Rings with a Semilinear Involution

Introduction

An involution of a ring is linear over the ring, and its twisting of the elements is only the reversal of products. When the ring carries an automorphism $\varsigma$, one may ask for a map that reverses products and intertwines $\varsigma$ with its inverse, a $\varsigma$-semilinear involution; the linear case is the case $\varsigma = \mathrm{id}$. The name is borrowed from the semilinear maps of a module, where a coefficient $\lambda$ is carried to $\varsigma(\lambda)$, and the point of the notion is that the naive identities then fail: the composite $\sigma\varsigma$ is an ordinary involution rather than the product of two, the map $\sigma$ commutes with $\varsigma$ only when $\varsigma^2 = \mathrm{id}$, and the fixed set of $\sigma$ need not be stable under $\varsigma$.

This article defines the semilinear involution of a ring, derives the identities that do hold, and records the failure of the ones a reader would expect; the map $\sigma\varsigma$ it produces is the ordinary involution to which the whole corpus applies. It assumes Involutive Rings for the involution, the fixed set and the symmetric and skew elements, and Ring and Field Automorphisms for the automorphism group; the semilinear involutions of a linear space and of an algebra over a field are treated with the linear structures of a later category and are named here only at the boundary. Throughout, $A$ is a ring with $1 \neq 0$, $\varsigma$ is an automorphism of $A$, and $\sigma$ is a $\varsigma$-semilinear involution; the ordinary involution attached to the pair is $\tau = \sigma\varsigma$. Nothing is measured, and no form or scalar product occurs.

The Definition

Definition. Let $\varsigma$ be an automorphism of the ring $A$. A $\varsigma$-semilinear involution of $A$ is a map $\sigma : A \to A$ such that for all $a, b \in A$

$$ \sigma(a+b) = \sigma(a)+\sigma(b), \quad \sigma(ab) = \sigma(b)\sigma(a), \quad \sigma(1) = 1, \quad \sigma^2 = \mathrm{id}, \quad \sigma\varsigma = \varsigma^{-1}\sigma . $$

The last condition is the semilinearity: $\sigma$ carries the automorphism $\varsigma$ to its inverse. When $\varsigma = \mathrm{id}$ the definition reduces to that of an ordinary involution, so the linear case is the case of a trivial twisting automorphism.

Proposition (elementary). Let $\sigma$ be $\varsigma$-semilinear. Then $\sigma$ is a bijection with $\sigma^{-1} = \sigma$, it is $\mathbb{Z}$-linear, it satisfies $\sigma(a^n) = \sigma(a)^n$, and it carries units to units. Moreover

$$ \sigma\varsigma = \varsigma^{-1}\sigma, \qquad \varsigma\sigma = \sigma\varsigma^{-1}, \qquad \sigma\varsigma^2 = \varsigma^{-2}\sigma, \qquad \sigma\varsigma^{k} = \varsigma^{-k}\sigma $$

for every integer $k$.

Proof. Bijectivity and the power law are as for an involution. The second display follows from the first by composing with $\varsigma$ on the left and using $\varsigma^{-1}\varsigma = \mathrm{id}$: $\varsigma\sigma\varsigma = \sigma$, so $\varsigma\sigma = \sigma\varsigma^{-1}$. The third and fourth are obtained by iterating the second, $\sigma\varsigma^k = \varsigma^{-k}\sigma$ for all $k$.

The Attached Ordinary Involution

Theorem. Let $\sigma$ be a $\varsigma$-semilinear involution and put $\tau = \sigma\varsigma$. Then $\tau$ is an ordinary involution of $A$, $\sigma = \tau\varsigma^{-1}$, and the automorphisms $\varsigma$ and $\sigma$ generate the dihedral group

$$ \langle \varsigma, \sigma \rangle = \{\varsigma^k, \sigma\varsigma^k : k \in \mathbb{Z}\}, \qquad \sigma^2 = \mathrm{id}, \qquad \sigma\varsigma\sigma^{-1} = \varsigma^{-1}. $$

Proof. $\tau$ is additive and carries $1$ to $1$; it is anti-multiplicative, $\tau(ab) = \sigma(\varsigma(ab)) = \sigma(\varsigma(a)\varsigma(b)) = \sigma(\varsigma(b))\sigma(\varsigma(a)) = \tau(b)\tau(a)$; and $\tau^2 = \sigma\varsigma\sigma\varsigma = \sigma(\varsigma\sigma)\varsigma = \sigma(\sigma\varsigma^{-1})\varsigma = \varsigma^{-1}\varsigma = \mathrm{id}$, using the second identity of the previous proposition. The formula $\sigma = \tau\varsigma^{-1}$ is then immediate. The relation $\sigma\varsigma\sigma^{-1} = \varsigma^{-1}$ is the semilinearity with $\sigma^{-1} = \sigma$, and together with $\sigma^2 = \mathrm{id}$ it presents $\langle \varsigma, \sigma \rangle$ as the semidirect product $\langle \varsigma \rangle \rtimes \langle \sigma \rangle$, the dihedral group of order $2m$ when $\varsigma$ has finite order $m$ and an infinite dihedral group otherwise.

Corollary (the ordinary case recovered). The map $\sigma$ satisfies $\sigma\varsigma = \varsigma\sigma$ exactly when $\varsigma^2 = \mathrm{id}$, and then $\tau = \sigma\varsigma$ is the composite of the two commuting involutions $\sigma$ and $\varsigma$ of $A$. The $\varsigma$-semilinear involutions of $A$ are exactly the maps $\tau\varsigma^{-1}$ with $\tau$ an ordinary involution of $A$ satisfying $\tau\varsigma = \varsigma^{-1}\tau$.

Proof. $\sigma\varsigma = \varsigma\sigma$ is exactly $\varsigma^{-1} = \varsigma$, that is $\varsigma^2 = \mathrm{id}$; then $\tau = \sigma\varsigma$ is the composite of the two commuting order-two maps $\sigma$ and $\varsigma$. For the parametrisation, an ordinary involution $\tau$ gives $\sigma = \tau\varsigma^{-1}$, and $\sigma^2 = \tau\varsigma^{-1}\tau\varsigma^{-1}$ is $\mathrm{id}$ exactly when $\tau\varsigma^{-1}\tau = \varsigma$, that is, composing on the left by $\tau$, when $\varsigma^{-1}\tau = \tau\varsigma$; the remaining axioms are immediate.

The Failure of the Naive Identities

The semilinear case is not the linear one with a decoration: four identities that hold when $\varsigma = \mathrm{id}$ fail as soon as the twisting is nontrivial.

Proposition. Let $\sigma$ be $\varsigma$-semilinear.

(a) $\sigma$ is $\varsigma$-linear ($\sigma\varsigma = \varsigma\sigma$) exactly when $\varsigma^2 = \mathrm{id}$; otherwise $\sigma$ intertwines $\varsigma$ with $\varsigma^{-1}$ and is not a semilinear map for $\varsigma$ in the usual sense of a module.

(b) The naive composite rule fails: whereas $\sigma^2\varsigma^2 = \varsigma^2$, the true composite is $(\sigma\varsigma)^2 = \mathrm{id}$ for every $\varsigma$.

(c) $\sigma$ commutes with $\varsigma^2$ exactly when $\varsigma^4 = \mathrm{id}$, because $\sigma\varsigma^2 = \varsigma^{-2}\sigma$.

(d) The fixed set $A^\sigma = \{a : \sigma(a) = a\}$ is an additive subgroup containing $1$, but it is stable under $\varsigma$ exactly when $\varsigma^2 = \mathrm{id}$ on $A^\sigma$; for $a \in A^\sigma$ one has $\sigma(\varsigma(a)) = \varsigma^{-1}(a)$, so $\varsigma(a)$ is again fixed exactly when $\varsigma^2(a) = a$.

Proof. (a) is the first identity of the previous proposition. (b) is the computation of the theorem. (c) is $\sigma\varsigma^2 = \varsigma^{-2}\sigma$, which is $\varsigma^2\sigma$ exactly when $\varsigma^{-2} = \varsigma^2$. (d) $A^\sigma$ is additive because $\sigma$ is, and contains $1$; for the stability, compute $\sigma(\varsigma(a)) = \varsigma^{-1}\sigma(a) = \varsigma^{-1}(a)$, which is $\varsigma(a)$ exactly when $\varsigma^2(a) = a$.

Proposition (the fixed set is not a subring in general). For $a, b \in A^\sigma$ one has $\sigma(ab) = \sigma(b)\sigma(a) = ba$, so $ab$ is fixed exactly when $a$ and $b$ commute. This is the same failure as for an ordinary involution of a noncommutative ring, and it is inherited from $\tau = \sigma\varsigma$: $A^\sigma$ is the fixed set of the ordinary involution $\tau$ composed with $\varsigma$.

Proof. The computation is the anti-multiplicativity of $\sigma$ on fixed elements; the fixed set of $\sigma$ is the image of the fixed set of $\tau$, $A^\sigma = \varsigma(A^\tau)$, because $\sigma(a) = a$ is equivalent to $\tau(\varsigma^{-1}(a)) = a$, that is to $\varsigma^{-1}(a) \in A^\tau$.

Remark. If $A$ is an $S$-algebra and the automorphism $\varsigma$ extends an automorphism $\varsigma_0$ of $S$ acting on the scalars, and if in addition $\sigma$ is $S$-semilinear in the module sense, $\sigma(sa) = \varsigma_0(s)\sigma(a)$, then $\sigma$ restricts to $\varsigma_0$ on the scalars and its fixed set is only an $S^{\varsigma_0}$-module; this is the semilinear involution of the linear structures of a later category, whose treatment is not repeated here.

Examples

(a) The trivial twist. $\varsigma = \mathrm{id}$ gives the ordinary involutions of Involutive Rings: the transpose, the conjugate transpose, the group-ring inversion. Every statement above specialises to the known one.

(b) The transpose twisted by a symmetric unit. On $A = M_n(R)$ let $\varsigma(X) = GXG^{-1}$ be the inner automorphism of a unit $G$ with $G^{\mathrm t} = G$. The transpose $T(X) = X^{\mathrm t}$ satisfies

$$ T\varsigma(X) = G^{-\mathrm t}X^{\mathrm t}G^{\mathrm t} = G^{-1}X^{\mathrm t}G = \varsigma^{-1}T(X), $$

since $G^{\mathrm t} = G$; so $T$ is a $\varsigma$-semilinear involution and $\tau = T\varsigma$ is the ordinary involution $X \mapsto G^{-1}X^{\mathrm t}G$. For $G$ with $G^{\mathrm t} = -G$, $G^2 = -1$ this $\tau$ is the symplectic involution of Matrix Rings with an Involution.

(c) A group ring. Let $A = K[G]$ and let $\varsigma$ be induced by an automorphism $\phi$ of $G$; the inversion map $\sigma(g) = g^{-1}$ extended linearly satisfies $\sigma\varsigma\sigma^{-1}(g) = \sigma\varsigma(g^{-1}) = \sigma(\phi(g)^{-1}) = \phi(g)$, so $\sigma\varsigma\sigma^{-1} = \varsigma$ and not $\varsigma^{-1}$ unless $\phi$ is the identity; thus inversion is $\varsigma$-semilinear only for the trivial twist, and the semilinear involutions of a group ring come from $\tau\varsigma^{-1}$ with $\tau$ the inverse composed with an anti-automorphism of $G$, as in Involutions of a Group Ring.

(d) A quadratic twist. On $A = K \times K$ with $\varsigma(a,b) = (b,a)$ of order two, set $\sigma(a,b) = (b,a)$; then $\sigma\varsigma = \mathrm{id}$-componentwise, $\sigma^2 = \mathrm{id}$ and $\sigma\varsigma = \varsigma\sigma = \varsigma$, so $\sigma$ is a $\varsigma$-semilinear involution with $\tau = \sigma\varsigma = \mathrm{id}$, the trivial ordinary involution.

Example (the dihedral reading). The relations $\sigma^2 = \mathrm{id}$, $\sigma\varsigma\sigma^{-1} = \varsigma^{-1}$ are the presentation of the dihedral group, so a ring with a $\varsigma$-semilinear involution is a ring with an action of a dihedral group, and the semilinear involution is the reflection of that action. This is the same reflection that Reflections as Signed Two-Sided Operators on a Ring reads as an operator, and the two are the element-level and the operator-level faces of one structure.

Summary

A $\varsigma$-semilinear involution of a ring $A$ is an additive anti-multiplicative map $\sigma$ with $\sigma(1) = 1$, $\sigma^2 = \mathrm{id}$ and $\sigma\varsigma = \varsigma^{-1}\sigma$, where $\varsigma$ is an automorphism; the ordinary involution is the case $\varsigma = \mathrm{id}$. The composite $\tau = \sigma\varsigma$ is an ordinary involution, $\sigma = \tau\varsigma^{-1}$, and $\varsigma, \sigma$ generate the dihedral group $\langle \varsigma, \sigma\rangle$ with $\sigma^2 = \mathrm{id}$ and $\sigma\varsigma\sigma^{-1} = \varsigma^{-1}$; the semilinear involutions are parametrised by the ordinary involutions $\tau$ with $\tau\varsigma\tau = \varsigma^{-1}$, a condition that is automatic when $\varsigma^2 = \mathrm{id}$.

The naive identities fail for a nontrivial twist: $\sigma$ commutes with $\varsigma$ exactly when $\varsigma^2 = \mathrm{id}$, the composite $\sigma\varsigma$ has order two although $\sigma^2\varsigma^2 = \varsigma^2$, the map $\sigma$ commutes with $\varsigma^2$ exactly when $\varsigma^4 = \mathrm{id}$, and the fixed set $A^\sigma$ is an additive subgroup containing $1$ that fails to be a subring as for any involution and fails to be $\varsigma$-stable unless $\varsigma^2 = \mathrm{id}$ on it. Over a coefficient ring the semilinearity is genuine: $\sigma$ restricts to $\varsigma$ on the scalars and its fixed set is only linear over the fixed ring of $\varsigma$.

Summary of Notation

Symbol	Meaning
$A$	Ring with $1 \neq 0$, not assumed commutative
$\varsigma$	Automorphism of $A$; the twisting automorphism
$\sigma$	$\varsigma$-semilinear involution
$\sigma\varsigma = \varsigma^{-1}\sigma$	Semilinearity
$\tau = \sigma\varsigma$	Attached ordinary involution; $\sigma = \tau\varsigma^{-1}$
$\sigma^2 = \mathrm{id}$, $\sigma\varsigma\sigma^{-1} = \varsigma^{-1}$	Dihedral relations; $\langle\varsigma,\sigma\rangle$ dihedral
$A^\sigma$	Fixed set of $\sigma$; additive subgroup, not stable under $\varsigma$ in general
$\sigma\varsigma^2 = \varsigma^{-2}\sigma$	$\sigma$ centralises $\varsigma^2$ iff $\varsigma^4 = \mathrm{id}$
$G^{\mathrm t} = G$	Condition on a unit making the transpose semilinear for the inner twist