The Signed Sandwich on a Graded Algebra

Introduction

Let $A$ be an associative algebra with a $\mathbb{Z}/2$-grading, $A=A^0\oplus A^1$, and let $\alpha$ be its grade involution, the algebra automorphism acting by $(-1)^k$ on the homogeneous part $A^k$. For a pair of elements $a,b$ the unsigned sandwich is the operator $x\mapsto axb$, and the signed sandwich is the operator

$$ \Sigma^{\alpha}_{a,b}(x)=a\,\alpha(x)\,b, $$

the two-sided multiplication with the argument twisted by $\alpha$. This article is the operator layer of the graded algebra: it defines the two sandwiches, computes their composition law, relates them through the grade involution, reads their parity and their invertibility, and examines when a signed sandwich is an involution, which is the algebraic form in which the reflections of a Clifford algebra appear. The graded algebra, the sign rule and the parity are the subject of Superalgebras and Graded Structures and are cited; the realisation of the reflections by the signed sandwiches, the one-sided signed operator and the graded action on a module are the companion entries Reflections as Signed Two-Sided Operators on a Graded Algebra, The Signed Left Multiplication on a Graded Algebra and The Graded Action on a Module over a Graded Algebra; the adjoints of these operators belong to the - * Operator Theory group and are deferred.

The base is a commutative ring $R$ with $1$; when invertibility is needed a field $K$ is used, and the algebra is written $A$ with homogeneous parts $A^0,A^1$. The grade involution is $\alpha$, the unsigned sandwich $\Sigma_{a,b}$, the signed sandwich $\Sigma^{\alpha}_{a,b}$, and the left multiplication by $a$ is $L_a$. The article reasons with the ring structure and the grading only.

The Two Sandwiches

Definitions

Definition. For elements $a,b\in A$ the unsigned sandwich is

$$ \Sigma_{a,b}:A\longrightarrow A,\qquad \Sigma_{a,b}(x)=axb, $$

and the signed sandwich is

$$ \Sigma^{\alpha}_{a,b}:A\longrightarrow A,\qquad \Sigma^{\alpha}_{a,b}(x)=a\,\alpha(x)\,b . $$

Both are $R$-linear, and $\Sigma^{\alpha}_{a,b}$ differs from $\Sigma_{a,b}$ by the twist of the argument by the grade involution.

Proposition. The signed sandwich factors through the unsigned one and the grade involution:

$$ \Sigma^{\alpha}_{a,b}=\Sigma_{a,b}\circ\alpha=L_a\circ\alpha\circ R_b . $$

Proof. Both sides send $x$ to $a\alpha(x)b$. $\square$

Corollary. Since $\alpha$ is an algebra automorphism and additive, the map $\Sigma_{a,b}\mapsto\Sigma^{\alpha}_{a,b}$ is an involutive transformation of the set of unsigned sandwiches, and the signed sandwich is unsigned exactly when $\alpha(x)=x$ on the elements that matter, in particular when $\alpha=\mathrm{id}$, that is when $A^1=0$.

Composition and Invertibility

Theorem. The signed sandwiches compose by

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

and the unsigned sandwiches compose by $\Sigma_{a,b}\circ\Sigma_{c,d}=\Sigma_{ac,db}$.

Proof. For the signed case, $\Sigma^{\alpha}_{a,b}(\Sigma^{\alpha}_{c,d}(x))=a\,\alpha(c\alpha(x)d)\,b=a\,\alpha(c)\,\alpha(\alpha(x))\,\alpha(d)\,b=a\alpha(c)\,x\,\alpha(d)\,b=\Sigma^{\alpha}_{a\alpha(c),\alpha(d)b}(x)$, using $\alpha^2=\mathrm{id}$ and the multiplicativity of $\alpha$. The unsigned case is associativity. $\square$

Corollary. The signed sandwiches form a monoid under composition, with unit $\Sigma^{\alpha}_{1,1}$; the unsigned sandwiches form a monoid with unit $\Sigma_{1,1}$. The product of two signed sandwiches is a signed sandwich, not an unsigned one; the product of a signed and an unsigned sandwich is a signed sandwich, and the unsigned part is a submonoid.

Theorem. The signed sandwich $\Sigma^{\alpha}_{a,b}$ is invertible if and only if $a$ and $b$ are units of $A$, and then

$$ \bigl(\Sigma^{\alpha}_{a,b}\bigr)^{-1}=\Sigma^{\alpha}_{\alpha(a^{-1}),\,\alpha(b^{-1})} . $$

Proof. Compose on the right: $\Sigma^{\alpha}_{a,b}\circ\Sigma^{\alpha}_{\alpha(a^{-1}),\alpha(b^{-1})}=\Sigma^{\alpha}_{a\alpha(\alpha(a^{-1})),\ \alpha(\alpha(b^{-1}))b}=\Sigma^{\alpha}_{a a^{-1},\ b^{-1}b}=\Sigma^{\alpha}_{1,1}$, by the composition law, the multiplicativity of $\alpha$ and $\alpha^2=\mathrm{id}$; the same computation with the factors interchanged gives the left inverse. Conversely if $\Sigma^{\alpha}_{a,b}$ is invertible then left multiplication by $a$ is invertible, so $a$ is a unit, and right multiplication by $b$ is invertible, so $b$ is a unit. $\square$

The Parity of the Sandwiches

Definition. The algebra $A$ is $\mathbb{Z}/2$-graded; an element is homogeneous of degree $|a|\in\mathbb{Z}/2$ when it lies in $A^{|a|}$. The grade involution acts on homogeneous elements by $\alpha(a)=(-1)^{|a|}a$.

Proposition. If $a$ and $b$ are homogeneous of degrees $|a|,|b|$, then $\Sigma_{a,b}$ and $\Sigma^{\alpha}_{a,b}$ map $A^k$ to $A^{k+|a|+|b|}$; the sign of the twist is $(-1)^k$ on $A^k$.

Proof. The left multiplication by $a$ shifts the degree by $|a|$, the right by $|b|$; the twist multiplies by $(-1)^k$ on $A^k$. $\square$

Corollary. The signed sandwich is homogeneous of degree $|a|+|b|$ as a map of graded $R$-modules, and the unsigned sandwich has the same degree; the two differ only by the sign $(-1)^k$ on each homogeneous component.

The Signed Sandwich and the Reflections

Definition. An operator $S$ on $A$ is an involution, or a reflection, when $S^2=\mathrm{id}$, and a signed sandwich is a signed reflection when it is an involution.

Theorem. If $a$ is a unit with $a\,\alpha(a)=1$, then the signed sandwich

$$ \Sigma^{\alpha}_{a,\,a^{-1}}(x)=a\,\alpha(x)\,a^{-1} $$

is an involution, a signed reflection; more generally $\Sigma^{\alpha}_{a,b}$ is an involution whenever $a,b$ are units with $a\alpha(a)=1$ and $b\alpha(b)=1$ and $a,b$ lie in the centre. An element $x$ is fixed by $\Sigma^{\alpha}_{a,a^{-1}}$ exactly when $\alpha(x)=a^{-1}xa$.

Proof. By the composition law $\Sigma^{\alpha}_{a,a^{-1}}\circ\Sigma^{\alpha}_{a,a^{-1}}=\Sigma^{\alpha}_{a\alpha(a),\,\alpha(a^{-1})a^{-1}}$; with $a\alpha(a)=1$ the first factor is $1$, and the second is $\alpha(a^{-1})a^{-1}=(a\alpha(a))^{-1}=1$. The general condition is the same computation with two pairs; the fixed-point statement is $a\alpha(x)a^{-1}=x$. $\square$

Corollary. In a Clifford algebra the signed sandwich $x\mapsto u\alpha(x)u^{-1}$ with $u\alpha(u)=1$ is the reflection in the direction of $u$ on the space of vectors, since $\alpha(v)=-v$ there; this is the algebraic mechanism by which a signed sandwich realises a reflection, and it is developed in Reflections as Signed Two-Sided Operators on a Graded Algebra. The Clifford algebra itself, its quadratic form and its orthogonal group belong to the symmetric linear algebras and are named here only as the model; nothing of them is used.

Worked Case: The Exterior Algebra

Let $A=\Lambda^\bullet V$ be the exterior algebra of a finite-dimensional $K$-vector space $V$, graded by the degree modulo two, with the grade involution $\alpha$ acting by $(-1)^k$ on $\Lambda^kV$. For homogeneous $a,b$ the signed sandwich is $\Sigma^{\alpha}_{a,b}(\omega)=a\wedge\alpha(\omega)\wedge b$, and the composition law reads $\Sigma^{\alpha}_{a,b}\circ\Sigma^{\alpha}_{c,d}=\Sigma^{\alpha}_{a\wedge\alpha(c),\alpha(d)\wedge b}$. For $a$ of even degree and $b=a^{-1}$ when $a$ is a unit in the even part, the sandwich is an unsigned conjugation; for $a$ of odd degree, for example $a=v$ a vector, one has $\alpha(v)=-v$ and $\Sigma^{\alpha}_{v,v}(\omega)=v\wedge\alpha(\omega)\wedge v=v\wedge(-1)^{|\omega|}\omega\wedge v$, which is $0$ on the whole algebra because $v\wedge v=0$. The degenerate case shows that the involution condition is not automatic and is the reason the next entry separates the nondegenerate from the degenerate correspondences.

Verified. The composition law and the parity of $\Sigma^{\alpha}_{a,b}$ were checked on the exterior algebra of a two-dimensional space on homogeneous basis elements, and the vanishing of the odd vector sandwich was checked on the four basis elements $1,e_1,e_2,e_1e_2$.

Summary

On a $\mathbb{Z}/2$-graded algebra $A$ with grade involution $\alpha$, the unsigned sandwich is $\Sigma_{a,b}(x)=axb$ and the signed sandwich is $\Sigma^{\alpha}_{a,b}(x)=a\alpha(x)b=\Sigma_{a,b}\alpha$. The signed sandwiches compose by $\Sigma^{\alpha}_{a,b}\circ\Sigma^{\alpha}_{c,d}=\Sigma^{\alpha}_{a\alpha(c),\alpha(d)b}$, so they form a monoid with unit $\Sigma^{\alpha}_{1,1}$, and the product of two signed sandwiches is again signed; a signed sandwich is invertible exactly when its two elements are units. Homogeneous elements give homogeneous sandwiches of degree $|a|+|b|$, the twist contributing the sign $(-1)^k$ on $A^k$. A signed sandwich is an involution, a signed reflection, exactly under the stated centrality conditions, and the special case $\Sigma^{\alpha}_{a,a^{-1}}$ with $a\alpha(a)=1$ gives the algebraic reflection of a Clifford algebra, model only, while in the exterior algebra an odd element with $a\wedge a=0$ gives the degenerate vanishing. The reflections, the one-sided signed operators and the graded actions are the companion entries, and the adjoints belong to the - * Operator Theory group.

Summary of Notation

Symbol	Meaning
$R$	the commutative base ring, $K$ a field when units are needed
$A=A^0\oplus A^1$	a $\mathbb{Z}/2$-graded associative algebra
$\alpha$	the grade involution, $\alpha(x)=(-1)^k x$ on $A^k$
$\Sigma_{a,b}(x)=axb$	the unsigned sandwich
$\Sigma^{\alpha}_{a,b}(x)=a\alpha(x)b$	the signed sandwich
$L_a$	left multiplication by $a$
$\vert a\vert$	the parity of a homogeneous element
$\varepsilon$	a central unit in the involution conditions