The Signed Sandwich on a Complex Vector Space

Introduction

Let $V$ be a complex vector space, $E = \operatorname{End}_{\mathbb C}(V)$ its endomorphism algebra, and $\alpha$ a grade involution of $E$: an automorphism of order two, given concretely by $\alpha(X) = TXT$ for a linear involution $T$ of $V$. The unsigned sandwich is the two-sided multiplication $$ \Phi_{a,b}(X) = aXb, $$ and the signed sandwich is its twist by the grade involution, $$ \Theta^{\alpha}_{a,b}(X) = a\,\alpha(X)\,b . $$ The signed sandwich is the most general operator built from two one-sided multiplications and the involution, and it is the object that realises the symmetries of the space: when the involution $T$ is unitary and self-adjoint for a Hermitian form $h$ on $V$ — so that $\alpha$ preserves the adjoint, $\alpha(X^{\dagger}) = \alpha(X)^{\dagger}$ — the inner signed sandwich $\Theta^{\alpha}_{u,u^{-1}} = \mathrm{Ad}_u\circ\alpha$ by a unitary $u$ is an isometry of the Hermitian trace form of $E$, and the unitary reflections of $V$ are the involutions it realises. Its composition laws reduce every composite of signed sandwiches to a signed sandwich, its square is $\mathrm{Ad}_{a\alpha(a)}$ so that it is an involution exactly when $a\alpha(a)$ is central, and its relation to the unsigned sandwich is the relation of the two involutive layers: the unsigned sandwich is the case of the trivial $\alpha$, and the signed family is the unsigned family composed with the involution.

The article has three sections: the sandwich and its composition laws; the reflections and the isometries it realises; and the relation to the unsigned sandwich and the degenerate cases. The unsigned and signed sandwiches, their laws and the associated involutive-subspace theory are The Signed Sandwich on a Linear Space, Involutive Linear Spaces and Involutive Linear Algebras; the endomorphism algebra, the double centraliser and the trace are Algebras of Endomorphisms; the Hermitian form, the unitary group and the involution are The Unitary and Symplectic Groups, Hermitian Geometry and the Unitary Group and The Left and Right Multiplication Operators on a Complex Vector Space, the previous article of this group. The correspondence between reflections and the elements acting by an involution is Reflections as Signed Two-Sided Operators on a Complex Vector Space, and the adjoint of the signed sandwich is The Signed Adjoint Sandwich on a Complex Vector Space, both of this group.

Throughout, $V$ is a finite-dimensional complex vector space of dimension $n$ with a positive-definite Hermitian form $h$, $E = \operatorname{End}_{\mathbb C}(V)$, $A^{\dagger}$ is the adjoint for $h$, $T$ is a unitary self-adjoint involution of $V$, $\alpha(X) = TXT$ is the grade involution it defines, $\Phi_{a,b}$ and $\Theta^{\alpha}_{a,b}$ are the unsigned and signed sandwiches on $E$, and $\langle X, Y\rangle = \operatorname{tr}(X^{\dagger}Y)$ is the Hermitian trace form of $E$. The characteristic is zero throughout, so $2 \neq 0$ and no collapse of the sign occurs.

The Sandwich and Its Composition Laws

Definition. For $a, b \in E$ the unsigned sandwich and the signed sandwich are $$ \Phi_{a,b}(X) = aXb, \qquad \Theta^{\alpha}_{a,b}(X) = a\,\alpha(X)\,b . $$ The signed sandwich is the composite $\Theta^{\alpha}_{a,b} = \Phi_{a,b}\circ\alpha = \alpha\circ\Phi_{\alpha(a),\alpha(b)}$, and the involution $\alpha$ is the only difference between the two families.

Proposition (composition). For all $a,b,c,d \in E$, $$ \Theta^{\alpha}_{a,b}\,\Theta^{\alpha}_{c,d} = \Theta^{\alpha}_{a\alpha(c),\,\alpha(d)b}, \qquad \Phi_{a,b}\,\Theta^{\alpha}_{c,d} = \Theta^{\alpha}_{ac,\,db}, \qquad \Theta^{\alpha}_{a,b}\,\Phi_{c,d} = \Theta^{\alpha}_{a\alpha(d),\,\alpha(c)b}. $$ Thus the set of signed sandwiches is closed under composition, and each product of a signed and an unsigned sandwich is again a signed sandwich; the unsigned sandwiches are closed among themselves, $\Phi_{a,b}\Phi_{c,d} = \Phi_{ac,db}$.

Proof. Direct computation: $\Theta^{\alpha}_{a,b}(\Theta^{\alpha}_{c,d}(X)) = a\alpha(c\alpha(X)d)b = a\,\alpha(c)\,\alpha(\alpha(X))\,\alpha(d)\,b = a\alpha(c)X\alpha(d)b = \Theta^{\alpha}_{a\alpha(c),\alpha(d)b}(X)$, using that $\alpha$ is an algebra automorphism with $\alpha^2 = \mathrm{id}$; the other two are the same computation with one factor unsigned, and the unsigned law is associativity.

Corollary (invertibility and inverse). $\Theta^{\alpha}_{a,b}$ is invertible if and only if $a$ and $b$ are invertible, and then $$ \bigl(\Theta^{\alpha}_{a,b}\bigr)^{-1} = \Theta^{\alpha}_{\alpha(a^{-1}),\,\alpha(b^{-1})} . $$

Proof. Invertibility follows from the factorisation $\Theta^{\alpha}_{a,b} = \Phi_{a,b}\circ\alpha$ and the invertibility of $\alpha$. The inverse is checked by the composition law: with $c = \alpha(a^{-1})$ and $d = \alpha(b^{-1})$ one has $a\alpha(c) = a\alpha(\alpha(a^{-1})) = aa^{-1} = \mathrm{id}$ and $\alpha(d)b = \alpha(\alpha(b^{-1}))b = b^{-1}b = \mathrm{id}$, so the composite is $\Theta^{\alpha}_{\mathrm{id},\mathrm{id}} = \mathrm{id}$, and likewise on the other side.

Definition. The inner signed sandwich of an invertible $a$ is $$ \Theta^{\alpha}_{a,a^{-1}} = \mathrm{Ad}_a\circ\alpha, \qquad \Theta^{\alpha}_{a,a^{-1}}(X) = a\,\alpha(X)\,a^{-1} . $$

Proposition (the square). $\Theta^{\alpha}_{a,a^{-1}}$ is an automorphism of $E$ and $$ \bigl(\Theta^{\alpha}_{a,a^{-1}}\bigr)^{2} = \mathrm{Ad}_{a\alpha(a)} , $$ so it is an involution exactly when $a\alpha(a)$ is central in $E$; in particular, when $a$ commutes with $\alpha$ it is an involution exactly when $a^2$ is central.

Proof. It is an automorphism as the composite of the automorphisms $\mathrm{Ad}_a$ and $\alpha$. For the square, $\mathrm{Ad}_a\alpha\mathrm{Ad}_a\alpha = \mathrm{Ad}_a\,\alpha\mathrm{Ad}_a\alpha = \mathrm{Ad}_a\mathrm{Ad}_{\alpha(a)}\alpha^2 = \mathrm{Ad}_{a\alpha(a)}$, using $\alpha\mathrm{Ad}_b\alpha = \mathrm{Ad}_{\alpha(b)}$ and $\alpha^2 = \mathrm{id}$; an inner automorphism is the identity exactly when its parameter is central, Algebras of Endomorphisms.

The Reflections and the Isometries Realised

Definition. For $u \in V$ with $h(u,u) \neq 0$ the unitary reflection in $u$ is $$ \rho_u(v) = v - 2\,\frac{h(v,u)}{h(u,u)}\,u ; $$ it is a $\mathbb{C}$-linear involution of $V$ of type $(n-1,1)$, fixing the hyperplane $u^{\perp}$ and negating the line $\mathbb{C}u$; it is unitary, $\rho_u^{\dagger} = \rho_u^{-1}$, and self-adjoint, $\rho_u^{\dagger} = \rho_u$, for $h$.

Proposition (the reflection is unitary and self-adjoint). $\rho_u^2 = \mathrm{id}$, $\rho_u^{\dagger} = \rho_u$, and $\rho_u$ is unitary; it is therefore an element $r \in E$ with $r^2 = \mathrm{id}$, $r^{\dagger} = r$, and the grade involution $\alpha_r(X) = rXr$ it defines is an isometry of the trace form of $E$.

Proof. $\rho_u$ fixes $u^{\perp}$ and sends $u$ to $-u$, so $\rho_u^2 = \mathrm{id}$; the self-adjointness is $h(\rho_uv,w) = h(v,\rho_uw)$ from the Hermitian symmetry of $h$ and the reality of the factor $h(v,u)/h(u,u)$ in the appropriate pairing; unitarity is self-adjointness together with the involution, $\rho_u^{\dagger}\rho_u = \rho_u^2 = \mathrm{id}$. Since $r$ is unitary, conjugation by $r$ preserves $\operatorname{tr}(A^\dagger B)$ because it preserves the adjoint and the trace, as in the proposition below.

Proposition (the inner signed sandwich by a unitary is an isometry). Let $u \in E$ be unitary, $u^{\dagger}u = uu^{\dagger} = \mathrm{id}$. Then $\Theta^{\alpha}_{u,u^{-1}}$ preserves the Hermitian trace form, $$ \bigl\langle \Theta^{\alpha}_{u,u^{-1}}(X),\, \Theta^{\alpha}_{u,u^{-1}}(Y)\bigr\rangle = \langle X, Y\rangle , $$ and it is an involution exactly when $u\alpha(u)$ is central. If moreover $\alpha$ comes from a unitary reflection $r$ and $u = r$, then $\Theta^{\alpha_r}_{r,r^{-1}} = \mathrm{id}$: a reflection is fixed by the signed inner sandwich of itself.

Proof. For the isometry, $\Theta(X)^{\dagger} = (u\alpha(X)u^{-1})^{\dagger} = u\,\alpha(X)^{\dagger}\,u^{-1} = u\,\alpha(X^{\dagger})\,u^{-1} = \Theta(X^{\dagger})$, using $u^{\dagger} = u^{-1}$ and $\alpha(X^{\dagger}) = T X^{\dagger} T = (TXT)^{\dagger} = \alpha(X)^{\dagger}$, which is the unitarity and self-adjointness of $T$; since $\Theta$ is an algebra automorphism, $\Theta(X^{\dagger})\Theta(Y) = \Theta(X^{\dagger}Y)$, and $\operatorname{tr}(\Theta(Z)) = \operatorname{tr}(u\alpha(Z)u^{-1}) = \operatorname{tr}(\alpha(Z)) = \operatorname{tr}(TZT) = \operatorname{tr}(Z)$, so the inner product is preserved. The involutivity criterion is the square proposition. For the last assertion, $\Theta^{\alpha_r}_{r,r^{-1}}(X) = r(rXr)r = X$ by $r^2 = \mathrm{id}$.

Remark (the reflections realised). The conjugations $\mathrm{Ad}_r$ by the unitary reflections $r$ are the involutions of $E$ that preserve the trace form and the complex structure; the signed inner sandwich $\Theta^{\alpha}_{u,u^{-1}}$ is an automorphism of $E$ that is the composite of such a conjugation with a grade involution, and it is an involution precisely when $u\alpha(u)$ is central. In this way the reflections of the Hermitian space are the involutions the signed sandwiches realise, and the unitary condition on $T$ is what makes the realised involutions isometries.

The Relation to the Unsigned Sandwich and the Degenerate Cases

Proposition (the unsigned sandwich as the trivial involution). The signed sandwich with $\alpha = \mathrm{id}$ is the unsigned sandwich, $\Theta^{\mathrm{id}}_{a,b} = \Phi_{a,b}$; for the general grade involution, $$ \Theta^{\alpha}_{a,b} = \Phi_{a,b}\circ\alpha = \alpha\circ\Phi_{\alpha(a),\alpha(b)}, $$ so the signed family is the unsigned family composed with the involution $\alpha$, and the diagonal case $b = a^{-1}$ is the inner signed sandwich $\Theta^{\alpha}_{a,a^{-1}} = \mathrm{Ad}_a\circ\alpha$, an inner automorphism followed by the involution.

Proof. The identity $\Theta^{\alpha}_{a,b} = \Phi_{a,b}\circ\alpha$ is the definition: $\Phi_{a,b}(\alpha(X)) = a\alpha(X)b$. For the second, $\alpha\circ\Phi_{\alpha(a),\alpha(b)}(X) = \alpha(\alpha(a)X\alpha(b)) = a\alpha(X)b = \Theta^{\alpha}_{a,b}(X)$, using $\alpha^2 = \mathrm{id}$. The diagonal case follows on setting $b = a^{-1}$.

Remark (the failure in the degenerate cases). Two degenerations break the correspondence between the signed sandwich and the geometry. When $\alpha = \mathrm{id}$ the signed and unsigned sandwiches coincide and the family carries no sign at all; this is the collapse of the $\mathbb Z/2$-grading, and the reflections are then ordinary conjugations. When the Hermitian form $h$ is degenerate, or when the vector $u$ is isotropic, $h(u,u) = 0$, the formula for $\rho_u$ is undefined and there is no unitary reflection in $u$; the signed sandwich by such an element still exists as an operator on $E$ but realises no reflection of $V$, so the correspondence between the elements acting by an involution and the reflections of the space fails exactly where the chosen form degenerates.

Corollary (the sandwich and the operator layer). The signed sandwich $\Theta^{\alpha}_{a,b}$ is the composite $L_a\circ\alpha\circ R_b$ of a left multiplication, the involution and a right multiplication, and every operator on $E$ built from the one-sided multiplications and one grade involution is of this form; with it the one-sided operators, the inner automorphisms and the reflections of the Hermitian space lie in a single family. This is the sense in which the signed sandwich is the central object of the operator layer of the category.

Summary

On the endomorphism algebra $E = \operatorname{End}_{\mathbb C}(V)$ of a complex vector space the signed sandwich $\Theta^{\alpha}_{a,b}(X) = a\alpha(X)b$ is the twist of the two-sided multiplication by the grade involution $\alpha(X) = TXT$, with the composition laws $\Theta^{\alpha}_{a,b}\Theta^{\alpha}_{c,d} = \Theta^{\alpha}_{a\alpha(c),\alpha(d)b}$, $\Phi_{a,b}\Theta^{\alpha}_{c,d} = \Theta^{\alpha}_{ac,db}$ and $\Phi_{a,b}\Phi_{c,d} = \Phi_{ac,db}$, and with inverse $(\Theta^{\alpha}_{a,b})^{-1} = \Theta^{\alpha}_{\alpha(a^{-1}),\alpha(b^{-1})}$. The inner signed sandwich $\Theta^{\alpha}_{a,a^{-1}} = \mathrm{Ad}_a\circ\alpha$ has square $\mathrm{Ad}_{a\alpha(a)}$, so it is an involution exactly when $a\alpha(a)$ is central, and when $T$ is unitary and self-adjoint and $u$ is unitary it preserves the Hermitian trace form $\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$; the unitary reflections $\rho_u(v) = v - 2h(v,u)h(u,u)^{-1}u$ of the Hermitian space are the elements whose conjugations are the realised involutions, and each reflection is fixed by its own signed inner sandwich. The signed family is the unsigned family $\Phi_{a,b}(X) = aXb$ composed with $\alpha$, the case $\alpha = \mathrm{id}$ being the unsigned one; the correspondence with the reflections fails when $\alpha = \mathrm{id}$ and when the form is degenerate or the vector isotropic. The laws, the involutive-subspace theory and the reflections are The Signed Sandwich on a Linear Space, Involutive Linear Spaces and Reflections as Signed Two-Sided Operators on a Complex Vector Space; the adjoint is The Signed Adjoint Sandwich on a Complex Vector Space.

Summary of Notation

Symbol	Meaning
$\Phi_{a,b}(X) = aXb$	the unsigned sandwich
$\Theta^{\alpha}_{a,b}(X) = a\alpha(X)b$	the signed sandwich
$\alpha(X) = TXT$	the grade involution, $T$ a unitary self-adjoint involution
$\Theta^{\alpha}_{a,a^{-1}} = \mathrm{Ad}_a\circ\alpha$	the inner signed sandwich
$(\Theta^{\alpha}_{a,a^{-1}})^2 = \mathrm{Ad}_{a\alpha(a)}$	the square, an involution iff $a\alpha(a)$ central
$\rho_u(v) = v - 2h(v,u)h(u,u)^{-1}u$	the unitary reflection in $u$
$\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$	the Hermitian trace form of $E$