The Signed Adjoint of the Reflection on an Algebra

Introduction

A reflection of an algebra with respect to a grade involution $\alpha$ is a signed conjugation $\rho_u(x)=u\,\alpha(x)\,u^{-1}$ of order two, the signed sandwich $S_{u,u^{-1}}$ of a reflector $u$, a unit whose product $u\alpha(u)$ is central. This article computes its adjoint: the twisted adjoint of the reflection by $u$ is the reflection by $\delta(u)=\sigma\alpha(u)$,

$$ \rho_u^{*_\sigma}=\rho_{\delta(u)}, \qquad \text{and for the plain pairing} \qquad \rho_u^{*}=\rho_{\alpha(u)}, $$

so the adjoint operation acts on the reflections by applying an anti-automorphism to the reflector. The reflection is self-adjoint for the twisted pairing exactly when $u^{-1}\delta(u)$ is central, that is when $\delta(u)$ is a central multiple of $u$, and this condition fails in interesting cases: it is automatic in the commutative algebras and when $\delta=\mathrm{id}$, and it is a genuine restriction for a matrix algebra with a non-symmetric reflector.

This article develops the adjoint of a reflection, the criterion for its self-adjointness, the cases in which the criterion is automatic, the unitary reflections beside the self-adjoint ones, and the interaction of the adjoint with the fixed subalgebra of the reflection. It assumes Reflections as Signed Two-Sided Operators on an Algebra for the reflectors $R^{\times}(A,\alpha)$ and the reflections $\mathrm{Ref}(A,\alpha)$, The Signed Sandwich on an Algebra for the signed conjugation and the relation $\rho_u^{2}=\iota_{u\alpha(u)}$, The Signed Adjoint Sandwich on an Algebra for the signed adjoint and the unitarity condition, The Adjoint of the Left Multiplication on an Algebra for the one-sided adjoints, Involutions of the Operator Algebra and The Adjoint in an Involutive Algebra for the adjoint operation and the two pairings, and Involutive Linear Algebras for the involutions. The graded reading is The Graded Adjoint Action on a Module over an Algebra; the geometric reflection and the metric form are Part II and Hilbert Algebras and Quadratic Forms and Clifford Algebras, named as the owner and not used. This article stays inside Part I: no distance, norm, form with a norm, topology or limit.

Throughout, $k$ is a field of characteristic not two, $A$ is a finite-dimensional unital associative $k$-algebra, $\tau$ is a trace whose pairing $\langle x,y\rangle=\tau(xy)$ is nondegenerate, $\sigma$ is an involution of $A$ with $\tau(\sigma(x))=\tau(x)$, and $\alpha$ is an involutive automorphism of $A$ commuting with $\sigma$ and preserving the trace. The twist is $\delta=\sigma\alpha$, the twisted pairing is $\{x,y\}=\tau(x\sigma(y))$, the unsigned sandwich is $T_{a,b}(x)=axb$, the signed sandwich is $S_{a,b}(x)=a\alpha(x)b$, the reflector is a unit $u$ with $u\alpha(u)\in Z(A)$, and the reflection is $\rho_u=S_{u,u^{-1}}$.

The Adjoint of a Reflection

Theorem. Let $u$ be a reflector, so that $\rho_u$ is an involutive automorphism. Then $\delta(u)$ is again a reflector, and

$$ \rho_u^{*_\sigma}=S_{\delta(u),\,\delta(u)^{-1}}=\rho_{\delta(u)}, \qquad \rho_u^{*}=S_{\alpha(u)^{-1},\,\alpha(u)}=\rho_{\alpha(u)} . $$

The twisted adjoint of the reflection by $u$ is the reflection by $\delta(u)$, and the plain adjoint is the reflection by $\alpha(u)$; both are reflections, and the adjoint operation carries the reflection set into itself.

Proof. The twisted adjoint of a signed sandwich is $S_{a,b}^{*_\sigma}=S_{\delta(a),\delta(b)}$ by The Signed Adjoint Sandwich on an Algebra; with $a=u$ and $b=u^{-1}$ this is $S_{\delta(u),\delta(u^{-1})}$, and $\delta(u^{-1})=\delta(u)^{-1}$ because $\delta$ is an anti-automorphism of order two. The result is the signed conjugation by $\delta(u)$, that is $\rho_{\delta(u)}$. For the plain pairing the adjoint of a signed sandwich is $S_{\alpha(b),\alpha(a)}$, which with $a=u$ and $b=u^{-1}$ is $S_{\alpha(u)^{-1},\alpha(u)}=\rho_{\alpha(u)}$. That $\delta(u)$ is a reflector: $\delta(u)\alpha(\delta(u))=\sigma(\alpha(u))\alpha(\sigma(\alpha(u)))=\sigma(\alpha(u))\sigma(u)=\sigma(u\alpha(u))$, which is central exactly when $u\alpha(u)$ is central, the involution $\sigma$ preserving centrality and being onto; so $\rho_{\delta(u)}$ is an involutive automorphism. That $\alpha(u)$ is a reflector is the same computation with $\delta$ replaced by $\alpha$: $\alpha(u)\alpha(\alpha(u))=\alpha(u)u=u\alpha(u)$, central.

Corollary. The adjoint operation acts on the set of reflections by the anti-automorphism $\delta$ for the twisted pairing and by the automorphism $\alpha$ for the plain one, and

$$ (\rho_u^{*_\sigma})^{*_\sigma}=\rho_u, \qquad (\rho_u^{*})^{*}=\rho_u, $$

so the action is an involution of the reflection set; the reflection by $1$ is $\alpha$, and it is fixed by both adjoints.

Proof. Order two is the order two of the adjoint operation and of $\delta$, respectively $\alpha$; the case $u=1$ gives $\rho_1=\alpha$, and $\alpha$ is self-adjoint for both pairings by The Signed Adjoint Sandwich on an Algebra.

Self-Adjointness

Theorem. A reflection $\rho_u$ is self-adjoint for the twisted pairing exactly when

$$ u^{-1}\delta(u) \in Z(A), $$

and self-adjoint for the plain pairing exactly when $u^{-1}\alpha(u) \in Z(A)$. Equivalently, $\rho_u^{*_\sigma}=\rho_u$ exactly when $\delta(u)=\lambda u$ for some central $\lambda \in Z(A)$.

Proof. Two signed conjugations by units $u$ and $v$ agree as operators exactly when $u^{-1}v$ is central: $\rho_v=\rho_u$ is $v\alpha(x)v^{-1}=u\alpha(x)u^{-1}$ for all $x$, that is $u^{-1}v\alpha(x)=\alpha(x)u^{-1}v$ for all $x$, and since $\alpha$ is onto this is the centrality of $u^{-1}v$. Applying this to the theorem gives the criterion with $v=\delta(u)$, respectively $v=\alpha(u)$; the last form is $\delta(u)=u\lambda$ with $\lambda=u^{-1}\delta(u)$ central.

Corollary. If the reflector $u$ is central then $\rho_u$ is self-adjoint for both pairings, since $\delta(u)$ and $\alpha(u)$ are then central as well and the quotients are central. If $u$ is fixed by $\delta$, respectively by $\alpha$, then the reflection is self-adjoint for the corresponding pairing.

Proof. A central $u$ has $u^{-1}\delta(u)$ central as the product of central elements, and a fixed $u$ has the quotient $1$, which is central.

Corollary (the reflection and the twisted involution). The reflection is self-adjoint for the twisted pairing if and only if the difference $\delta(u)-u$ is a central multiple of $u$, and the self-adjoint reflections by reflectors of the form $u=\lambda v$ with $\lambda$ central are exactly the reflections of the $v$ with $\delta(v)=v$; in particular the self-adjointness depends only on the reflector class modulo the central units.

Proof. $\delta(\lambda v)=\delta(v)\sigma(\lambda)$ for a central $\lambda$; for the twisted pairing and a central reflector the centrality bookkeeping is the statement that $\rho_{\lambda v}=\rho_{v}$ up to the central factor, and the criterion $u^{-1}\delta(u)$ central is unchanged when $u$ is multiplied by a central unit.

The Degenerate Cases

Theorem (the grade involution is the identity). If $\alpha=\mathrm{id}$ then the reflections are the inner involutions $\rho_u(x)=uxu^{-1}$ with $u^{2}$ central, $\rho_u^{*_\sigma}=\rho_{\sigma(u)}$ and $\rho_u^{*}=\rho_u$; every reflection is self-adjoint for the plain pairing, and it is self-adjoint for the twisted pairing exactly when $u^{-1}\sigma(u)$ is central.

Proof. With $\alpha=\mathrm{id}$ the signed sandwich is the unsigned one and the signed conjugation is the inner automorphism; the reflector condition $u\alpha(u)=u^{2}\in Z(A)$ is the classical one, and the two adjoint formulas specialize to $\rho_{\sigma(u)}$ and $\rho_{\alpha(u)}=\rho_u$. The last statement is the self-adjointness criterion with $\alpha=\mathrm{id}$.

Theorem (the anti-automorphism is the grade involution). If $\delta=\mathrm{id}$, equivalently $\sigma=\alpha$, then every signed sandwich is self-adjoint for the twisted pairing, $S_{a,b}^{*_\sigma}=S_{a,b}$, and in particular every reflection is self-adjoint. The case occurs exactly when $\sigma=\alpha$ and the algebra is commutative.

Proof. The twisted adjoint of a signed sandwich is $S_{\delta(a),\delta(b)}$, which is $S_{a,b}$ when $\delta=\mathrm{id}$; the equation $\sigma=\alpha$ between an anti-automorphism and an automorphism gives $\alpha(x)\alpha(y)=\alpha(y)\alpha(x)$ for all $x,y$, so the image of $\alpha$, which is $A$, is commutative, and conversely in a commutative algebra the identity is both an automorphism and an anti-automorphism.

Theorem (the commutative algebra). If $A$ is commutative then every unit is a reflector, every reflection is the grade involution $\alpha$, and every reflection is self-adjoint for both pairings.

Proof. In a commutative algebra $u\alpha(u)$ is central for every unit $u$, and $\rho_u(x)=u\alpha(x)u^{-1}=\alpha(x)$ because $u$ commutes with everything; the reflections are therefore all equal to $\alpha$, which is self-adjoint, and the general criterion is automatic because every element is central.

The two theorems describe the failure of the self-adjointness criterion as a degeneracy: it holds trivially when the algebra is commutative or when the twist $\delta$ is the identity, and it is a genuine condition only for a noncommutative algebra with $\delta\neq\mathrm{id}$. In that non-degenerate case the reflectors split into those whose reflection is self-adjoint and those whose reflection is not, and the split is by the criterion $u^{-1}\delta(u)\in Z(A)$.

The Unitary Reflections

Theorem. A reflection is unitary for the twisted pairing exactly when $\sigma(u)u$ is central and fixed by $\alpha$; a reflection that is both self-adjoint and unitary satisfies $\rho_u^{2}=\mathrm{id}$ as an operator and $\rho_u^{*_\sigma}=\rho_u=\rho_u^{-1}$.

Proof. The unitarity criterion is the corollary of The Signed Adjoint Sandwich on an Algebra for $b=\alpha(a)^{-1}$, that is $\sigma(u)u\in Z(A)$ and $\alpha(\sigma(u)u)=\sigma(u)u$. A self-adjoint unitary operator has $T^{*}=T$ and $T^{*}=T^{-1}$, so $T=T^{-1}$; for the reflection, $T^{2}=\mathrm{id}$ holds by the reflector condition.

Corollary. For a reflection that is self-adjoint, the unitarity is the condition that $\sigma(u)u$ be a central element of the even part; the orthogonal reflections of a matrix algebra with the transpose are the case in which $\sigma(u)u$ is a scalar.

Proof. The centrality and the $\alpha$-fixedness are the two clauses of the criterion, and an element fixed by the grade involution is even; for the matrix algebra with the transpose, a scalar $\sigma(u)u$ is central and fixed.

The Fixed Subalgebra of a Reflection

Theorem. Let $u$ be a reflector. The fixed set of the reflection,

$$ \mathrm{Fix}(\rho_u)=\{x\in A : u\,\alpha(x)=x\,u\}, $$

is a unital subalgebra of $A$, on which $\rho_u$ acts as the identity, and it contains the central elements fixed by the grade involution. The reflection preserves the trace and the plain pairing of the category,

$$ \tau(\rho_u(x))=\tau(x), \qquad \langle \rho_u(x),\rho_u(y)\rangle=\langle x,y\rangle , $$

so a reflection is an automorphism of the algebra that is also an automorphism of its pairing.

Proof. The reflection is a signed conjugation by the unit $u$, hence an automorphism, so its fixed set is a subalgebra; the unit is fixed because $\rho_u(1)=u\alpha(1)u^{-1}=uu^{-1}=1$, and a central $z$ with $\alpha(z)=z$ satisfies $u\alpha(z)=uz=zu$. For the trace, $\tau(u\alpha(x)u^{-1})=\tau(\alpha(x)u^{-1}u)=\tau(\alpha(x))=\tau(x)$ by the cyclic property and the invariance of $\tau$ under $\alpha$; then $\langle\rho_u(x),\rho_u(y)\rangle=\tau(u\alpha(x)u^{-1}u\alpha(y)u^{-1})=\tau(u\alpha(xy)u^{-1})=\tau(\alpha(xy))=\tau(xy)$, using the multiplicativity of $\alpha$ and the trace computation again.

Corollary. The fixed subalgebra of the twisted adjoint is the fixed subalgebra of the reflected reflector, $\mathrm{Fix}(\rho_u^{*_\sigma})=\mathrm{Fix}(\rho_{\delta(u)})$, and that of the plain adjoint is $\mathrm{Fix}(\rho_u^{*})=\mathrm{Fix}(\rho_{\alpha(u)})$; hence self-adjointness is sufficient for the equality of the fixed subalgebras, $\rho_u^{*_\sigma}=\rho_u$ implying $\mathrm{Fix}(\rho_u)=\mathrm{Fix}(\rho_{\delta(u)})$, and the criterion $u^{-1}\delta(u)\in Z(A)$ of the earlier section gives the exact condition.

Proof. The adjoints are the reflections $\rho_{\delta(u)}$ and $\rho_{\alpha(u)}$ by the theorem of the first section, and their fixed sets are computed as above; two equal operators have equal fixed sets, which is the implication.

Examples

(a) The matrix algebra with the transpose. $A=M_n(k)$, $\sigma$ the transpose, $\alpha$ the grade involution of a $\mathbb{Z}/2$-grading; a reflector $u$ has $\rho_u^{*_\sigma}=\rho_{\delta(u)}$ and $\rho_u$ is self-adjoint exactly when $\delta(u)$ is a scalar multiple of $u$. For $\alpha=\mathrm{id}$ this is the condition $u^{\mathsf{T}}=\lambda u$ with $\lambda$ scalar, the symmetry of $u$ up to a scalar; a generic reflector fails it.

(b) The group algebra with a parity. $A=k[G]$ with a parity homomorphism $\chi$ and $\sigma(g)=g^{-1}$; a reflector is a group element $g$ with $g\alpha(g) = \chi(g)\,g^{2}$ central, and $\rho_g^{*_\sigma}=\rho_{\delta(g)}$ with $\delta(g)=\chi(g)g^{-1}$. The reflection is self-adjoint exactly when $g^{-1}\delta(g)=\chi(g)g^{-2}$ is central, that is when $g^{2}\in Z(k[G])$, since $\chi(g)$ is the scalar $\pm1$.

(c) The Clifford algebra. For a Clifford algebra with its grading and an odd element $v$ of square a scalar, the reflection $\rho_v$ is the reflection in the vector $v$; it is self-adjoint for the twisted pairing when $\sigma(v)v$ is central, and the classical statement that a vector reflection is self-adjoint for the bilinear form is the case in which that product is a scalar. The metric reading of the form is Part II.

(d) The commutative case. For a commutative algebra every reflection is the grade involution and is self-adjoint for both pairings, so the criterion of this article is vacuous; this is the extreme degenerate case, and it shows that the self-adjointness is a condition on the noncommutativity of the algebra as much as on the reflection.

Summary

The reflection $\rho_u=S_{u,u^{-1}}$ by a reflector $u$ has twisted adjoint $\rho_u^{*_\sigma}=\rho_{\delta(u)}$ and plain adjoint $\rho_u^{*}=\rho_{\alpha(u)}$, and both are again reflections, so the adjoint operation acts on the reflection set by the two order-two maps $\delta$ and $\alpha$; the reflection by $1$ is the grade involution $\alpha$, self-adjoint for both pairings. The reflection is self-adjoint for the twisted pairing exactly when $u^{-1}\delta(u)$ is central, equivalently when $\delta(u)$ is a central multiple of $u$, and for the plain pairing exactly when $u^{-1}\alpha(u)$ is central; central reflectors and reflectors fixed by the twist always give self-adjoint reflections. The criterion is automatic in the degenerate cases — $\alpha=\mathrm{id}$, where the reflections are the inner involutions; $\delta=\mathrm{id}$, where every signed sandwich is self-adjoint; and the commutative algebras, where every reflection is the grade involution — and it is a genuine condition only for a noncommutative algebra with $\delta\neq\mathrm{id}$. A reflection is unitary for the twisted pairing exactly when $\sigma(u)u$ is central and even, and a reflection that is both self-adjoint and unitary is its own inverse adjoint. The fixed set of a reflection is a unital subalgebra, it contains the central elements fixed by the grade involution, and the reflection preserves the trace and the plain pairing of the category; self-adjointness forces the fixed subalgebra of the twisted adjoint to coincide with that of the reflection. The geometric reflection and the metric form are Part II.

Summary of Notation

Symbol	Meaning
$S_{a,b}(x)=a\alpha(x)b$	the signed sandwich
$\rho_u=S_{u,u^{-1}}$, $\rho_u(x)=u\alpha(x)u^{-1}$	the reflection by the reflector $u$
$R^{\times}(A,\alpha)$, $\mathrm{Ref}(A,\alpha)$	the reflectors and the reflections
$\alpha$, $\sigma$, $\delta=\sigma\alpha$	the grade involution, the involution, and their twist
$\rho_u^{*_\sigma}=\rho_{\delta(u)}$	the twisted adjoint of a reflection
$\rho_u^{*}=\rho_{\alpha(u)}$	the plain adjoint of a reflection
$u^{-1}\delta(u)\in Z(A)$	self-adjointness for the twisted pairing
$u^{-1}\alpha(u)\in Z(A)$	self-adjointness for the plain pairing
$\mathrm{Fix}(\rho_u)=\{x:u\alpha(x)=xu\}$	the fixed subalgebra of a reflection
$\tau(\rho_u(x))=\tau(x)$, $\langle\rho_ux,\rho_uy\rangle=\langle x,y\rangle$	a reflection preserves the trace and the pairing
$\alpha=\mathrm{id}$	the reflections are the inner involutions
$\delta=\mathrm{id}$	every signed sandwich is self-adjoint
$\sigma(u)u\in Z(A)$, $\alpha$-fixed	the unitary reflections