Reflections as Signed Two-Sided Operators on a Bimodule over an Algebra

Introduction

A reflection of a graded bimodule is a two-sided operator of order two whose middle factor carries the grade involution. The Signed Sandwich on a Bimodule over an Algebra defined the operator $r_u=S^{\alpha}_{u,u^{-1}}$ and found that its square is the inner conjugation by $u\alpha(u)$; this article reads the construction backwards. Given an operator of order two that is built from a unit and the grade involution, which unit realises it, and when is the operator genuinely an involution? The answer pairs the reflections with the elements acting by an involution, that is, the units $u$ with $u\alpha(u)$ in the centralizer of the bimodule, and it fails in two ways: when $u\alpha(u)$ is not in that centralizer the operator has infinite order, and when the bimodule is not faithful different units realise the same reflection.

The article assumes the graded bimodule of The Signed Sandwich on a Bimodule over an Algebra, and the operator layer of Left and Right Multiplication of a Module and Module Endomorphisms. The signed operators built from the involution of the elements and from the adjoint belong to the * Operator Theory group and do not occur here. The article stays inside Part I: no distance, norm, form or limit, and reflection means an operator of order two without any geometric reading. Throughout, $A$ is a unital associative $R$-algebra with a grade involution $\alpha$, ${}_A M_A$ is a graded $(A,A)$-bimodule, $C_A(M)=\{w \in A : wx=xw \text{ for all } x \in M\}$ is the centralizer of the bimodule, and $r_u=S^{\alpha}_{u,u^{-1}}$.

Reflections and the Signed Sandwich

Definition

Definition. Let $u \in A^{\times}$. The signed inner conjugation by $u$ is the operator

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

A reflection of the graded bimodule is a signed two-sided operator $r_u$ that is an involution, $r_u^{2}=\mathrm{id}_M$.

Proposition. The signed inner conjugations are invertible, with

$$ r_u^{-1}=r_{\alpha(u)^{-1}}=r_{\alpha(u^{-1})}, $$

and they satisfy the product rule

$$ r_u\,r_v=\operatorname{conj}_{u\alpha(v)} \qquad (u,v \in A^{\times}), $$

where $\operatorname{conj}_w(x)=wxw^{-1}$. The product of two reflections is therefore an inner conjugation, and it is a reflection only when it can be rewritten in the form $r_w$; in general the reflections are not closed under composition.

Proof. The inverse is the corollary on invertibility of The Signed Sandwich on a Bimodule over an Algebra. For the product, the composition law gives $r_ur_v=S_{u\alpha(v),\,\alpha(v^{-1})u^{-1}}$; since $\alpha(v^{-1})=\alpha(v)^{-1}$ the second parameter is $\alpha(v)^{-1}u^{-1}=(u\alpha(v))^{-1}$, so the composite is the unsigned diagonal sandwich $S_{w,w^{-1}}$ with $w=u\alpha(v)$, that is $\operatorname{conj}_w$. A reflection as a set of operators is the image of $A^{\times}$ under $u\mapsto r_u$, which need not be a subgroup. $\square$

The criterion

Theorem. For $u \in A^{\times}$ the operator $r_u$ is a reflection if and only if $u\alpha(u) \in C_A(M)$:

$$ r_u^{2}=\operatorname{conj}_{u\alpha(u)}=\mathrm{id}_M \iff u\,\alpha(u) \in C_A(M). $$

When the condition holds, $r_u$ is the inner automorphism $\operatorname{conj}_{u\alpha(u)}$ of order two; when it fails, $r_u$ is not a reflection, its order being whatever the order of the coset of $u\alpha(u)$ in $A^{\times}/C_A(M)^{\times}$ is.

Proof. The square is $\operatorname{conj}_{u\alpha(u)}$ by the proposition, and an inner conjugation is the identity exactly when the conjugating element commutes with every element of $M$, which is the definition of $C_A(M)$. If the condition fails, the square is a nontrivial inner conjugation, so $r_u$ is not of order two and is not a reflection. Its order may be finite or infinite: $\operatorname{conj}_w^{k}=\operatorname{conj}_{w^{k}}$ is the identity exactly when $w^{k} \in C_A(M)$, and a unit outside $C_A(M)$ may still have a power inside it. $\square$

The theorem names the units that realise reflections.

Definition. A unit $u \in A^{\times}$ acts by an involution when the signed inner conjugation $r_u$ is an involution, that is, when $u\alpha(u) \in C_A(M)$.

Thus the reflections are exactly the operators $r_u$ with $u$ acting by an involution, and the correspondence to be studied is between the reflections and the cosets of these units.

The special case $u\alpha(u)=1$

Proposition. If $u\alpha(u)=1$, that is $\alpha(u)=u^{-1}$, then $u$ acts by an involution and

$$ r_u(x)=u\,\alpha(x)\,u^{-1}=u\,\alpha(x)\,\alpha(u)=u\,\alpha(xu). $$

Proof. $\alpha(u)=u^{-1}$ gives $u\alpha(u)=1 \in C_A(M)$, so the criterion holds; the displayed rewriting uses $u^{-1}=\alpha(u)$ and the automorphism property $\alpha(a)\alpha(b)=\alpha(ab)$. $\square$

The elements with $\alpha(u)=u^{-1}$ are the $\alpha$-antisymmetric units; they always act by an involution, and they form a subgroup of $A^{\times}$ when the centre contains a suitable element, as the examples show.

The Correspondence

The map from units to reflections

Theorem. For units $u,v \in A^{\times}$,

$$ r_u=r_v \iff v^{-1}u \in C_A(M). $$

Consequently the map $u \mapsto r_u$ is constant on the right cosets of the subgroup $C_A(M)^{\times}=C_A(M)\cap A^{\times}$ and separates distinct cosets; its image is the set of reflections, and the criterion "$u$ acts by an involution" depends only on the coset.

Proof. $r_u=r_v$ says $u\alpha(x)u^{-1}=v\alpha(x)v^{-1}$ for all $x$, that is $w\alpha(x)w^{-1}=\alpha(x)$ with $w=v^{-1}u$, that is $w\alpha(x)=\alpha(x)w$ for all $x$. Since $\alpha$ is bijective on $M$ this says $wy=yw$ for all $y \in M$, which is $w \in C_A(M)$. The subgroup statement is the proposition below, and the dependence on the coset is the proposition after it. $\square$

Proposition. $C_A(M)$ is a subalgebra of $A$ containing $Z(A)$, and $C_A(M)^{\times}$ is a subgroup of $A^{\times}$.

Proof. $C_A(M)$ is closed under addition, multiplication and the scalars, and contains the centre because central elements commute with everything. If $w \in C_A(M)$ is a unit then $w^{-1}x=w^{-1}xww^{-1}=w^{-1}wxw^{-1}=xw^{-1}$ for all $x$, so $w^{-1} \in C_A(M)$; hence the units of $C_A(M)$ form a subgroup of $A^{\times}$. $\square$

Proposition. If $u$ acts by an involution and $c \in C_A(M)^{\times}$, then $uc$ acts by an involution and $r_{uc}=r_u$.

Proof. $r_{uc}=r_u$ by the theorem, since $u^{-1}(uc)=c \in C_A(M)$; and the criterion is a property of the operator $r_u$, so it holds at $uc$ as well. $\square$

The quotient description

The correspondence is therefore a bijection of sets

$$ \{u \in A^{\times} : u\alpha(u) \in C_A(M)\} \big/ C_A(M)^{\times} \;\longleftrightarrow\; \{\text{reflections of } M\}, $$

sending the coset of $u$ to $r_u$. For a faithful bimodule $C_A(M)=Z(A)$ and the correspondence is between reflections and the cosets of $Z(A)^{\times}$ in the units that act by an involution.

Corollary (the faithful case). If $M$ is faithful over $A$, then $r_u=r_v$ exactly when $v^{-1}u \in Z(A)$, and $r_u$ is a reflection exactly when $u\alpha(u) \in Z(A)$.

Proof. $C_A(M)=Z(A)$ when $M$ is faithful, by the definition of the centralizer. $\square$

The elements of order two

Proposition. Suppose $\alpha(u)=u$ and $u^{2} \in C_A(M)$. Then $u$ acts by an involution and $r_u=\operatorname{conj}_u$. In particular every involution $u$ of $A$ with $u^{2}=1$ that commutes with the action on $M$ realises a reflection.

Proof. $u\alpha(u)=u^{2} \in C_A(M)$; and $r_u(x)=u\alpha(x)u^{-1}=uxu^{-1}$ because $\alpha(u)=u$, so $r_u=\operatorname{conj}_u$, whose square is $\operatorname{conj}_{u^{2}}=\mathrm{id}$. $\square$

The Degenerate Cases

Failure of the involution property

The first failure is the non-involutive operator.

Proposition. If $u\alpha(u) \notin C_A(M)$ then $r_u$ is not a reflection: its square is the nontrivial inner conjugation $\operatorname{conj}_{u\alpha(u)}$, so $r_u^{2}\neq\mathrm{id}_M$. Its order is the order of the class of $u\alpha(u)$ in $A^{\times}/C_A(M)^{\times}$ and may be finite or infinite.

Proof. Immediate from the criterion and the computation of the powers of $\operatorname{conj}_w$, namely $\operatorname{conj}_w^{k}=\operatorname{conj}_{w^{k}}$. $\square$

Example. Let $A=M_2(k)$ with $\alpha=\mathrm{id}$, $M=A$, and $u=E_{12}+I$. Then $u^{2}=I+2E_{12}\notin Z(A)=C_A(M)$, so $r_u=\operatorname{conj}_u$ has $r_u^{2}=\operatorname{conj}_{u^{2}}\neq\mathrm{id}$: the operator has infinite order and the unit does not act by an involution.

Failure of uniqueness

The second failure is the non-faithful module.

Proposition. If $C_A(M) \supsetneq Z(A)$ — for instance when $M$ is not faithful, or when the action of $A$ on $M$ factors through a quotient with a larger centre — then there are units $u \neq v$ with $r_u=r_v$: the element is not recoverable from the reflection.

Proof. Take $v=uc$ with $c \in C_A(M)^{\times}$, $c \neq 1$. Then $v \neq u$ but $r_v=r_u$ by the theorem. $\square$

Example. Let $A=M_2(k)$ act on $M=k^2$ through the quotient $k$ — that is, $A$ acts by scalars through an algebra homomorphism $A \to k$ — so that every element of $M_2(k)$ acts as a scalar and $C_A(M)$ is the whole algebra. Then every unit of $A$ realises the same signed inner conjugation on $M$, and the correspondence collapses: the reflection is one operator and the class of realising units is all of $A^{\times}$.

The trivial grade involution

Proposition. If $\alpha=\mathrm{id}$ then the criterion is $u^{2} \in C_A(M)$, the correspondence sends $u$ to $\operatorname{conj}_u$, and the reflections are the inner conjugations by units with $u^{2} \in C_A(M)$.

Proof. $u\alpha(u)=u^{2}$ and $r_u(x)=uxu^{-1}$ when $\alpha=\mathrm{id}$. $\square$

The trivial action of the grade involution

Proposition. If $\alpha$ fixes every element of $M$ then $r_u=\operatorname{conj}_u$ for every $u$, whatever $\alpha$ does on $A$; the signed and the unsigned inner conjugations coincide, and the criterion reduces to $u^{2} \in C_A(M)$.

Proof. $\alpha(x)=x$ for $x \in M$, so $r_u(x)=uxu^{-1}$, and $u\alpha(u)$ acts on $M$ as $u^{2}$. $\square$

Examples

(a) The regular bimodule. For $M={}_A A_A$ the bimodule is faithful and $C_A(M)=Z(A)$; the reflections are the $r_u$ with $u\alpha(u) \in Z(A)$, and $r_u=r_v$ exactly when $v^{-1}u \in Z(A)$. This is the criterion of the ring-level article, now read as a statement about cosets.

(b) The matrix algebra with an even/odd grading. For $A=M_2(k)$, $J=\operatorname{diag}(1,-1)$, $\alpha(X)=JXJ^{-1}$, and $M=A$, the unit $u=E_{12}+E_{21}$ has $\alpha(u)=-u$ and $u^{2}=I$, so $u\alpha(u)=-I \in Z(A)$ and $r_u$ is a reflection; the units $-u$ and $u$ realise the same reflection because $(-u)^{-1}u=-I \in Z(A)$.

(c) The antisymmetric units. For $A=\mathbb{H}$ with the grade involution $\alpha$ given by quaternion conjugation, a pure imaginary unit $u$ satisfies $\alpha(u)=-u$ and $u^{2}=-1$, so $u\alpha(u)=-u^{2}=1$ is central and every such unit acts by an involution: $r_u(x)=u\alpha(x)u^{-1}$ is a reflection. The correspondence is a bijection from the units of $\mathbb{H}$ modulo $\mathbb{R}^{\times}$ onto these reflections.

(d) A non-faithful module. For $A=M_n(k)$ acting on $k$ through a character $A \to k$, $C_A(M)=A$, and all units realise the same reflection; the correspondence is maximally degenerate.

Summary

For a graded bimodule ${}_A M_A$ with grade involution $\alpha$, the signed inner conjugation by a unit $u$ is $r_u(x)=u\alpha(x)u^{-1}$, with $r_u^{-1}=r_{\alpha(u)^{-1}}$ and $r_ur_v=\operatorname{conj}_{u\alpha(v)}$. It is a reflection exactly when $u\alpha(u)$ lies in the centralizer $C_A(M)$ of the bimodule, and the units with this property are the units that act by an involution; the elements with $\alpha(u)=u^{-1}$ are the basic examples. The map $u\mapsto r_u$ is constant on the right cosets of $C_A(M)^{\times}$ and separates distinct cosets, so the reflections correspond bijectively to the cosets of $C_A(M)^{\times}$ in the set of units acting by an involution; for a faithful bimodule the centralizer is the centre $Z(A)$. Two degenerations occur: a unit with $u\alpha(u) \notin C_A(M)$ gives an operator of infinite order and no reflection, and a non-faithful bimodule makes the centralizer larger than the centre, so distinct units realise the same reflection and the element cannot be recovered from the operator. When $\alpha=\mathrm{id}$ or when $\alpha$ fixes the module, the signed conjugations reduce to the ordinary ones and the criterion is $u^{2} \in C_A(M)$.

Summary of Notation

Symbol	Meaning
$A$	unital associative $R$-algebra with grade involution $\alpha$
${}_A M_A$	graded $(A,A)$-bimodule
$\alpha$	grade involution, $\alpha^{2}=\mathrm{id}$, compatible with the bimodule
$r_u=S^{\alpha}_{u,u^{-1}}$	signed inner conjugation by a unit, $x\mapsto u\alpha(x)u^{-1}$
$r_u^{2}=\operatorname{conj}_{u\alpha(u)}$	the square is an inner conjugation
$C_A(M)$	centralizer of the bimodule, $\{w : wx=xw \text{ for all } x\}$
$C_A(M)^{\times}$	the units of $C_A(M)$, a subgroup of $A^{\times}$
$u$ acts by an involution	$u\alpha(u) \in C_A(M)$, equivalently $r_u$ is a reflection
$r_ur_v=\operatorname{conj}_{u\alpha(v)}$	product of two signed inner conjugations
$Z(A)$	the centre of $A$, equal to $C_A(M)$ for a faithful $M$
$\alpha(u)=u^{-1}$	the antisymmetric units, always acting by an involution