The Adjoint of the Left Multiplication on a Group

Introduction

The left multiplication $x\mapsto ax$ is the simplest operator a group carries, and its adjoint with respect to the natural pairing of the category is explicit: it is the left multiplication by the inverse of $a$. The case in which the operator is self-adjoint is exactly the case in which the parameter has order at most two, so the involutions of the group reappear as the self-adjoint left multiplications. This article gives the adjoint, the self-adjointness criterion, and the compatibility of the adjoint operation with the involution of the group; the two-sided case and the interpretation of the inversion as the adjoint of the left multiplication are in Involutions on the Operator Layer and The Group Inversion as an Adjoint, and the left multiplication itself is in Left and Right Multiplication in a Group. It is the third article of the * Operator Theory group.

Throughout, $k[G]$ is the group algebra with orthonormal basis $G$, $\sigma$ is an involution of $G$ with linear extension $\Sigma_{\sigma}$, and the adjoint is defined by $\langle Tx,y\rangle=\langle x,T^{*}y\rangle$ for the natural pairing $\langle g,h\rangle=\delta_{g,h}$.

The Adjoint

Theorem (the adjoint of the left multiplication). For every $a\in G$,

$$ (L_a)^{*}=L_{a^{-1}}. $$

Proof. In the basis $G$, $(L_ag)_h=\delta_{ag,h}$, and the matrix of the adjoint is the transpose, so $(L_a^{*}h)_g=\delta_{ag,h}=\delta_{g,a^{-1}h}$; hence $L_a^{*}(h)=a^{-1}h$ and $L_a^{*}=L_{a^{-1}}$. The same computation applies to any group of operators generated by the left multiplications.

Corollary (iterates and products). $(L_a^{*})^{*}=L_a$, $(L_aL_b)^{*}=L_{(ab)^{-1}}$, and $(L_a^{k})^{*}=L_{a^{-k}}$ for every integer $k$. The left multiplications form a $*$-closed set: the adjoint of a left multiplication is a left multiplication.

Proof. The adjoint is an involution, so $(L_a^{*})^{*}=L_a$; the product rule $(L_aL_b)^{*}=L_b^{*}L_a^{*}=L_{b^{-1}}L_{a^{-1}}=L_{(ab)^{-1}}$ uses the anti-multiplicativity and the composition law; the power formula follows by induction, with the negative powers from the invertibility of the adjoint.

Proposition (the matrices). In the basis $G$ the matrix of $L_a$ has the single $1$ in each column, at the row $ag$ of the column $g$; the matrix of $L_a^{*}=L_{a^{-1}}$ has its $1$ in the row $a^{-1}g$ of the column $g$. The two matrices are transpose to one another, as they must be.

Proof. The column $g$ of $L_a$ has coefficient $1$ at $ag$ and $0$ elsewhere; the transpose has the $1$ at the row $g$ of the column $ag$, which is to say at the row $a^{-1}h$ of the column $h$ after setting $h=ag$.

Self-Adjointness

Theorem (the self-adjoint left multiplications are the involutions). $L_a$ is self-adjoint if and only if $a^{2}=e$. Consequently the self-adjoint left multiplications are exactly the left translations by the elements of order at most two, they form a set in bijection with the $2$-torsion of $G$, and they include the identity.

Proof. $L_a$ is self-adjoint exactly when $L_a=L_{a^{-1}}$, and $a\mapsto L_a$ is injective since $L_ag=g$ forces $a=e$; so the condition is $a=a^{-1}$, that is $a^{2}=e$. The set of such elements is the $2$-torsion of $G$, which contains $e$.

Example. In $C_4=\langle r\rangle$ the self-adjoint left multiplications are $L_e$ and $L_{r^{2}}$, since $r^{2}$ is the unique element of order two. In $V_4$ all four left multiplications are self-adjoint, because every element is its own inverse. In $S_3$ the self-adjoint left multiplications are the four with parameter in $\{e,(1\,2),(1\,3),(2\,3)\}$. In $Q_8$ they are $L_{\pm1}$.

Proposition (the skew-adjoint left multiplications). There is no nonzero skew-adjoint left multiplication unless $\operatorname{char}k=2$; in characteristic not two, $L_a$ is skew-adjoint if and only if $L_a=0$, which never happens for a translation.

Proof. Skew-adjointness is $L_a^{*}=-L_a$, that is $L_{a^{-1}}=-L_a$. Evaluating at $e$ gives $a^{-1}=-a$, which in characteristic not two would make $L_{a^{-1}}$ and $-L_a$ have entries $1$ and $-1$ in the same positions, a contradiction; in characteristic two the two signs agree and the condition is $a^{2}=e$, so every left multiplication is at once self-adjoint and skew-adjoint.

Remark. The self-adjoint criterion is the operator-level form of the fixed-point criterion of the involutive part of the category: an element of order at most two is a fixed point of the inversion, and the left multiplication by such an element is a fixed point of the adjoint. The correspondence is exact and is the reason the two structures are read together.

Compatibility with the Involution

Proposition (the adjoint commutes with the extension of the involution). For every involution $\sigma$ of $G$ and every $a\in G$,

$$ (L_{\sigma(a)})^{*}=L_{\sigma(a)^{-1}}=L_{\sigma(a^{-1})}=\Sigma_{\sigma}\,L_{a^{-1}}\,\Sigma_{\sigma}, $$

and the extension $\Sigma_{\sigma}$ commutes with the adjoint operation: $(\Sigma_{\sigma}T\Sigma_{\sigma})^{*}=\Sigma_{\sigma}T^{*}\Sigma_{\sigma}$.

Proof. The first chain is the adjoint formula together with $\sigma(a)^{-1}=\sigma(a^{-1})$; the last equality is the conjugation formula for a translation under the extension of $\sigma$, which acts on the basis by $\Sigma_{\sigma}g=\sigma(g)$. For the commutation, $\Sigma_{\sigma}$ is self-adjoint by The Group Inversion as an Adjoint, so $(\Sigma_{\sigma}T\Sigma_{\sigma})^{*}=\Sigma_{\sigma}^{*}T^{*}\Sigma_{\sigma}^{*}=\Sigma_{\sigma}T^{*}\Sigma_{\sigma}$.

Proposition (the left regular representation is a $*$-representation). The map $a\mapsto L_a$ is a representation of $G$ into the algebra $\operatorname{End}_k(k[G])$ with the adjoint as involution, and it satisfies $L_a^{*}=L_{a^{-1}}$, that is, it is a $*$-representation for the inversion of $G$.

Proof. The multiplicativity $L_aL_b=L_{ab}$ is the composition law of the translations; the adjoint relation is the theorem; and the defining property of a $*$-representation, $\rho(a)^{*}=\rho(a^{-1})$, is exactly the adjoint formula.

Corollary (the commutant is closed under the adjoint). The commutant of $L(G)$ in $\operatorname{End}_k(k[G])$, which is the set of right multiplications $R(G)$, together with the scalars, is closed under the adjoint: $(R_b)^{*}=R_{b^{-1}}\in R(G)$.

Proof. The commutant is $R(G)$ by the centraliser theorem of Left and Right Multiplication in a Group, and the adjoint of $R_b$ is $R_{b^{-1}}$, which lies in $R(G)$.

Summary

The adjoint of the left multiplication is the left multiplication by the inverse, $(L_a)^{*}=L_{a^{-1}}$; equivalently the matrix of $L_a$ has its $1$'s at the positions $g\mapsto ag$ and the adjoint at $g\mapsto a^{-1}g$, the transpose. The adjoint commutes with products through the anti-multiplicativity, $(L_aL_b)^{*}=L_{(ab)^{-1}}$, and the left multiplications are closed under the adjoint.

The self-adjoint left multiplications are exactly those with parameter of order at most two: $L_a$ is self-adjoint if and only if $a^{2}=e$, so the self-adjoint translations correspond to the $2$-torsion of $G$; there are no nonzero skew-adjoint left multiplications in characteristic not two, and in characteristic two every left multiplication is both. The adjoint commutes with the extension of any group involution $\sigma$, and $a\mapsto L_a$ is a $*$-representation of $G$ for the inversion, whose commutant $R(G)$ is likewise closed under the adjoint.

Summary of Notation

Symbol	Meaning
$(L_a)^{*}=L_{a^{-1}}$	the adjoint of a left multiplication
$(L_aL_b)^{*}=L_{(ab)^{-1}}$	the adjoint inverts a product
$L_a$ self-adjoint $\iff a^{2}=e$	the self-adjoint translations are the involutions
$L_{\sigma(a)}^{*}=L_{\sigma(a^{-1})}$	compatibility with a group involution
$\rho(a)^{*}=\rho(a^{-1})$	the left regular representation is a $*$-representation
$(R_b)^{*}=R_{b^{-1}}$	the commutant is $*$-closed