Reflections as Signed Two-Sided Operators on a Ring

Introduction

Among the signed two-sided operators of a graded ring, the ones of order two have a special place: they are the reflections. If $A$ carries a grade involution $\alpha$ and $u$ is a unit, the signed inner conjugation $$r_u(x) = u\,\alpha(x)\,u^{-1}$$ is an automorphism of $A$ for every unit $u$, and it is an involution precisely when $u\alpha(u)$ is central. This article studies the reflections as a class: the correspondence between a reflection and the unit that defines it, the subgroup and coset structure they carry inside $\operatorname{Aut}(A)$, the criterion that singles out the involutive ones, and the degenerate cases in which the correspondence fails.

A reflection is not a new kind of operator — it is the diagonal signed sandwich $S^{\alpha}_{u,u^{-1}} = \operatorname{conj}_u\circ\alpha$ of The Signed Sandwich on a Ring — but it is the one class of two-sided operators that acts on the elements as an automorphism of order two, and it is the algebraic skeleton of the geometric reflection of a quadratic space. That geometric reading requires forms and belongs to Part II and Part VI; nothing here measures an angle, a length or a distance, and "reflection" is a purely operator-theoretic name for an order-two automorphism of this shape.

Throughout, $A$ is a ring with $1 \neq 0$, not assumed commutative; $\alpha$ is a grade involution, $\alpha^2 = \mathrm{id}$; $A^\times$ and $Z(A)$ are the unit group and the centre; $\operatorname{conj}_u(x) = uxu^{-1}$ is the inner automorphism; and $r_u = \operatorname{conj}_u\circ\alpha$ is the reflection by the unit $u$. The set-theoretic and coset facts about $\operatorname{Inn}(A)$ are from Inner Automorphisms of a Ring.

Reflections as Operators

The reflection by a unit

Definition. Let $u \in A^\times$. The reflection by $u$ is the operator

$$ r_u : A \to A, \qquad r_u(x) = u\,\alpha(x)\,u^{-1} . $$

It is the signed inner conjugation of The Signed Sandwich on a Ring, the diagonal signed sandwich $S^{\alpha}_{u,u^{-1}}$.

Proposition. For every unit $u$, $r_u$ is a unital ring automorphism of $A$, equal to the composite

$$ r_u = \operatorname{conj}_u \circ \alpha = \alpha \circ \operatorname{conj}_{\alpha(u)}, $$

with inverse $r_u^{-1} = r_{\alpha(u)^{-1}}$; and for all units $u, v$,

$$ r_u \circ r_v = \operatorname{conj}_{u\,\alpha(v)}, \qquad r_u = r_v \iff uv^{-1} \in Z(A). $$

Proof. $r_u = \operatorname{conj}_u\alpha$ is a composite of automorphisms, hence an automorphism. The identity $\operatorname{conj}_u\alpha = \alpha\operatorname{conj}_{\alpha(u)}$ is $\alpha\operatorname{conj}_{\alpha(u)}(x) = \alpha(\alpha(u)x\alpha(u)^{-1}) = u\alpha(x)u^{-1}$. The inverse: $\alpha\operatorname{conj}_{\alpha(u)}$ composed with $\alpha\operatorname{conj}_{\alpha(u)^{-1}}$... equivalently $r_u r_{\alpha(u)^{-1}}(x) = u\alpha(\alpha(u)^{-1}\alpha(x)\alpha(u))u^{-1} = u\alpha(\alpha(u)^{-1})x\alpha(\alpha(u))u^{-1} = u\alpha(u)^{-1}x\alpha(u)u^{-1} = x$. For the product, $r_ur_v = \operatorname{conj}_u\alpha\operatorname{conj}_v\alpha = \operatorname{conj}_u\operatorname{conj}_{\alpha(v)}\alpha^2 = \operatorname{conj}_{u\alpha(v)}$. Finally $r_u=r_v \iff \operatorname{conj}_u = \operatorname{conj}_v$, and that holds exactly when $uv^{-1}\in Z(A)$.

Corollary. The assignment $u \mapsto r_u$ is a well-defined injective map $A^\times/Z(A)^\times \to \operatorname{Aut}(A)$ whose image is the coset $\operatorname{Inn}(A)\,\alpha$ of the subgroup $\operatorname{Inn}(A)$ generated with $\alpha$: the reflections are the elements of $\operatorname{Aut}(A)$ of the form (an inner automorphism)$\circ\alpha$. In particular, the number of distinct reflections equals the order of the inner automorphism group at the units, $|A^\times / Z(A)^\times|$.

Remark. The reflections do not form a subgroup of $\operatorname{Aut}(A)$: the product of two of them is the inner automorphism $\operatorname{conj}_{u\alpha(v)}$, not a reflection, by the proposition. They form a coset of $\operatorname{Inn}(A)$ in the subgroup generated by $\operatorname{Inn}(A)$ and $\alpha$, so a reflection composed with a reflection is never a reflection (unless $\operatorname{Inn}(A)$ is trivial).

When a Reflection is an Involution

The criterion

Theorem (the order-two criterion). Let $u \in A^\times$. The reflection $r_u$ is an involution, $r_u^2 = \mathrm{id}$, if and only if

$$ u\,\alpha(u) \in Z(A). $$

When this holds, $r_u$ is the inner automorphism $\operatorname{conj}_{u\alpha(u)}$ of order two.

Proof. From the product formula, $r_u^2 = r_u r_u = \operatorname{conj}_{u\alpha(u)}$, and an inner automorphism is the identity exactly when its conjugating element is central, by Inner Automorphisms of a Ring. Hence $r_u^2 = \mathrm{id} \iff u\alpha(u) \in Z(A)$.

Corollary. If $u$ is fixed by $\alpha$ then $u\alpha(u) = u^2$ and $r_u$ is an involution exactly when $u^2$ is central; if $u$ is negated by $\alpha$ and $2$ is invertible then $u\alpha(u) = -u^2$, again central exactly when $u^2$ is central. In both parities the criterion reduces to $u^2 \in Z(A)$.

The correspondence with the elements

The criterion organises the units into those that act by an involution and those that act by an automorphism of larger order.

Definition. A unit $u$ is a reflecting element of $(A,\alpha)$ when $u\alpha(u) \in Z(A)$; equivalently, when the reflection $r_u$ is an involution.

Proposition. The reflecting elements form a subgroup $A^\times_{\mathrm{refl}}$ of $A^\times$ containing $Z(A)^\times$, and the map $u \mapsto r_u$ induces a bijection

$$ A^\times_{\mathrm{refl}} / Z(A)^\times \;\longrightarrow\; \{\text{involutive reflections}\}, $$

so the involutive reflections are in bijection with the reflecting units modulo the centre. The larger set of all reflections is in bijection with $A^\times/Z(A)^\times$.

Proof. If $u\alpha(u)$ and $v\alpha(v)$ are central, then $(uv)\alpha(uv) = u v \alpha(u)\alpha(v) = u\alpha(u)\,v\alpha(v)$ is central, since $\alpha$ is a homomorphism and central elements commute; and $u^{-1}\alpha(u^{-1}) = (u\alpha(u))^{-1}$ is central. So the reflecting elements form a subgroup, and the bijection is the restriction of the map of the proposition above, whose kernel is $Z(A)^\times$.

Example. In $A = M_2(k)$ with $J = \operatorname{diag}(1,-1)$ and $\alpha = \operatorname{conj}_J$, the unit $u = E_{12} + E_{21}$ has $\alpha(u) = -u$ and $u\alpha(u) = -u^2 = -I$, which is central; so $u$ is a reflecting element and $r_u$ is an involution. The unit $I + E_{12}$ acts by $\alpha = \mathrm{id}$, and $(I+E_{12})^2 = I + 2E_{12}$ is not central, so it is not a reflecting element and $r_{I+E_{12}} = \operatorname{conj}_{I+E_{12}}$ has infinite order in characteristic $0$.

Degenerate Cases

Non-units

The construction uses $u^{-1}$, and the criterion uses $u\alpha(u)$. If $u$ is a zero divisor the operator $x \mapsto u\alpha(x)u^{-1}$ is not defined, and the multiplication $x \mapsto u\alpha(x)$ alone is the signed left multiplication of The Signed Left Multiplication on a Ring, not a reflection. The correspondence between reflections and elements therefore has domain the units, and it does not extend to the zero divisors.

Non-central $u\alpha(u)$

If $u$ is a unit but $u\alpha(u) \notin Z(A)$, then $r_u$ is an automorphism and not an involution: its square is the nontrivial inner automorphism $\operatorname{conj}_{u\alpha(u)}$, and $r_u$ has infinite order in characteristic $0$ as soon as some power of $u\alpha(u)$ is non-central. The reflection exists — it is a signed two-sided operator, and $r_u = \operatorname{conj}_u\alpha$ — but it is not of order two, and the name "reflection" must then be carried only by the criterion and not by the operator.

The trivial grading

If $\alpha = \mathrm{id}$, every reflection is the inner conjugation $r_u = \operatorname{conj}_u$, and the criterion reads $u^2 \in Z(A)$; the involutive reflections are the self-inverse inner automorphisms. For a commutative ring $\operatorname{Inn}(A) = 1$, all reflections are the identity, and the class is degenerate. In characteristic $2$ the decomposition into even and odd parts fails, since $-1 = 1$ and the equation $\alpha(x) = -x$ is $\alpha(x) = x$; the only grade involution that is not the identity on the $p$-th-power structure is the Frobenius-related one of The Frobenius Operator on a Field of Characteristic p, and a reflection with $\alpha = \mathrm{id}$ is again an inner conjugation.

The fixed subring

Proposition. For a unit $u$, the fixed set of the reflection $r_u$ is the subring

$$ A^{r_u} = \{x \in A : \alpha(x) = \operatorname{conj}_{u^{-1}}(x)\}, $$

which contains the central even elements $Z(A) \cap A_{\bar 0}$.

Proof. The fixed set of an automorphism is a subring: if $r_u(x) = x$ and $r_u(y) = y$ then $r_u(x \pm y) = x \pm y$ and $r_u(xy) = xy$, and $r_u(1) = 1$. Rewriting $r_u(x) = x$ as $u\alpha(x)u^{-1} = x$ and multiplying by $u^{-1}$ on the left and $u$ on the right gives the displayed description. A central element $z$ with $\alpha(z) = z$ satisfies $\operatorname{conj}_{u^{-1}}(z) = z = \alpha(z)$, so it lies in $A^{r_u}$.

Summary

For a ring $A$ with grade involution $\alpha$ and a unit $u$, the reflection $r_u(x) = u\alpha(x)u^{-1}$ is the diagonal signed sandwich $S^{\alpha}_{u,u^{-1}}$ and the composite $\operatorname{conj}_u\circ\alpha$; it is an automorphism with inverse $r_{\alpha(u)^{-1}}$, two reflections agree exactly when their units differ by a central unit, and the product of two reflections is the inner automorphism $\operatorname{conj}_{u\alpha(v)}$. The reflections are therefore the coset $\operatorname{Inn}(A)\alpha$ inside the subgroup generated by the inner automorphisms and $\alpha$, indexed by $A^\times/Z(A)^\times$, and they do not form a subgroup.

The reflection $r_u$ is an involution exactly when $u\alpha(u)$ is central; the units satisfying this are the reflecting elements, they form a subgroup containing the centre, and the involutive reflections are in bijection with the reflecting elements modulo the centre. The correspondence fails at three points: non-units are excluded because the construction needs $u^{-1}$; a unit with non-central $u\alpha(u)$ gives a reflection that is not of order two; and when $\alpha = \mathrm{id}$ (as in a commutative ring, or in characteristic $2$ with the trivial grading) the reflections collapse to the inner conjugations. The fixed set of a reflection is the subring where $\alpha$ and $\operatorname{conj}_{u^{-1}}$ agree. No form, length or angle is used; the geometric reflection of a quadratic space is a Part II and Part VI reading of the same operator.

Summary of Notation

Symbol	Meaning
$A$, $\alpha$	Ring with $1 \neq 0$ and its grade involution, $\alpha^2 = \mathrm{id}$
$r_u(x) = u\alpha(x)u^{-1}$	Reflection by a unit $u$; the signed inner conjugation
$r_u = \operatorname{conj}_u\circ\alpha = S^{\alpha}_{u,u^{-1}}$	The reflection as an inner automorphism composed with $\alpha$
$r_ur_v = \operatorname{conj}_{u\alpha(v)}$	Product of two reflections is inner
$r_u = r_v \iff uv^{-1} \in Z(A)$	Reflections indexed by $A^\times/Z(A)^\times$
$\operatorname{Inn}(A)\alpha$	The reflections as a coset of the inner automorphism group
$r_u^2 = \operatorname{conj}_{u\alpha(u)}$	Square of a reflection
$u\alpha(u) \in Z(A)$	Criterion for $r_u$ to be an involution
$A^\times_{\mathrm{refl}}$	Reflecting elements: units with $u\alpha(u)$ central
$\{x : \alpha(x) = \operatorname{conj}_{u^{-1}}(x)\}$	Fixed set of the reflection $r_u$