The Signed Adjoint of the Reflection on a Lie Algebra

Introduction

A reflection of a graded Lie algebra is the signed conjugation $\rho_a = e^{\operatorname{ad}_a}\alpha$, and its adjoint with respect to an invariant Hermitian form that the grade involution preserves is its inverse: $\rho_a^{*} = \alpha e^{-\operatorname{ad}_a} = \rho_a^{-1}$. The reflection is therefore always a unitary operator of the form, and it is self-adjoint exactly when it is an involution, that is when $a + \alpha(a)$ is central; the reflection by an odd element satisfies this automatically, while the reflection by an even element need not. The article computes the adjoint, identifies the self-adjoint reflections with the involutive ones, and examines the three ways in which the self-adjointness fails.

This article treats the adjoint of a reflection read as a signed operator on a Lie algebra, the self-adjointness of the reflections and its failure in the degenerate case. It is the fourth article of the - * Operator Theory group of the category; the reflection and its involution condition are Reflections as Signed Two-Sided Operators on a Lie Algebra, the signed sandwich which realises it is The Signed Sandwich on a Lie Algebra, both above in the category, the adjoint of the sandwich is The Signed Adjoint Sandwich on a Lie Algebra, above, the invariant form is Hermitian Forms on a Lie Algebra, and the remaining adjoints are The Signed Adjoint of the Left Multiplication on a Lie Algebra and The Graded Adjoint Action on a Module over a Lie Algebra, below in this group.

The article assumes the reflection, the signed conjugation and the carrying elements from Reflections as Signed Two-Sided Operators on a Lie Algebra, the adjoint operation and the adjoints of the one-sided multiplications from The Signed Adjoint Sandwich on a Lie Algebra, the invariant Hermitian form and its positive definite case from Hermitian Forms on a Lie Algebra, the inner automorphisms and the exponential from The Lie Correspondence and the Adjoint Representation, and the grading and the grade involution from Graded Lie Algebras with an Involution. The form is used as an instrument; the geometric reflection in a hyperplane, which needs a metric, is Part IV.

The Adjoint of a Reflection

The Computation

Definition. Let $H$ be a non-degenerate invariant Hermitian form on the Lie algebra $\mathrm{G}$, preserved by the grade involution, and let $\rho_a = e^{\operatorname{ad}_a}\alpha$ be the signed conjugation. The adjoint is the operator $\rho_a^{*}$ with $H(\rho_ax,y) = H(x,\rho_a^{*}y)$.

Theorem (the reflection is unitary). The adjoint of the signed conjugation is its inverse,

$$ \rho_a^{*} = \alpha e^{-\operatorname{ad}_a} = \rho_a^{-1} , $$

so that $\rho_a^{*}\rho_a = \rho_a\rho_a^{*} = \mathrm{id}$: every reflection preserves the form $H$.

Proof. The adjoint of the composite is the composite of the adjoints in the reverse order, $\rho_a^{*} = \alpha^{*}\bigl(e^{\operatorname{ad}_a}\bigr)^{*}$; the isometry $\alpha$ has $\alpha^{*} = \alpha$, and the exponential of the skew-adjoint $\operatorname{ad}_a$ has adjoint the exponential of $-\operatorname{ad}_a$, $\bigl(e^{\operatorname{ad}_a}\bigr)^{*} = e^{-\operatorname{ad}_a}$; hence $\rho_a^{*} = \alpha e^{-\operatorname{ad}_a}$, which is the inverse of $e^{\operatorname{ad}_a}\alpha$ computed in Reflections as Signed Two-Sided Operators on a Lie Algebra.

Corollary. The reflection is a unitary operator of the form and its adjoint is another reflection, the signed conjugation by $-\alpha(a)$,

$$ \rho_a^{*} = \rho_{-\alpha(a)} , $$

since $\alpha e^{-\operatorname{ad}_a} = e^{-\operatorname{ad}_{\alpha(a)}}\alpha = e^{\operatorname{ad}_{-\alpha(a)}}\alpha$.

Proof. Move $\alpha$ past the exponential with $\alpha e^{-\operatorname{ad}_a} = e^{-\operatorname{ad}_{\alpha(a)}}\alpha$.

Self-Adjointness

Theorem (the self-adjoint reflections). The reflection is self-adjoint if and only if it is an involution,

$$ \rho_a^{*} = \rho_a \iff \rho_a^{2} = \mathrm{id} \iff a + \alpha(a)\in Z(\mathrm{G}) , $$

and the self-adjoint reflections are exactly the involutive reflections; in particular the reflection by an odd element is self-adjoint.

Proof. The reflection is unitary, so $\rho_a^{*} = \rho_a^{-1}$; the identity $\rho_a^{-1} = \rho_a$ is the involution, whose criterion is the centrality of $a+\alpha(a)$ from Reflections as Signed Two-Sided Operators on a Lie Algebra. For an odd element $a + \alpha(a) = 0$, which is central.

Corollary. The reflection by an even element is self-adjoint exactly when $2a$ is central; the reflection by a mixed pair reduces to the even and the odd cases componentwise, and the reflection by the zero element is the grade involution $\alpha$, which is self-adjoint and an involution.

Proof. For an even element $\alpha(a) = a$ and the criterion is $2a$ central; the componentwise statement is the linearity of the map $a\mapsto a+\alpha(a)$; the zero case is immediate.

The Fixed Subalgebra and the Form

Theorem. The fixed subalgebra of a self-adjoint reflection is the orthogonal complement of the image of the operator $\mathrm{id} - \rho_a$,

$$ \operatorname{Fix}(\rho_a) = \bigl(\operatorname{im}(\mathrm{id}-\rho_a)\bigr)^{\perp} , $$

and for an involutive reflection the algebra decomposes into the fixed subalgebra and its complement on which the reflection acts by $-\mathrm{id}$.

Proof. A self-adjoint operator has its fixed set equal to the orthogonal complement of the image of $\mathrm{id}-\rho_a$, because $H((\mathrm{id}-\rho_a)x, y) = H(x,(\mathrm{id}-\rho_a)y)$; for an involution the operator $\mathrm{id}-\rho_a$ is twice the projection onto the anti-fixed part.

The Failure of Self-Adjointness

The Even Carrying Element

Proposition. For an even carrying element $a$ with $2a$ not central the reflection is unitary but not self-adjoint; the defect

$$ \rho_a - \rho_a^{*} = \rho_a - \rho_a^{-1} $$

does not vanish, and the reflection is a unitary operator whose inverse is a different reflection, $\rho_a^{*} = \rho_{-\alpha(a)} = \rho_{-a}$.

Proof. The self-adjointness criterion is not satisfied, and the adjoint is the reflection by the negative of the parameter; the two reflections differ exactly when the involution condition fails.

The Inner Involution

Proposition. If the grade involution is inner, $\alpha = e^{\operatorname{ad}_w}$, then the reflection is the inner automorphism $e^{\operatorname{ad}_{a+w}}$, which is unitary; it is self-adjoint exactly when it is an involution, that is when $2(a+w)$ is central, and in the generic case it is unitary but not self-adjoint.

Proof. Substitute $\alpha = e^{\operatorname{ad}_w}$; the reflection is the inner automorphism of the sum, and the criterion is the centrality of the doubled generator.

The Degenerate Form

Proposition. If the form $H$ is degenerate, the adjoint of a reflection need not exist, and if the grade involution is not an isometry of $H$ the adjoint of the reflection is not its inverse: in the first case the operator has no adjoint on the radical, and in the second the reflection fails to preserve the form, so it is not unitary; the reflection is unitary and self-adjoint exactly in the non-degenerate isometric case and under the involution condition.

Proof. The adjoint exists for every operator exactly when $H$ is non-degenerate; the unitarity of $\rho_a$ uses $\alpha^{*} = \alpha$, which is the isometry of the involution. The Heisenberg algebra with its degenerate form is the standard example, as in The Signed Adjoint Sandwich on a Lie Algebra.

The Non-Unitary Reflection

Proposition. If the involution is not an isometry, the adjoint of the reflection is $\rho_a^{*} = \alpha^{*}e^{-\operatorname{ad}_a}$, which differs from $\rho_a^{-1}$ by the defect $\alpha^{*}\alpha^{-1}$, and the reflection is then only a similarity of the form with the defect of the involution; the reflection is unitary exactly when the involution is.

Proof. Compute the product $\rho_a^{*}\rho_a = \alpha^{*}e^{-\operatorname{ad}_a}e^{\operatorname{ad}_a}\alpha = \alpha^{*}\alpha$, which is the identity exactly when $\alpha$ is an isometry.

Examples

The Cartan Involution

For $a = 0$ the reflection is the grade involution $\rho_0 = \alpha$, which is an involution and an isometry, hence self-adjoint and unitary; this is the Cartan involution of The Cartan Decomposition and the Cartan Involution read as the simplest reflection, and its fixed subalgebra is the maximal compact subalgebra.

The Odd Reflections

Let $\mathrm{G} = \mathrm{sl}_2(\mathbb{R})$ with the Cartan involution $\alpha$ as the grading and the invariant form of signature $(2,1)$; for an odd element $a$, that is a symmetric traceless matrix, the reflection $\rho_a = e^{\operatorname{ad}_a}\alpha$ is a self-adjoint unitary involution, its fixed subalgebra is one-dimensional, and the reflection is a hyperbolic involution of the algebra whose geometric reading is Part IV.

The Even Reflection

Let $\mathrm{G} = \mathrm{gl}(n)$ with $\alpha = \mathrm{id}$ and $H(X,Y) = \operatorname{tr}(X\overline{Y})$; the reflection is the inner automorphism $e^{\operatorname{ad}_a}$, which is unitary for the form; it is self-adjoint exactly when $2a$ is central, so a reflection by a traceless diagonal element with distinct eigenvalues is unitary but not self-adjoint.

Summary

With respect to a non-degenerate invariant Hermitian form preserved by the grade involution, the reflection $\rho_a = e^{\operatorname{ad}_a}\alpha$ has the adjoint $\rho_a^{*} = \alpha e^{-\operatorname{ad}_a} = \rho_a^{-1}$, so every reflection is unitary and its adjoint is the reflection by $-\alpha(a)$; the reflection is self-adjoint exactly when it is an involution, which is the centrality of $a+\alpha(a)$, and this holds automatically for every odd element, while for an even element it requires $2a$ central. The self-adjoint reflection has its fixed subalgebra equal to the orthogonal complement of the image of $\mathrm{id}-\rho_a$, and the algebra then splits into the fixed part and the anti-fixed part. The self-adjointness fails in the degenerate forms of the construction: for an even carrying element with $2a$ not central the reflection is unitary but not self-adjoint — the reflection by a generic inner automorphism — and for a degenerate form the adjoint need not exist, while for a non-isometric involution the reflection is not unitary at all, its defect being that of the involution. The Cartan involution is the reflection by the zero element, self-adjoint and unitary, and the reflections by the odd elements are the self-adjoint unitary involutions of the algebra.

Summary of Notation

Symbol	Meaning
$H$	the invariant Hermitian form, preserved by the involution
$\rho_a = e^{\operatorname{ad}_a}\alpha$	the reflection, the signed conjugation
$\rho_a^{*} = \alpha e^{-\operatorname{ad}_a} = \rho_a^{-1}$	the adjoint, the inverse
$\rho_a^{*} = \rho_{-\alpha(a)}$	the adjoint as a reflection
$\rho_a^{*} = \rho_a \iff \rho_a^{2} = \mathrm{id}$	the self-adjointness criterion
$a+\alpha(a)\in Z(\mathrm{G})$	the involution condition
$\operatorname{Fix}(\rho_a) = (\operatorname{im}(\mathrm{id}-\rho_a))^{\perp}$	the fixed subalgebra
$\rho_a^{}\rho_a = \alpha^{}\alpha$	the defect of the non-isometric involution
degenerate $H$	the failure of the existence of the adjoint
$\rho_0 = \alpha$	the Cartan involution as a reflection