\mathrm{sl}(2,\mathbb{R})$

Let $\mathrm{G}_0=\mathrm{sl}(2,\mathbb{R})$ with basis $e,h,f$ and $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$. The Cartan involution is $\theta(x)=-x^{t}$, the negative transpose, which fixes

$$ \mathrm{K}=\left\{\begin{pmatrix}0&b\\-b&0\end{pmatrix}\right\}=\langle e-f\rangle, $$

a one-dimensional algebra, and negates the complementary space $\mathrm{P}=\langle h,\ e+f\rangle$ of dimension two; the pair $(\mathrm{G}_0,\mathrm{K})$ is the symmetric pair of the upper half-plane, whose group theory belongs to a later Part. The form $\kappa_\theta(x,y)=-\kappa(x,\theta y)$ is positive definite on $\mathrm{G}_0$, as the theory requires; the decomposition is an eigenspace decomposition of the involution and no metric reading is taken.

Verified. The involution $\theta(x)=-x^{t}$ was checked to be an algebra automorphism of $\mathrm{sl}(2,\mathbb{R})$ on the three brackets, and the eigenspace dimensions were checked to be $1$ and $2$; the bracket relations of the decomposition were verified on the basis.

Summary

A Cartan involution of a real semisimple Lie algebra $\mathrm{G}_0$ is an involution $\theta$ with $\kappa_\theta(x,y)=-\kappa(x,\theta y)$ positive definite; it is unique up to inner automorphisms. Its Cartan decomposition $\mathrm{G}_0=\mathrm{K}\oplus\mathrm{P}$ is the eigenspace decomposition for the eigenvalues $+1$ and $-1$, with $\mathrm{K}$ a subalgebra, $\mathrm{P}$ a module over it, and the brackets $[\mathrm{K},\mathrm{K}]\subseteq\mathrm{K}$, $[\mathrm{K},\mathrm{P}]\subseteq\mathrm{P}$, $[\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}$ making $\mathrm{G}_0$ a $\mathbb{Z}/2$-graded Lie algebra whose grade involution is $\theta$. The fixed subalgebra $\mathrm{K}$ is the maximal compact subalgebra, the Lie algebra of a maximal compact subgroup of the adjoint group, a compactness statement named and deferred to the Lie groups. The pair $(\mathrm{G}_0,\mathrm{K})$ is a symmetric pair, treated in Symmetric Pairs of a Lie Algebra and Graded Lie Algebras with an Involution. For $\mathrm{sl}(2,\mathbb{R})$ the involution is the negative transpose, $\mathrm{K}$ is one-dimensional and $\mathrm{P}$ two-dimensional. The operator layer of the involution, its adjoints and the Casimir operator under the involution belong to the - * Operator Theory group.

Summary of Notation

Symbol	Meaning
$\mathrm{G}_0$	a real semisimple Lie algebra
$\kappa$	the Killing form
$\theta$	a Cartan involution
$\kappa_\theta(x,y)=-\kappa(x,\theta y)$	the associated positive definite form
$\mathrm{K}=\ker(\theta-\mathrm{id})$	the fixed subalgebra, the maximal compact subalgebra
$\mathrm{P}=\ker(\theta+\mathrm{id})$	the complement
$(\mathrm{G}_0,\mathrm{K})$	the symmetric pair