The Signed Left Multiplication on a Complex Vector Space

Introduction

The one-sided companion of the signed sandwich is the signed left multiplication on the endomorphism algebra $E = \operatorname{End}_{\mathbb C}(V)$, $$ \Lambda^{\alpha}_{a} = L_a\circ\alpha, \qquad \Lambda^{\alpha}_{a}(X) = a\,\alpha(X), $$ obtained by taking the right factor of the signed sandwich $\Theta^{\alpha}_{a,b}(X) = a\alpha(X)b$ to be the identity. It is the composite of the ordinary left multiplication with the grade involution, so it is the case $b = 1$ of the signed sandwich, its products are ordinary left multiplications, its inverse is again signed, and its fixed elements are the solutions of $a\alpha(X) = X$. When the grade involution comes from a unitary self-adjoint involution $T$ of a Hermitian space $V$ and the element $a$ is unitary, the signed left multiplication is an isometry of the Hermitian trace form of $E$, so that the one-sided signed operators of the geometry are the isometric ones; the unsigned left multiplication is the case of the trivial involution, and the difference between the two is the sign carried by $\alpha$.

The article has three sections: the signed left multiplication and its laws; the fixed elements and their computation through the involution; and the relation to the unsigned left multiplication, with the isometries. The unsigned and signed sandwiches are The Signed Sandwich on a Complex Vector Space, the previous articles of this group; the endomorphism algebra and its trace are Algebras of Endomorphisms; the fixed part of an involution and its dimension are Involutive Linear Spaces; the unitary group and the involution are The Unitary and Symplectic Groups and Hermitian Geometry and the Unitary Group. The unsigned case with its adjoint is The Adjoint of the Left Multiplication on a Complex Vector Space, the signed adjoint is The Signed Adjoint of the Left Multiplication on a Complex Vector Space, and the module-level variant is The Graded Action on a Module over a Complex Vector Space, all of this group. The group-level analogue is The Signed Left Multiplication on a Symmetry 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, $L_a(X) = aX$ and $R_b(X) = Xb$ are the one-sided multiplications, $\Lambda^{\alpha}_{a} = L_a\circ\alpha$ is the signed left multiplication, and $\langle X, Y\rangle = \operatorname{tr}(X^{\dagger}Y)$ is the Hermitian trace form of $E$.

The Signed Left Multiplication and Its Laws

Definition. For $a \in E$ the signed left multiplication is $$ \Lambda^{\alpha}_{a} = L_a\circ\alpha, \qquad \Lambda^{\alpha}_{a}(X) = a\,\alpha(X) . $$

Proposition (the two factorisations). $\Lambda^{\alpha}_{a}$ is the signed sandwich with right factor the identity, $\Lambda^{\alpha}_{a} = \Theta^{\alpha}_{a,1}$, and $$ \Lambda^{\alpha}_{a} = L_a\circ\alpha = \alpha\circ L_{\alpha(a)} , $$ so the signed left multiplication is the ordinary left multiplication by $\alpha(a)$ conjugated by $\alpha$.

Proof. The first identity is the definition of the sandwich with $b = 1$; for the second, $\alpha L_{\alpha(a)}(X) = \alpha(\alpha(a)X) = a\alpha(X) = \Lambda^{\alpha}_{a}(X)$.

Proposition (composition). For all $a,b \in E$, $$ \Lambda^{\alpha}_{a}\Lambda^{\alpha}_{b} = L_{a\alpha(b)}, \qquad L_a\Lambda^{\alpha}_{b} = \Lambda^{\alpha}_{ab}, \qquad \Lambda^{\alpha}_{a}L_b = R_{\alpha(b)}\Lambda^{\alpha}_{a} . $$ In particular the composite of two signed left multiplications is an ordinary left multiplication, and the set $\{L_a : a \in E\} \cup \{\Lambda^{\alpha}_{a} : a \in E\}$ is closed under composition.

Proof. $\Lambda^{\alpha}_{a}\Lambda^{\alpha}_{b}(X) = a\alpha(b\alpha(X)) = a\alpha(b)\alpha(\alpha(X)) = a\alpha(b)X = L_{a\alpha(b)}(X)$; $L_a\Lambda^{\alpha}_{b}(X) = ab\alpha(X) = \Lambda^{\alpha}_{ab}(X)$; $\Lambda^{\alpha}_{a}L_b(X) = a\alpha(X)\alpha(b) = R_{\alpha(b)}(\Lambda^{\alpha}_{a}(X))$. The closure is the three identities together.

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

Proof. Invertibility follows from the factorisation $\Lambda^{\alpha}_{a} = L_a\alpha$ and the invertibility of $\alpha$; the inverse is checked by the composition law: $\Lambda^{\alpha}_{a}\Lambda^{\alpha}_{\alpha(a^{-1})} = L_{a\alpha(\alpha(a^{-1}))} = L_{aa^{-1}} = \mathrm{id}$, and likewise on the other side.

The Fixed Elements and the Involution

Proposition (fixed elements). The fixed space of $\Lambda^{\alpha}_{a}$ is $$ \ker(\Lambda^{\alpha}_{a} - \mathrm{id}) = \{X \in E : a\alpha(X) = X\} ; $$ equivalently, when $a$ is invertible, $X$ is fixed exactly when $\alpha(X) = a^{-1}X$. For $a = 1$ the fixed space is the fixed part $E^{+}$ of $\alpha$, and in a basis of $V$ in which $T$ is diagonal with $p$ entries $+1$ and $q$ entries $-1$, $p+q = n$, its dimension is $$ \dim E^{+} = p^{2} + q^{2}. $$

Proof. The first display is the eigenvalue-one equation. For $a = 1$ it is $\alpha(X) = X$, whose solution space is $E^{+}$; the dimension is the computation for a diagonal involution: the matrix-unit $E_{ij}$ is fixed exactly when the two signs agree, $(-1)^{i+j} = 1$, which happens for the $p^2$ units with both indices in the $+1$ block and the $q^2$ units with both indices in the $-1$ block. This is Involutive Linear Spaces.

Proposition (the fixed space of a reflection's involution). If $T = r$ is a unitary reflection of type $(n-1,1)$, with the $-1$ eigenspace spanned by $u$, then the fixed part $E^{+}$ of $\alpha_r$ has dimension $(n-1)^2 + 1$: it consists of the endomorphisms $X$ with $X = rXr$, that is those that preserve the decomposition $V = u^{\perp}\oplus\mathbb{C}u$, act on the hyperplane $u^{\perp}$ by an arbitrary endomorphism and on the line $\mathbb{C}u$ by a scalar.

Proof. With $p = n-1$ and $q = 1$ the dimension formula gives $(n-1)^2 + 1$; the description is the block form in the orthogonal decomposition $V = u^{\perp}\oplus\mathbb{C}u$, in which $T = \operatorname{diag}(1,\dots,1,-1)$ and $X = rXr$ means that $X$ is block diagonal for this decomposition with an arbitrary top block and a scalar bottom block.

Example (matrix units as eigenvectors). For $T = \operatorname{diag}(1,-1)$ on $\mathbb{C}^2$ and $a = \operatorname{diag}(\lambda, \mu)$ with $\lambda\mu \neq 0$, the signed left multiplication on the two-by-two matrices is $X \mapsto a\,TXT$, and the matrix units $E_{ij}$ are eigenvectors with eigenvalue $a_{ii}(-1)^{i+j}$; the fixed space is spanned by $E_{11}$ and $E_{22}$ and has dimension $2 = 1^2+1^2$, and the off-diagonal units are eigenvectors with eigenvalues $-\lambda$ and $-\mu$.

The Relation to the Unsigned Left Multiplication and the Isometries

Proposition (the unsigned case). The signed left multiplication with $\alpha = \mathrm{id}$ is the ordinary left multiplication, $\Lambda^{\mathrm{id}}_{a} = L_a$, and for the general $\alpha$ the two families are exchanged by the involution, $$ \Lambda^{\alpha}_{a} = L_a\circ\alpha, \qquad L_{\alpha(a)} = \Lambda^{\alpha}_{a}\circ\alpha , $$ the first the definition and the second its consequence $\Lambda^{\alpha}_{a}\alpha = L_a\alpha\alpha = L_a$; the signed family is the ordinary family twisted by $\alpha$, exactly as the signed sandwich is the unsigned sandwich twisted by $\alpha$.

Proof. The first identity is the definition; composing it on the right with $\alpha$ and using $\alpha^2 = \mathrm{id}$ gives $\Lambda^{\alpha}_{a}\alpha = L_a$, which after replacing $a$ by $\alpha(a)$ is the second. Setting $\alpha = \mathrm{id}$ returns $L_a$ in both.

Proposition (the unitary case is isometric). Let $a = u$ be unitary and let $T$ be unitary and self-adjoint. Then $\Lambda^{\alpha}_{u}$ preserves the Hermitian trace form, $$ \bigl\langle \Lambda^{\alpha}_{u}(X),\, \Lambda^{\alpha}_{u}(Y)\bigr\rangle = \langle X, Y\rangle , $$ and $\Lambda^{\alpha}_{u}$ is invertible with $(\Lambda^{\alpha}_{u})^{-1} = \Lambda^{\alpha}_{\alpha(u^{-1})} = \Lambda^{\alpha}_{u^{\dagger}}$.

Proof. $\langle \Lambda_u(X),\Lambda_u(Y)\rangle = \operatorname{tr}((u\alpha(X))^{\dagger}u\alpha(Y)) = \operatorname{tr}(\alpha(X)^{\dagger}u^{\dagger}u\alpha(Y)) = \operatorname{tr}(\alpha(X)^{\dagger}\alpha(Y)) = \operatorname{tr}(\alpha(X^{\dagger}Y)) = \operatorname{tr}(X^{\dagger}Y) = \langle X,Y\rangle$, using $u^{\dagger} = u^{-1}$, $\alpha(Z)^{\dagger} = \alpha(Z^{\dagger})$ (from the unitarity and self-adjointness of $T$) and $\operatorname{tr}(\alpha(Z)) = \operatorname{tr}(TZT) = \operatorname{tr}(Z)$. The inverse is the corollary of the first section, together with $u^{-1} = u^{\dagger}$.

Remark (the fixed elements in the geometric reading). The fixed space of $\Lambda^{\alpha}_{a}$ is the set of endomorphisms that $\alpha$ carries to a fixed multiple of $a^{-1}$; for $a = 1$ it is the fixed part $E^{+}$ of the involution, of dimension $p^2+q^2$, and for a reflection's involution of type $(n-1,1)$ it has dimension $(n-1)^2+1$. The signed left multiplication by a unitary is an isometry of the trace form, and its fixed space is nontrivial exactly when the unitary element has an eigenvalue-one direction compatible with the involution, which is the geometric content of the eigenvalue-one equation.

Summary

On the endomorphism algebra $E = \operatorname{End}_{\mathbb C}(V)$ of a complex vector space the signed left multiplication is $\Lambda^{\alpha}_{a} = L_a\circ\alpha$, the signed sandwich with right factor one, equal to $\alpha\circ L_{\alpha(a)}$. Its product with another is the ordinary left multiplication $\Lambda^{\alpha}_{a}\Lambda^{\alpha}_{b} = L_{a\alpha(b)}$; it satisfies $L_a\Lambda^{\alpha}_{b} = \Lambda^{\alpha}_{ab}$ and $\Lambda^{\alpha}_{a}L_b = R_{\alpha(b)}\Lambda^{\alpha}_{a}$, so the signed and ordinary left multiplications together form a closed monoid; it is invertible exactly for invertible $a$, with inverse $\Lambda^{\alpha}_{\alpha(a^{-1})}$; and its fixed space is the solution set of $a\alpha(X) = X$, equal to the fixed part $E^{+}$ of $\alpha$ when $a = 1$, of dimension $p^2+q^2$ for an involution with $p$ signs $+1$ and $q$ signs $-1$, and of dimension $(n-1)^2+1$ for a reflection's involution. When $T$ is unitary and self-adjoint and $a = u$ is unitary, the signed left multiplication is an isometry of the Hermitian trace form $\langle X,Y\rangle = \operatorname{tr}(X^\dagger Y)$. The unsigned case with its adjoint, the signed adjoint and the module-level graded variant are the companion articles of this group; the abstract laws and the involutive-subspace theory are The Signed Sandwich on a Complex Vector Space and Involutive Linear Spaces.

Summary of Notation

Symbol	Meaning
$E=\operatorname{End}_{\mathbb C}(V)$, $\alpha(X)=TXT$	the endomorphism algebra and the grade involution
$L_a(X)=aX$, $R_b(X)=Xb$	the one-sided multiplications
$\Lambda^{\alpha}_{a}=L_a\circ\alpha$	the signed left multiplication
$\Lambda^{\alpha}_{a}=L_a\alpha=\alpha L_{\alpha(a)}$	the two factorisations
$\Lambda^{\alpha}_{a}\Lambda^{\alpha}_{b}=L_{a\alpha(b)}$	the composition law
$(\Lambda^{\alpha}_{a})^{-1}=\Lambda^{\alpha}_{\alpha(a^{-1})}$	the inverse
$\dim E^{+}=p^2+q^2$	the dimension of the fixed part of an involution