The Adjoint of the Left Multiplication on a Clifford Algebra

Introduction

The left multiplication $L_a$ acts on the Clifford algebra by $x\mapsto ax$, and its adjoint for the standard form $\langle x,y\rangle=\operatorname{Sc}(\hat xy)$ is the left multiplication by the Clifford conjugate,

$$ L_a^{*} = L_{\hat a} , \qquad \hat a = \alpha(\tilde a) , $$

an operator of the same family with the parameter conjugated. The adjoint therefore defines an involution on the parameters, and its consequences are geometric: the self-adjoint left multiplications are those with $\hat a=a$, the orthogonal ones are the units $a$ with $\hat aa=1$, and the multiplication by a vector is skew-adjoint, $L_u^*=-L_u$. The article collects these consequences and the corresponding statements for the twisted form of The Twisted Adjoint on a Clifford Algebra, on which the article depends for the form and the involution.

The boundaries. The operator and its calculus are The Left and Right Multiplication Operators on a Clifford Algebra; the form, the Clifford conjugation and the twisted form are The Twisted Adjoint on a Clifford Algebra; the right-hand mirror of the article is The Adjoint of the Right Multiplication; the two-sided operators are The Adjoint of the Sandwich and The Adjoint of the Two-Sided Multiplication Operator. The unitary and versor conditions in the geometry of the orthogonal group are The Clifford, Pin and Spin Groups with Signed Inner Conjugation. The base is a field $F$ of characteristic not $2$ with a non-degenerate $q$, $q(u)=B(u,u)$, $uv+vu=2B(u,v)$.

The Adjoint and Its Involution

Theorem. For every $a\in\mathrm{Cl}(V,q)$ the adjoint of the left multiplication for the standard form is $L_a^*=L_{\hat a}$; the map $a\mapsto\hat a$ is an anti-automorphism and an involution of the algebra, $\widehat{a+b}=\hat a+\hat b$, $\widehat{ab}=\hat b\hat a$, $\hat{\hat a}=a$, and $\hat u=-u$ for a vector $u$.

Proof. $\langle ax,y\rangle=\operatorname{Sc}(\widehat{ax}y)=\operatorname{Sc}(\hat x\hat ay)=\langle x,\hat ay\rangle$, using the anti-automorphism property of the conjugation and the cyclic invariance of the scalar part; the involution and anti-automorphism statements are those of the conjugation, and the vector value is $\hat u=\alpha(\tilde u)=-u$. The computation is the one-factor case of The Twisted Adjoint on a Clifford Algebra.

Proposition (self-adjointness). $L_a$ is self-adjoint, $L_a^*=L_a$, if and only if $\hat a=a$; the elements with $\hat a=a$ form a subspace of the algebra and include the scalars, the unit, and every product of an even number of anticommuting vectors in the definite case.

Proof. $L_a^*=L_{\hat a}$ and the left multiplications are faithful, so $L_a^*=L_a$ is equivalent to $\hat a=a$ by The Left and Right Multiplication Operators on a Clifford Algebra; the fixed subspace of an involution is a subspace, and the examples are immediate from $\hat u=-u$ and multiplicativity.

Proposition (orthogonality). $L_a$ is orthogonal for the standard form, $L_a^*L_a=\operatorname{id}$, if and only if $\hat aa=1$; such $a$ are units. The orthogonal left multiplications form a subgroup of the group of units, containing $\pm1$ and closed under $a\mapsto\hat a$.

Proof. $L_a^*L_a=L_{\hat a}L_a=L_{\hat aa}$, and $L_c=\operatorname{id}$ only for $c=1$, so orthogonality is $\hat aa=1$; the condition makes $a$ invertible with $a^{-1}=\hat a$, the set is closed under inversion because $\hat{\hat a}a=1$, and closed under products because $\widehat{ab}ab=\hat b\hat aab$, which is $1$ when $\hat aa=\hat bb=1$ and $\hat b$ commutes with $a$ in the scalar case $1$.

Remark (the norm). The quantity $\langle ax,ax\rangle$ is the quadratic form of the algebra transported by left multiplication; the orthogonality condition $\hat aa=1$ is the statement that the conjugation is the inverse, and for a unit vector $u$ with $u^2=q(u)$ it reads $(-u)u=-q(u)=1$, which holds for the negative-definite normalisation $q(u)=-1$. The passage between the two normalisations changes which vectors are orthogonal but not the structure of the statements.

The Twisted Adjoint

Proposition. For the twisted form the adjoint of $L_a$ is the signed left multiplication

$$ (L_a)^{*\alpha} = L_{\hat a}\,\alpha , $$

as The Twisted Adjoint on a Clifford Algebra records; the operator is one-sided signed and differs from the standard adjoint by the grade involution. Consequently a left multiplication is self-adjoint for the twisted form exactly when $L_{\hat a}\alpha=L_a$, that is when $a\alpha(x)=\alpha(x)a$ for all $x$, which holds only for the central scalars in the central-simple case.

Proof. The formula is the corollary of the twisted-adjoint identity $A^{*\alpha}=\alpha A^*\alpha$ and the commutation $L_u\alpha=\alpha L_{\alpha(u)}$; the self-adjointness computation reduces to $\hat a=a$ together with $\alpha$ centralising $L_a$, and in a central-simple algebra only central elements $\alpha$-commute with every left multiplication.

Worked Cases

The Quaternions

For $\mathrm{Cl}\cong\mathbb H$ with $e_1^2=e_2^2=-1$, the conjugation is the quaternionic conjugate, and $L_{e_1}^*=L_{-e_1}=-L_{e_1}$: the left multiplication by a pure quaternion unit is skew-adjoint. The orthogonal left multiplications are $L_a$ with $\hat aa=|a|^2=1$, the unit sphere, the group of unit quaternions.

A Unit Vector

For any vector $u$ with $q(u)\ne0$, $L_u^*=L_{-u}=-L_u$, so the multiplication by a vector is always skew-adjoint, independent of the signature; the same computation is the manifold relation $c(v)^*=-c(v)$ of The Adjoint of the Clifford Multiplication.

An Even Element

For $a=e_1e_2$ in the negative-definite plane, $\hat a=(-e_2)(-e_1)=e_2e_1=-e_1e_2=-a$, and $L_a^*=L_{-a}=-L_a$: an even element can be skew-adjoint too. The parity of $a$ does not decide self-adjointness; the involution $\hat{}$ does.

Summary

For the standard form $\langle x,y\rangle=\operatorname{Sc}(\hat xy)$ the adjoint of the left multiplication is the left multiplication by the Clifford conjugate, $L_a^*=L_{\hat a}$ with $\hat a=\alpha(\tilde a)$ an anti-automorphism and an involution acting on a vector by $\hat u=-u$. Hence $L_a$ is self-adjoint exactly when $\hat a=a$ and orthogonal exactly when $\hat aa=1$, the latter defining the group of orthogonal left multiplications inside the units; the multiplication by a vector is always skew-adjoint, $L_u^*=-L_u$. For the twisted form the adjoint is the signed left multiplication $(L_a)^{*\alpha}=L_{\hat a}\alpha$, and a left multiplication is self-adjoint for the twisted form only for central scalars in the central-simple case. The operator is The Left and Right Multiplication Operators on a Clifford Algebra, the form is The Twisted Adjoint on a Clifford Algebra, the mirror article is The Adjoint of the Right Multiplication, and the manifold case is The Adjoint of the Clifford Multiplication.

Summary of Notation

Symbol	Meaning
$L_a(x)=ax$	Left multiplication
$\hat a=\alpha(\tilde a)$	Clifford conjugation; anti-automorphism, involution
$\langle x,y\rangle=\operatorname{Sc}(\hat xy)$	Standard form
$L_a^*=L_{\hat a}$	Adjoint of the left multiplication
$\hat u=-u$	Conjugation on a vector; $L_u^*=-L_u$
$\hat a=a$	Self-adjointness condition
$\hat aa=1$	Orthogonality condition; group of orthogonal left multiplications
$(L_a)^{*\alpha}=L_{\hat a}\alpha$	Twisted adjoint; signed left multiplication