The Signed Left Multiplication on a Clifford Algebra

Introduction

The left multiplication $L_a$ of a Clifford algebra acts by $y\mapsto ay$, and it is multiplicative in the written order. Composing it with the grade involution gives the signed left multiplication

$$ \mathrm{L}^{\alpha}_a = L_a\circ\alpha , \qquad \mathrm{L}^{\alpha}_a(y) = a\,\alpha(y) , $$

the ordinary left multiplication applied to the twisted argument. It agrees with $L_a$ on the even part of the algebra and is $-L_a$ there where the argument is odd; it composes according to $L_{a\alpha(c)}$, so two signed left multiplications multiply to an ordinary one; and it is one half of the signed sandwich of The Signed Sandwich on a Clifford Algebra, the other half being an ordinary right multiplication. The article treats this operator, its fixed elements, and the precise sense in which a one-sided operator is half a motion of the quadratic space.

A word of disambiguation begins the account, because the corpus writes two different operators with the adjective signed in front of "left multiplication". In this article the twist is on the argument, $\mathrm{L}^{\alpha}_a=L_a\alpha$; in One-Sided Operators on a Clifford Algebra with Signed Inner Conjugation and The Graded Multiplication Operators the twist is on the parameter, $\Lambda^{\alpha}_x(y)=\alpha(x)y$. The two agree only on the even part, and their purposes differ: the parameter twist belongs to the inner conjugation and the versor theory, the argument twist to the signed product and the reflections. Only the argument twist is treated here.

The boundaries. The one-sided calculus, the commutant and the composition of the ordinary families are One-Sided Operators on a Clifford Algebra; the signed product, its composition law and its fixed elements are One-Sided Operators with the Signed Product, and the two-sided version is Two-Sided Operators with the Signed Product; the parameter-twisted family is The Graded Multiplication Operators. The signed sandwich formed from the two factors is The Signed Sandwich on a Clifford Algebra, and the reflections are Reflections as Signed Two-Sided Operators on a Clifford Algebra. The base is a field $F$ of characteristic not $2$, $q$ a non-degenerate quadratic form with $q(u)=B(u,u)$ and $uv+vu=2B(u,v)$.

The Operator

Definition and Elementary Properties

Definition. For $a\in\mathrm{Cl}(V,q)$ the signed left multiplication is the $F$-linear operator

$$ \mathrm{L}^{\alpha}_a = L_a\circ\alpha : \mathrm{Cl}(V,q)\longrightarrow\mathrm{Cl}(V,q), \qquad \mathrm{L}^{\alpha}_a(y) = a\,\alpha(y) . $$

Proposition. The following hold.

(a) $\mathrm{L}^{\alpha}_a = L_a\alpha = \alpha L_{\alpha(a)}$.

(b) For a homogeneous element $a$ of degree $k$ one has $\mathrm{L}^{\alpha}_a = (-1)^{k}L_a$ on the odd part of the algebra and $\mathrm{L}^{\alpha}_a = L_a$ on the even part.

(c) $\mathrm{L}^{\alpha}_1 = \alpha$, and $\mathrm{L}^{\alpha}_a$ is invertible exactly when $a$ is a unit.

(d) $\mathrm{L}^{\alpha}_a$ is a homogeneous operator of parity $|a|$.

Proof. (a) is $\alpha L_{\alpha(a)}(y)=\alpha(\alpha(a)y)=a\alpha(y)$. (b) For $y$ homogeneous of degree $j$, $\alpha(y)=(-1)^jy$; on the even part $j=0$ and on the odd part $j=1$, so the operator differs from $L_a$ by $-1$ on the odd part. (c) is the value at $1$ and the invertibility of $L_a$. (d) The shift of the grading is that of $L_a$, because $\alpha$ preserves the grading.

Remark. The operator $\mathrm{L}^{\alpha}_a$ is the composition of a multiplication operator with the parity operator $\Gamma(y)=\alpha(y)$; it is therefore the conjugate $L_a\alpha=\Gamma L_{\alpha(a)}\Gamma$, and the twisting enters through $\Gamma$, as The Graded Multiplication Operators records for the parameter-twisted family. Since $\Gamma$ is an involution that is the identity on the even part, every statement about $\mathrm{L}^{\alpha}_a$ is a statement about $L_a$ read separately on the two parity components.

Composition and Fixed Elements

Proposition (composition). For all $a,c$,

$$ \mathrm{L}^{\alpha}_a\circ\mathrm{L}^{\alpha}_c = L_{a\alpha(c)} , \qquad \mathrm{L}^{\alpha}_a\circ L_c = \mathrm{L}^{\alpha}_{ac} , \qquad L_a\circ\mathrm{L}^{\alpha}_c = \mathrm{L}^{\alpha}_{a\alpha(c)} . $$

Two signed left multiplications compose to an ordinary left multiplication, the two grade involutions cancelling; a signed and an ordinary one compose to a signed one. The signed left family is therefore a coset $L(\Gamma)\alpha$ of the group of invertible left multiplications and not a group, exactly as for the two-sided family.

Proof. $\mathrm{L}^{\alpha}_a\mathrm{L}^{\alpha}_c=L_a\alpha L_c\alpha=L_aL_{\alpha(c)}\alpha^2=L_{a\alpha(c)}$; the other two are the same computation with one factor untwisted.

Proposition (the fixed elements). The fixed space of $\mathrm{L}^{\alpha}_a$ is $\{y : a\alpha(y)=y\}$. For the scalar $a=\lambda$ it is the even part for $\lambda=1$, the odd part for $\lambda=-1$, and $\{0\}$ for every other $\lambda$. For a general $a$ it is the translate of a subspace by the equation $a\alpha(y)=y$, and it is not a graded subspace unless $a$ is scalar or the equation degenerates.

Proof. The defining equation is linear in $y$; for $a=\lambda$ it reads $\lambda\alpha(y)=y$, which on the even component is $(\lambda-1)y=0$ and on the odd component is $(-\lambda-1)y=0$, giving the three cases. The general case is the same equation with a non-scalar coefficient, and its solution space is not stable under the grading because $a$ need not be even.

The Operator as Half a Motion

Proposition (the pairing with the right factor). For a unit $x$ the composition of the signed left multiplication with the ordinary right multiplication by the inverse of the twisted parameter is the signed sandwich,

$$ \mathrm{L}^{\alpha}_x\circ R_{\alpha(x)^{-1}} = T^{\alpha}_{x,\alpha(x)^{-1}} , $$

and for a vector $u$ with $q(u)\ne0$ this is the reflection $\rho_u$; the signed left multiplication alone is not a motion of $V$.

Proof. $(\mathrm{L}^{\alpha}_xR_{\alpha(x)^{-1}})(y)=x\alpha(y)\alpha(x)^{-1}$, which is the signed sandwich of The Signed Sandwich on a Clifford Algebra with parameters $(x,\alpha(x)^{-1})$; for $x=u$ a vector, $\alpha(u)^{-1}=u^{-1}$, and the reflection formula gives $\rho_u$. The last claim is the next proposition.

Proposition (no one-sided operator preserves the space). Let $u\in V$ with $q(u)\ne0$. Then $\mathrm{L}^{\alpha}_u$ does not map $V$ into itself: it sends the unit $1$ to $u$ and the vector $u$ to $u\alpha(u)=-q(u)$, a scalar, so the image of the even part of the algebra meets the odd part and vice versa. Consequently no signed left multiplication by a non-scalar element preserves the quadratic space.

Proof. $\mathrm{L}^{\alpha}_u(1)=u$ and $\mathrm{L}^{\alpha}_u(u)=u\,(-u)=-q(u)\in F$; the two values lie in different parity components, so $\mathrm{L}^{\alpha}_u$ does not preserve the subspace $V$ of the algebra. A one-sided multiplication by $a$ preserves the parity grading only when $a$ is even, and then it does not preserve $V$ either unless $a$ is a scalar.

Remark (the geometry of the pairing). The signed left multiplication is the left half of the signed sandwich, and only the pairing with a right multiplication restores the grading and produces an operator of $V$. This is the one-sided form of the statement that the reflections are two-sided operators, and it is why the geometry of the category is carried by the two-sided family. The parameter-twisted signed left multiplication of One-Sided Operators on a Clifford Algebra with Signed Inner Conjugation has the same one-sided deficiency, and the pairing there is with the inverse right multiplication.

Worked Cases

A Vector in the Negative-Definite Three-Space

For $V$ of dimension three with $e_j^2=-1$ and $u=e_1$, the operator $\mathrm{L}^{\alpha}_{e_1}$ sends $1\mapsto e_1$, $e_1\mapsto -e_1^2=1$, $e_2\mapsto e_1(-e_2)=-e_1e_2$; the even and odd parts are exchanged, and the image is not contained in $V$. The pairing with $R_{e_1^{-1}}=R_{-e_1}$ returns the reflection $\rho_{e_1}$ of the previous article.

The Scalar $-1$

For $a=-1$ the operator is $\mathrm{L}^{\alpha}_{-1}(y)=-\alpha(y)$, which is the identity on the odd part and minus the identity on the even part: the fixed space is the odd part, an example of the fixed-element proposition, and the operator is an involution.

An Even Element

For $a=x=e_1e_2$, even, $\mathrm{L}^{\alpha}_x=L_x$ because $\alpha$ is the identity on the even part; the signed and the ordinary left multiplications coincide, and the pairing with $R_{x^{-1}}$ is the inner conjugation $\mathrm{Ad}_x$, a rotation of the plane. The signed family differs from the ordinary one only through the odd parameters, which is where the reflections live.

Summary

The signed left multiplication $\mathrm{L}^{\alpha}_a=L_a\circ\alpha$, $\mathrm{L}^{\alpha}_a(y)=a\alpha(y)$, is the ordinary left multiplication of the twisted argument; it satisfies $\mathrm{L}^{\alpha}_a=\alpha L_{\alpha(a)}$, it coincides with $L_a$ on the even part of the algebra and is $-L_a$ on the odd part, it is invertible exactly for units, and it has the parity of $a$. Its products satisfy $\mathrm{L}^{\alpha}_a\mathrm{L}^{\alpha}_c=L_{a\alpha(c)}$, so the signed left family is a coset of the group of invertible left multiplications; its fixed space is $\{y : a\alpha(y)=y\}$, which for a scalar is the even part, the odd part or nothing according as the scalar is $1$, $-1$ or neither. The operator is one half of the signed sandwich, $\mathrm{L}^{\alpha}_xR_{\alpha(x)^{-1}}=T^{\alpha}_{x,\alpha(x)^{-1}}$, which for a vector is the reflection $\rho_u$; the signed left multiplication alone never preserves the quadratic space, because it exchanges the two parity components, and only the pairing with a right multiplication restores the grading. The parameter-twisted family $\alpha(x)y$ is a different operator, by The Graded Multiplication Operators; the one-sided calculus is One-Sided Operators on a Clifford Algebra, and the signed product is One-Sided Operators with the Signed Product.

Summary of Notation

Symbol	Meaning
$\mathrm{L}^{\alpha}_a=L_a\circ\alpha$	Signed left multiplication, $y\mapsto a\alpha(y)$
$\Lambda^{\alpha}_x(y)=\alpha(x)y$	The different parameter-twisted family, not treated here
$\mathrm{L}^{\alpha}_a=\alpha L_{\alpha(a)}$	Relation to the ordinary left multiplication
$\mathrm{L}^{\alpha}_a=L_a$ on the even part, $-L_a$ on the odd part	The parity split
$\mathrm{L}^{\alpha}_a\mathrm{L}^{\alpha}_c=L_{a\alpha(c)}$	Composition law; coset of the left-multiplication group
$\{y : a\alpha(y)=y\}$	Fixed space; even part, odd part or nothing for a scalar
$\mathrm{L}^{\alpha}_xR_{\alpha(x)^{-1}}=T^{\alpha}_{x,\alpha(x)^{-1}}$	Pairing to the signed sandwich
$\mathrm{L}^{\alpha}_u(1)=u$, $\mathrm{L}^{\alpha}_u(u)=-q(u)$	Failure to preserve $V$