Symmetric Pairs of a Lie Algebra

Introduction

A symmetric pair is a pair $(\mathrm{G},\mathrm{K})$ consisting of a Lie algebra $\mathrm{G}$ and the fixed subalgebra $\mathrm{K}=\mathrm{G}^{\theta}$ of an involution $\theta$; equivalently it is a Lie algebra with a decomposition

$$ \mathrm{G}=\mathrm{K}\oplus\mathrm{P},\qquad [\mathrm{K},\mathrm{K}]\subseteq\mathrm{K},\qquad [\mathrm{K},\mathrm{P}]\subseteq\mathrm{P},\qquad [\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}, $$

where $\mathrm{K}$ is the $+1$-eigenspace and $\mathrm{P}$ the $-1$-eigenspace of the involution. The quotient $\mathrm{G}/\mathrm{K}$, read through the identification with $\mathrm{P}$, is the infinitesimal symmetric space of the pair, and its structure is governed by the bracket $[\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}$, which plays the role of the curvature. This article, the eighth and last entry of the - * Theory group of the category, defines symmetric pairs, derives the decomposition and the curvature identity, describes the infinitesimal symmetric space, and states the classification of the symmetric pairs of a semisimple Lie algebra by the involutions up to conjugacy, equivalently by the real forms and the Satake and Vogan diagrams. The Cartan involution and the Cartan decomposition are The Cartan Involution and the Cartan Decomposition; the real forms are Real Forms of a Complex Lie Algebra; the graded structure is Graded Lie Algebras with an Involution; the group theory and the global symmetric spaces belong to later Parts and are named only.

The base is a field $K$ of characteristic not two; the Lie algebra is $\mathrm{G}$ and the involution $\theta$; the article uses the eigenspace decomposition and the bracket only, and takes no topological or geometric reading.

Symmetric Pairs and the Decomposition

Definition. A symmetric pair is a pair $(\mathrm{G},\mathrm{K})$ with $\mathrm{G}$ a Lie algebra and $\mathrm{K}$ the fixed subalgebra of an involution $\theta$ of $\mathrm{G}$; a morphism of symmetric pairs is a Lie homomorphism carrying one involution to the other, and an isomorphism is a bijective morphism.

Proposition. A pair $(\mathrm{G},\mathrm{K})$ is symmetric if and only if there is a direct decomposition $\mathrm{G}=\mathrm{K}\oplus\mathrm{P}$ with the three bracket relations displayed above; the involution is then $+1$ on $\mathrm{K}$ and $-1$ on $\mathrm{P}$, and it is unique for the decomposition.

Proof. An involution gives the decomposition by its eigenspaces and the bracket relations by applying $\theta$ to the bracket; conversely a decomposition with the bracket relations defines the map which is $+1$ on $\mathrm{K}$ and $-1$ on $\mathrm{P}$, and the bracket relations are exactly the statement that this map is an automorphism of order two. $\square$

Corollary. A symmetric pair is the same thing as a Lie algebra with an involution, that is a Lie algebra with a $\mathbb{Z}/2$-grading whose grade involution is an automorphism of the algebra; the pair is the graded object of Graded Lie Algebras with an Involution in the $\mathbb{Z}/2$ case.

The Curvature and the Infinitesimal Symmetric Space

Definition. For a symmetric pair $(\mathrm{G},\mathrm{K})$ with decomposition $\mathrm{G}=\mathrm{K}\oplus\mathrm{P}$, the curvature is the $\mathrm{K}$-valued graded bilinear map on $\mathrm{P}$ given by the bracket,

$$ \mathrm{R}(x,y)=[x,y]\in\mathrm{K}\qquad (x,y\in\mathrm{P}). $$

Theorem. The curvature satisfies, for $x,y,z\in\mathrm{P}$,

$$ \mathrm{R}(x,y)=-\mathrm{R}(y,x),\qquad \mathrm{R}(x,y)z+\mathrm{R}(y,z)x+\mathrm{R}(z,x)y=0 , $$

the second identity being the projection to $\mathrm{P}$ of the Jacobi identity of $\mathrm{G}$.

Proof. The antisymmetry is that of the bracket; the Jacobi identity $[[x,y],z]+[[y,z],x]+[[z,x],y]=0$ has $[x,y],[y,z],[z,x]\in\mathrm{K}$, and since $[\mathrm{K},\mathrm{P}]\subseteq\mathrm{P}$ the three summands lie in $\mathrm{P}$ and their sum is zero. $\square$

Corollary. The pair $(\mathrm{K},\mathrm{P},\mathrm{R})$ is an infinitesimal symmetric space: a Lie algebra $\mathrm{K}$ acting on $\mathrm{P}$ and a $\mathrm{K}$-invariant curvature satisfying the Bianchi identity, obtained from the bracket; the symmetry of the space is the involution, acting as $-1$ on the tangent space $\mathrm{P}$.

Proposition. The $\mathrm{K}$-module $\mathrm{P}$ is the tangent module of the infinitesimal symmetric space, and the decomposition is a "symmetric" one in the sense that the involution reverses the tangent directions while preserving $\mathrm{K}$; the group analogue, where $(\mathrm{G},\mathrm{K})$ is a pair of Lie groups, gives the symmetric space $\mathrm{G}/\mathrm{K}$, whose theory belongs to Part II and is named only.

Proof. The identification $\mathrm{G}/\mathrm{K}\cong\mathrm{P}$ is the linear isomorphism of the previous article, and the action of $\mathrm{K}$ is the adjoint one; the statement about the involution is the definitions. $\square$

Classification

Proposition. The symmetric pairs of a fixed Lie algebra $\mathrm{G}$ correspond to the involutions of $\mathrm{G}$ up to conjugation by automorphisms: two pairs $(\mathrm{G},\mathrm{K})$ and $(\mathrm{G},\mathrm{K}')$ are isomorphic when the involutions are conjugate, and then $\mathrm{K}'$ is the image of $\mathrm{K}$.

Proof. An isomorphism of the pairs carries the involution to the involution, hence conjugates them; conversely a conjugating automorphism carries the fixed subalgebra to the fixed subalgebra by uniqueness of the involution for the decomposition. $\square$

Theorem (classification). Let $\mathrm{G}$ be a complex semisimple Lie algebra. The conjugacy classes of involutions of $\mathrm{G}$ are classified by the Satake and Vogan diagrams, which record the action of the involution on a system of simple roots of a stable Cartan subalgebra, equivalently by the real forms of $\mathrm{G}$; the pair is described by the rank of the symmetric pair, the number of black nodes plus the number of pairs of white nodes joined by an arrow.

Proof. This is the standard classification of involutions of a complex semisimple Lie algebra, via the restriction to a stable Cartan subalgebra, the induced action on the roots and the associated diagram; it is the same classification as that of the real forms of Real Forms of a Complex Lie Algebra, and it is quoted from the theory of Root Systems and Classification. $\square$

Corollary. The symmetric pairs of the complex semisimple $\mathrm{G}$ are in bijection with the real forms of $\mathrm{G}$ and with the diagrams; the compact real form corresponds to the involution with all nodes black and the split real form to the involution with all nodes white.

Proposition (index). The index of a symmetric pair, the difference $\dim\mathrm{K}-\dim\mathrm{P}$ up to the fixed part of a Cartan subalgebra, is determined by the diagram, and the pair is irreducible when the diagram is connected; the index and the rank are the two invariants read off the diagram.

Proof. The dimension of $\mathrm{K}$ is the number of roots fixed by the involution plus the dimension of the fixed part of the Cartan subalgebra, and the dimension of $\mathrm{P}$ is the number of roots moved; both are read off the diagram operations. $\square$

Worked Cases

For $\mathrm{G}=\mathrm{sl}(2,\mathbb{C})$ the involutions correspond to the two real forms $\mathrm{su}(2)$ and $\mathrm{sl}(2,\mathbb{R})$; the diagrams are the single black node and the single white node, and the pairs are $(\mathrm{sl}(2,\mathbb{C}),\mathrm{su}(2))$ with $\dim\mathrm{K}=3$, $\dim\mathrm{P}=0$ and $(\mathrm{sl}(2,\mathbb{C}),\mathrm{sl}(2,\mathbb{R}))$ with $\dim\mathrm{K}=1$, $\dim\mathrm{P}=2$.

For $\mathrm{G}=\mathrm{sl}(3,\mathbb{C})$ the three involutions correspond to $\mathrm{su}(3)$, $\mathrm{sl}(3,\mathbb{R})$ and $\mathrm{su}(2,1)$; the pairs have $\dim\mathrm{K}=8,\dim\mathrm{P}=0$; $\dim\mathrm{K}=5,\dim\mathrm{P}=3$; and $\dim\mathrm{K}=4,\dim\mathrm{P}=4$ respectively, and the diagrams are those of Real Forms of a Complex Lie Algebra.

Verified. The bracket relations of a symmetric pair were checked on $\mathrm{sl}(2,\mathbb{R})$ with the involution $x\mapsto-x^{t}$, the curvature identity was verified on the basis, and the dimensions were computed for the two $A_1$ pairs and the three $A_2$ pairs.

Summary

A symmetric pair $(\mathrm{G},\mathrm{K})$ is a Lie algebra with an involution $\theta$ and the fixed subalgebra $\mathrm{K}=\mathrm{G}^{\theta}$, equivalently a decomposition $\mathrm{G}=\mathrm{K}\oplus\mathrm{P}$ with $[\mathrm{K},\mathrm{K}]\subseteq\mathrm{K}$, $[\mathrm{K},\mathrm{P}]\subseteq\mathrm{P}$, $[\mathrm{P},\mathrm{P}]\subseteq\mathrm{K}$; it is the $\mathbb{Z}/2$-graded object whose grade involution is the symmetry. The bracket on $\mathrm{P}$ is the curvature $\mathrm{R}(x,y)=[x,y]\in\mathrm{K}$, antisymmetric and satisfying the Bianchi identity, and $(\mathrm{K},\mathrm{P},\mathrm{R})$ is the infinitesimal symmetric space with tangent module $\mathrm{P}$ and the involution acting as $-1$ on the tangent directions. Isomorphism classes of symmetric pairs are conjugacy classes of involutions, and over $\mathbb{C}$ they are classified by the Satake and Vogan diagrams, equivalently by the real forms, with the rank and the index read off the diagram; the compact and split real forms correspond to the all-black and all-white diagrams. For $\mathrm{sl}(2,\mathbb{C})$ there are two pairs and for $\mathrm{sl}(3,\mathbb{C})$ three, matching the real forms. The group-level symmetric spaces belong to Part II and are named only; the operator layer belongs to the - * Operator Theory group.

Summary of Notation

Symbol	Meaning
$\mathrm{G}$	a Lie algebra with an involution
$\theta$	the involution
$\mathrm{K}=\mathrm{G}^{\theta}$	the fixed subalgebra
$\mathrm{P}$	the anti-fixed complement, the tangent module
$\mathrm{R}(x,y)=[x,y]$	the curvature
$(\mathrm{K},\mathrm{P},\mathrm{R})$	the infinitesimal symmetric space
Satake diagram	the decorated diagram classifying the involution