The Signed Left Multiplication on a Topological Group

Introduction

The one-sided operators on a topological group are the left translations $x \mapsto ax$ and their right-handed mirror images, and the grade involution twists one of them: the signed left multiplication is the operator $x \mapsto a\,\alpha(x)$, the composite of the left translation with the continuous involutive automorphism. It is a homeomorphism, but it is not an automorphism of the group; the family it forms is a coset of the left regular subgroup inside $\operatorname{Homeo}(G)$, its products with the unsigned translations are unsigned, and the elements it fixes form a closed set that is generally not a subgroup. This article defines the operator, computes its laws and its relation to the unsigned left multiplication, and describes its fixed set and the case in which it is an involution.

The article assumes the continuous involutive automorphism $\alpha$, the fixed and inverted subgroups and the dictionary between them from Involutive Topological Groups; the abstract signed left multiplication, its relation to the unsigned one and its fixed elements from The Signed Left Multiplication on a Group; the signed sandwich and its composition laws from The Signed Sandwich on a Topological Group; and the operator layer, the compact-open topology and the natural pairing from Operators on a Topological Group. The signed adjoint of this operator is The Signed Adjoint of the Left Multiplication on a Topological Group. No adjoint is used here, and nothing analytic or geometric occurs.

Throughout, $G$ is a Hausdorff topological group with identity $e$, $\alpha$ is a continuous involutive automorphism, $\sigma = \iota\alpha$ is the associated topological involution, $G^\alpha$ is the fixed subgroup of $\alpha$ and $I(\sigma) = G^\alpha$ its other face. The signed left multiplication by $a$ is $\Lambda_a$ and the signed right multiplication by $b$ is $\Lambda^{\mathrm{R}}_b$:

$$ \Lambda_a(x) = a\,\alpha(x), \qquad \Lambda^{\mathrm{R}}_b(x) = \alpha(x)\,b . $$

The Signed Left Multiplication

Definition. For $a \in G$ the signed left multiplication is the map

$$ \Lambda_a : G \longrightarrow G, \qquad \Lambda_a(x) = a\,\alpha(x) = (L_a \circ \alpha)(x) . $$

Theorem (it is a homeomorphism). $\Lambda_a = L_a\circ\alpha$ is a homeomorphism of $G$, with inverse

$$ \Lambda_a^{-1} = \alpha \circ L_{a^{-1}} = \Lambda^{\mathrm{R}}_{\alpha(a^{-1})} , $$

that is $\Lambda_a^{-1}(y) = \alpha(a^{-1}y) = \alpha(y)\,\alpha(a)^{-1}$.

Proof. $L_a$ and $\alpha$ are homeomorphisms, so their composite is one. For the inverse, $\Lambda_a(y) = a\alpha(y)$, so $\Lambda_a^{-1}(y)$ is the unique $x$ with $a\alpha(x) = y$, namely $x = \alpha(a^{-1}y) = \alpha(a^{-1})\alpha(y)$ by the anti-multiplicativity of $\alpha$; and $\alpha(a^{-1}y) = \alpha(y)\alpha(a)^{-1}$.

Proposition (it is not an automorphism). If $\alpha \neq \mathrm{id}$ then $\Lambda_a$ is not an automorphism of $G$, and it is not a translation; the only values of $a$ for which $\Lambda_a$ is a homomorphism are those with $\alpha = \mathrm{id}$ or $\alpha(a) = e$.

Proof. $\Lambda_a(xy) = a\alpha(y)\alpha(x)$ and $\Lambda_a(x)\Lambda_a(y) = a\alpha(x)a\alpha(y)$; these agree for all $x, y$ exactly when $a = e$ or $\alpha$ is the identity. Similarly $\Lambda_a = L_b$ would give $\alpha = L_{a^{-1}b}$, and a left translation is an automorphism only when $b = a$, hence $\alpha = \mathrm{id}$.

So the signed left multiplication is an affine operator: it permutes the group without preserving its multiplication, exactly as a reflection or a translation does, and its study belongs with the operators that act on $G$ as a set with topology rather than as a group.

Composition and the Generated Family

Proposition (the composition laws). For all $a, b \in G$,

$$ \Lambda_a \circ \Lambda_b = L_{a\alpha(b)}, \qquad L_a \circ \Lambda_b = \Lambda_{ab}, \qquad \Lambda_a \circ L_b = \Lambda_{a\alpha(b)} . $$

Proof. $\Lambda_a(\Lambda_b(x)) = a\alpha(b\alpha(x)) = a\alpha(b)x = L_{a\alpha(b)}(x)$; the second and third are the same computation with the appropriate factor unsigned, using $\alpha(xb) = \alpha(b)\alpha(x)$.

Corollary. A product of two signed left multiplications is an unsigned left translation, a product of an unsigned and a signed left multiplication is signed, and the sign of a product is the product of the signs. The signed family is the coset

$$ \Lambda(G) = \{\Lambda_a : a \in G\} = L(G)\,\alpha = \alpha\,L(G), $$

and the group generated by $L(G)$ and $\alpha$ inside $\operatorname{Homeo}(G)$ is $L(G) \sqcup L(G)\alpha$.

Proof. The composition law gives closure and the parity; $\Lambda_a = L_a\alpha$ gives the coset. Since $\alpha^2 = \mathrm{id}$ and $\alpha L_b\alpha = L_{\alpha(b)}$ by The Signed Sandwich on a Topological Group, the product of any word in the generators is an element of $L(G)$ or of $L(G)\alpha$, so the generated subgroup is the union.

Proposition (the signed family is disjoint from the unsigned one). If $\alpha \neq \mathrm{id}$ then $\Lambda(G) \cap L(G) = \varnothing$; if $\alpha = \mathrm{id}$ then $\Lambda_a = L_a$ for every $a$ and the two families coincide.

Proof. $\Lambda_a = L_b$ would make $\alpha = L_{a^{-1}b}$, an automorphism that is a left translation, which forces $\alpha = \mathrm{id}$ and then $b = a$. Since $\alpha$ is not the identity, no such pair exists.

Corollary (continuity of the parametrisation). The map

$$ G \longrightarrow \operatorname{Homeo}(G), \qquad a \mapsto \Lambda_a = L_a\alpha , $$

is continuous for the compact-open topology; when $G$ is compact it is a homeomorphism onto its image, and the signed family $\Lambda(G)$ is a compact, hence closed, subset of $\operatorname{Homeo}(G)$.

Proof. It is the composite of the continuous map $a\mapsto L_a$ and the composition-by-$\alpha$ map, the latter being continuous in the compact-open topology because $\alpha$ is a homeomorphism. For compact $G$, $G$ is compact and $a \mapsto \Lambda_a$ is injective with inverse $\Lambda_a \mapsto \Lambda_a(e) = a\alpha(e) = a$, a continuous evaluation map; a continuous bijection from a compact space onto a Hausdorff image is a homeomorphism, and the image of a compact space is compact and closed.

The Elements it Fixes

Definition. The fixed set of the signed left multiplication $\Lambda_a$ is

$$ \operatorname{Fix}_L(a) = \{x \in G : a\,\alpha(x) = x\} = \{x \in G : \alpha(x) = a^{-1}x\} . $$

Proposition. $\operatorname{Fix}_L(a)$ is closed in $G$; it is nonempty exactly when $a$ is a product $x\,\sigma(x)^{-1}$ with $x$ ranging over $G$; and it is a subgroup only in the degenerate cases. The element $x$ is fixed by $\Lambda_a$ exactly when the conjugate product $x^{-1}a$ equals $\alpha(x)^{-1}\alpha(a^{-1})$; equivalently, on setting $u = \alpha(x)$, the fixed points correspond to the solutions of $a = x\,\alpha(x)^{-1}$.

Proof. The fixed set is the equalizer of the continuous maps $L_a\alpha$ and $\mathrm{id}$, hence closed in the Hausdorff group. The equation $a\alpha(x) = x$ is equivalent to $a = x\alpha(x)^{-1} = x\,\sigma(x)^{-1}$, since $\alpha(x)^{-1} = \iota\alpha(x) = \sigma(x)$. When $\alpha = \mathrm{id}$ the equation is $a = e$, so the fixed set is all of $G$ if $a = e$ and empty otherwise; when $\alpha$ has a large fixed set the equation may have many solutions but the set need not contain $e$ and is not a subgroup.

Example (the inversion). Let $G$ be abelian and $\alpha = \iota$, so that $\Lambda_a(x) = ax^{-1}$ and $\sigma = \mathrm{id}$. Then $\operatorname{Fix}_L(a)$ is the set of solutions of $x^2 = a$, the square roots of $a$; it is closed, it consists of one element when $G$ is uniquely $2$-divisible and it is empty when $a$ has no square root, as on $S^1$ for $a$ outside the image of the squaring map. On the circle every $a$ has two square roots, and the fixed set of $\Lambda_a$ is the pair $\{\pm x_0\}$ for the principal root $x_0$.

Example (the trivial grade involution). If $\alpha = \mathrm{id}$ then $\Lambda_a = L_a$ is the left translation, which fixes $e$ only for $a = e$: the fixed set is empty for $a\neq e$ and is $G$ for $a = e$. The signed operator is then the unsigned one, and the family is not a new one.

Proposition (when the operator is an involution). The square of the signed left multiplication is the left translation

$$ \Lambda_a^2 = L_{a\alpha(a)} , $$

so $\Lambda_a$ is an involution of the set $G$ if and only if $a\alpha(a) = e$, that is if and only if $\alpha(a) = a^{-1}$, equivalently $a \in I(\alpha) = G^\sigma$. In that case $\Lambda_a$ is a homeomorphism of order two whose fixed set is closed; by the dictionary of Involutive Topological Groups the carrier condition is the same one that makes the signed conjugation $\rho_a$ a reflection with central product $a\alpha(a) = e$.

Proof. $\Lambda_a^2 = L_{a\alpha(a)}$ by the composition law with $b = a$, using $\alpha(a)\cdot\alpha(\alpha(a))$; the left translation $L_c$ is the identity exactly when $c = e$. The identification $a\alpha(a) = e \iff \alpha(a) = a^{-1} \iff a \in I(\alpha) = G^\sigma$ is the dictionary.

Remark (relation to the sandwich). The signed left multiplication is the one-sided part of the signed sandwich: $\Sigma^{\alpha}_{a,b} = L_a\circ\Lambda^{\mathrm{R}}_b = \Lambda_a\circ R_{\alpha(b)}$, and the signed left multiplication is recovered from the sandwich by putting $b = e$, since $\alpha(e) = e$. The signed sandwich is the two-sided operator of the category, and the one-sided operators are the specialisations of it in which one of the two factors is the identity.

Summary

The signed left multiplication $\Lambda_a(x) = a\alpha(x) = L_a\alpha$ on a graded topological group is a homeomorphism with inverse $\Lambda_a^{-1} = \alpha L_{a^{-1}} = \Lambda^{\mathrm{R}}_{\alpha(a^{-1})}$, and it is not an automorphism unless $\alpha = \mathrm{id}$: it is an affine operator that permutes the group without preserving the multiplication. Its composition laws are $\Lambda_a\Lambda_b = L_{a\alpha(b)}$, $L_a\Lambda_b = \Lambda_{ab}$ and $\Lambda_aL_b = \Lambda_{a\alpha(b)}$, so the signed left multiplications form the coset $\Lambda(G) = L(G)\alpha = \alpha L(G)$, a product of two of them is unsigned, and the generated subgroup of $\operatorname{Homeo}(G)$ is $L(G)\sqcup L(G)\alpha$; the signed family is disjoint from the left translations when $\alpha \neq \mathrm{id}$, and the parametrisation $a \mapsto \Lambda_a$ is continuous, a homeomorphism onto a compact closed image when $G$ is compact. The fixed set of $\Lambda_a$ is closed and consists of the solutions of $a = x\alpha(x)^{-1}$, which is the set of square roots of $a$ when $\alpha$ is the inversion and abelian; it is not a subgroup in general. The square of the operator is $\Lambda_a^2 = L_{a\alpha(a)}$, so the signed left multiplication is an involution exactly when $a \in G^\sigma$, the fixed set of the associated involution, and otherwise it is a homeomorphism whose order is the order of the left translation by $a\alpha(a)$. The signed left multiplication is the one-sided specialisation of the signed sandwich, recovered by putting one factor equal to the identity.

Summary of Notation

Symbol	Meaning
$\alpha$	the grade involution, a continuous involutive automorphism
$\sigma = \iota\alpha$	the associated topological involution
$\Lambda_a(x) = a\alpha(x) = L_a\alpha$	the signed left multiplication, a homeomorphism
$\Lambda^{\mathrm{R}}_b(x) = \alpha(x)b$	the signed right multiplication
$\Lambda_a^{-1} = \Lambda^{\mathrm{R}}_{\alpha(a^{-1})}$	its inverse
$\Lambda_a\Lambda_b = L_{a\alpha(b)}$	the product of two signed left multiplications is unsigned
$\Lambda(G) = L(G)\alpha$	the signed left multiplications as a coset
$L(G)\sqcup L(G)\alpha$	the group generated by the translations and $\alpha$
$\Lambda_a^2 = L_{a\alpha(a)}$	the square of the signed left multiplication
$a\alpha(a) = e \iff a \in G^\sigma$	criterion for $\Lambda_a$ to be an involution
$\operatorname{Fix}_L(a) = \{x : \alpha(x) = a^{-1}x\}$	the fixed set, closed
$G^\alpha$, $I(\sigma)$	the fixed subgroup of $\alpha$, and the inverted subgroup of $\sigma$