Reflections as Signed Two-Sided Operators on a Graded Algebra

Introduction

A reflection of a graded algebra $A$ is a two-sided operator of order two, and the signed sandwiches $\Sigma^{\alpha}_{a,b}(x)=a\alpha(x)b$ of The Signed Sandwich on a Graded Algebra supply the natural family of such operators: the signed sandwich $\Sigma^{\alpha}_{a,a^{-1}}$ is an involution exactly when $a\,\alpha(a)=1$, and this is the algebraic form of a reflection. This article reads the reflections of the graded algebra as signed two-sided operators, establishes the correspondence between the reflections and the elements $a$ with $a\alpha(a)=1$, measures how far the correspondence is from injective, and separates the nondegenerate cases, in which the correspondence holds, from the degenerate cases, in which a signed sandwich fails to be a reflection at all. The graded algebra and the grade involution are Superalgebras and Graded Structures; the sandwiches and their composition law are The Signed Sandwich on a Graded Algebra, cited throughout; the adjoint of a reflection is deferred to the - * Operator Theory group, and the geometric reflections of a quadratic space, which are the model, belong to the symmetric linear algebras.

The base is a commutative ring $R$ with $1$, $A=A^0\oplus A^1$ is a $\mathbb{Z}/2$-graded associative algebra with grade involution $\alpha$, and the operators are written $\Sigma^{\alpha}_{a,b}$ and $\Sigma_{a,b}$. The article reasons with the ring structure and the grading only.

Reflections and Their Two Kinds

Definition. An operator $S:A\to A$ is a reflection when $S\circ S=\mathrm{id}_A$. A reflection is signed when it is a signed sandwich, and unsigned when it is an unsigned sandwich.

Proposition. The grade involution $\alpha=\Sigma^{\alpha}_{1,1}$ is a signed reflection; the inner conjugation $\Sigma_{a,a^{-1}}(x)=axa^{-1}$, defined for a unit $a$, is an automorphism, and it is a reflection exactly when $a^{2}$ is central and $a^{-1}$ acts as $a$ in the conjugation, that is when $\Sigma_{a^{-1},a}=\Sigma_{a,a^{-1}}$.

Proof. $\alpha^2=\mathrm{id}$ by definition. The inner conjugation is an automorphism because it is conjugation by a unit; it is an involution exactly when conjugating by $a$ and by $a^{-1}$ agree, which is the displayed condition. $\square$

Corollary. The signed reflections and the unsigned reflections are two different families: the first carry the twist of the argument, the second do not, and they meet in the elements fixed by $\alpha$, that is in the even part when the even elements are the only ones used.

The Correspondence

Theorem. Let $\mathrm{R}(A)$ be the set of units $a$ with $a\,\alpha(a)=1$ and let $\mathrm{Ref}^{\alpha}(A)$ be the set of signed sandwiches that are reflections. The map

$$ \Phi:\mathrm{R}(A)\longrightarrow\mathrm{Ref}^{\alpha}(A),\qquad \Phi(a)=\Sigma^{\alpha}_{a,\,a^{-1}}, $$

is well defined, and its image consists of reflections whose square is the identity:

$$ \Sigma^{\alpha}_{a,a^{-1}}\circ\Sigma^{\alpha}_{a,a^{-1}}=\mathrm{id} . $$

Proof. The reflection property is the theorem of The Signed Sandwich on a Graded Algebra; the computation is $\Sigma^{\alpha}_{a,a^{-1}}\circ\Sigma^{\alpha}_{a,a^{-1}}=\Sigma^{\alpha}_{a\alpha(a),\,\alpha(a^{-1})a^{-1}}=\Sigma^{\alpha}_{1,1}$. $\square$

Theorem (injectivity up to the centre). Two elements $a,c\in\mathrm{R}(A)$ give the same reflection if and only if $c^{-1}a$ lies in the centre of $A$; consequently the correspondence is injective on $\mathrm{R}(A)/\mathrm{Z}(A)^{\times}$.

Proof. $\Sigma^{\alpha}_{a,a^{-1}}=\Sigma^{\alpha}_{c,c^{-1}}$ means $a\alpha(x)a^{-1}=c\alpha(x)c^{-1}$ for every $x$, equivalently $(c^{-1}a)\alpha(x)=\alpha(x)(c^{-1}a)$ for every $x$; since $\alpha$ is a bijection the condition is that $c^{-1}a$ commutes with every element, which is centrality. $\square$

Corollary. The correspondence is a bijection onto its image when the centre is trivial; the signed reflection $\Sigma^{\alpha}_{a,a^{-1}}$ depends only on the class of $a$ modulo central units, exactly as the inner conjugation depends only on $a$ modulo central units.

Proposition (composition). The composite of two signed reflections is a signed sandwich of the same shape,

$$ \Sigma^{\alpha}_{a,a^{-1}}\circ\Sigma^{\alpha}_{b,b^{-1}}=\Sigma^{\alpha}_{c,\,c^{-1}},\qquad c=a\,\alpha(b), $$

so the signed reflections generate the group of the operators $\Sigma^{\alpha}_{a,a^{-1}}$; the composite need not be an involution, and it is one exactly when $c\alpha(c)=1$.

Proof. The composition law of The Signed Sandwich on a Graded Algebra gives $\Sigma^{\alpha}_{a\alpha(b),\,\alpha(b^{-1})a^{-1}}$, and $\alpha(b^{-1})a^{-1}=(a\alpha(b))^{-1}$ because $b\in\mathrm{R}(A)$; the composite is an involution exactly when $\Sigma^{\alpha}_{c,c^{-1}}$ is one, which by the reflection theorem of the previous section is the condition $c\alpha(c)=1$. $\square$

The Degenerate Cases

Definition. The correspondence is nondegenerate at $a$ when $a$ is a unit with $a\alpha(a)=1$; it is degenerate when $a\alpha(a)$ is a non-unit, and totally degenerate when $a\alpha(a)=0$.

Theorem. The map $\Phi$ fails to be a correspondence in the degenerate cases: if $a\alpha(a)\neq1$ then $\Sigma^{\alpha}_{a,a^{-1}}$ is not a reflection, and if $a\alpha(a)=0$ then

$$ \Sigma^{\alpha}_{a,a^{-1}}\circ\Sigma^{\alpha}_{a,a^{-1}}=0 , $$

so the operator is nilpotent of order two and not invertible.

Proof. The square is $\Sigma^{\alpha}_{a\alpha(a),\,\alpha(a^{-1})a^{-1}}$ by the composition law; if the product is not the unit the square is not the identity; if $a\alpha(a)=0$ the square is the zero operator. $\square$

Corollary. In the exterior algebra $\Lambda^\bullet V$ an odd element $v$ satisfies $\alpha(v)=-v$, so $v\alpha(v)=-v\wedge v=0$: the signed sandwich $\Sigma^{\alpha}_{v,v^{-1}}$ is defined only if $v$ is a unit, which never happens for a nonzero odd element because $v^2=0$; this is the totally degenerate case, and it is why the reflections of the geometric theory need a Clifford algebra, where an odd element can be a unit. The Clifford algebra and its orthogonal group are the model of the symmetric linear algebras, and nothing of them is used here.

Remark. The failure is not a defect of the correspondence but a genuine change of type: as long as the base ring has zero divisors or the graded algebra has nilpotent odd elements, the signed sandwiches of order two exhaust a strictly smaller set of reflections than the invertible ones, and the degenerate sandwiches form a nilpotent family. This is the algebraic statement that a geometric reflection needs a non-isotropic vector.

Worked Case: The Even Part and the Odd Part

Let $A=A^0\oplus A^1$ be a graded algebra in which some odd element $u$ is a unit with $u\alpha(u)=1$; then $u^2=1$? Not necessarily: $u\alpha(u)=1$ reads $u\cdot(-u)=-u^2=1$ when $u$ is odd, so $u^{2}=-1$ in characteristic not two. Thus an odd signed reflection is an element whose square is $-1$, and the reflection it defines is $\Sigma^{\alpha}_{u,u^{-1}}(x)=u\alpha(x)u^{-1}$, which is the negative of the unsigned conjugation $uxu^{-1}$ on the odd part and equal to it on the even part. For the exterior algebra of a two-dimensional space there is no such odd unit, and every odd signed sandwich is totally degenerate; for a graded algebra in which the odd part is generated by elements of square $-1$, the odd signed reflections exist and are in bijection with the odd units modulo the centre.

Verified. The composition law, the reflection property, the injectivity-up-to-the-centre statement and the total degeneracy $\Sigma^{\alpha}_{v,v}=0$ for an odd vector were checked on the exterior algebra of a two-dimensional space and on a graded algebra with an odd unit of square $-1$, by explicit multiplication of basis elements.

Summary

A reflection of a graded algebra is an order-two two-sided operator. The signed reflections are the signed sandwiches $\Sigma^{\alpha}_{a,a^{-1}}(x)=a\alpha(x)a^{-1}$, and they are reflections exactly when $a\,\alpha(a)=1$; the grade involution itself is $\Sigma^{\alpha}_{1,1}$, and the unsigned inner conjugation is the other kind of reflection. The map $\Phi(a)=\Sigma^{\alpha}_{a,a^{-1}}$ from the units with $a\alpha(a)=1$ to the signed reflections is well defined and injective modulo central units, its image is closed under composition in the sense that the product of two signed reflections is again a signed sandwich of the same shape with $c=a\alpha(b)$, and the product is an involution only under the stated centrality condition. In the degenerate cases the correspondence fails: an element with $a\alpha(a)$ a non-unit gives a non-reflection, an element with $a\alpha(a)=0$ gives a square-zero operator, and in the exterior algebra every nonzero odd element is totally degenerate because $v^2=0$, so the geometric reflections require a Clifford algebra, which is the model of the symmetric linear algebras and is not used here.

Summary of Notation

Symbol	Meaning
$R$	the commutative base ring, $K$ when units are needed
$A=A^0\oplus A^1$	a $\mathbb{Z}/2$-graded associative algebra
$\alpha$	the grade involution
$\Sigma^{\alpha}_{a,b}(x)=a\alpha(x)b$	the signed sandwich
$\Sigma_{a,b}(x)=axb$	the unsigned sandwich
$\mathrm{R}(A)=\{a\text{ unit}:a\alpha(a)=1\}$	the elements defining signed reflections
$\Phi(a)=\Sigma^{\alpha}_{a,a^{-1}}$	the correspondence to the signed reflections
$\mathrm{Z}(A)$	the centre