The Signed Sandwich on a Jordan Algebra

Introduction

A two-sided operator dresses an element between a left and a right factor. For an associative algebra the sandwich is $x\mapsto axb$, and with a grade involution $\alpha$ of order two available it can be twisted in the middle to $x\mapsto a\,\alpha(x)\,b$; this is the signed sandwich, and its theory for a ring — the composition law $S_{a,b}S_{c,d} = S_{ac,db}$, the twisted law $S^{\alpha}_{a,b}S^{\alpha}_{c,d} = S^{\alpha}_{a\alpha(c),b\alpha(d)}$, and the criterion for the diagonal $r_u = S^{\alpha}_{u,u^{-1}}$ to be an involution — belongs to The Signed Sandwich on a Ring. The product of a Jordan algebra is not associative, so the expression $axb$ has no intrinsic meaning; it exists only when the Jordan algebra is special, $J = A^+$ for an associative algebra $A$, in which case $axb$ is an operator on the underlying module of $J$ and the intrinsic Jordan two-sided operator is its symmetrisation, the quadratic representation $U_{a,b}$ of The Left and Right Multiplication Operators on a Jordan Algebra.

This article carries the two-sided operators to the Jordan setting. It defines the unsigned sandwich as an operator on a special Jordan algebra, proves that its symmetrisation is the quadratic representation, and shows that the symmetrised operators are exactly the two-sided operators that live in the multiplication algebra. It then defines the signed sandwich $S^{\alpha}_{a,b}(x) = a\,\alpha(x)\,b$ and its symmetrisation $\Sigma^{\alpha}_{a,b} = U_{a,b}\circ\alpha$, and it isolates the failure of the ring composition law: the composition of two symmetrised signed sandwiches is a product $U_{a,b}U_{c,d}$ of quadratic representations, which is not in general a quadratic representation, so the monoid law of the ring case does not survive the symmetrisation. The diagonal signed sandwich $r_u = S^{\alpha}_{u,u^{-1}}$ is the signed conjugation $x \mapsto u\alpha(x)u^{-1}$, and its square is the conjugation by $u\alpha(u)$; it is an involution, a reflection, exactly when $u\alpha(u)$ is central. The reflection correspondence is developed in Reflections as Signed Two-Sided Operators on a Jordan Algebra, next in this group.

The article assumes The Signed Sandwich on a Ring for the definitions of the grade involution, the unsigned and the signed sandwich on a ring and their composition laws (used, not reproved), The Left and Right Multiplication Operators on a Jordan Algebra for the quadratic representation and the fundamental formula, Jordan Algebras for the special Jordan algebra $A^+$, and The Polarisation Operator for the polarised form $U_{a,b} = \tfrac12(U_{a+b}-U_a-U_b)$. Throughout, $J = A^+$ is a special unital Jordan algebra over a commutative ring $R$, $\alpha$ is a grade involution of the associative algebra $A$ (an automorphism of order two, hence an automorphism of $J$), and $S_{a,b}$, $S^{\alpha}_{a,b}$ denote the operators on the underlying module $J$ defined below. No norm, form or distance occurs; the "reflection" is an operator of order two, without any geometric reading, and the symmetric-space theory of reflections is met in Part IV.

The Unsigned Sandwich on a Special Jordan Algebra

Definition

Definition. For $a, b \in J = A^+$ the unsigned sandwich is the operator

$$ S_{a,b} : J \longrightarrow J, \qquad S_{a,b}(x) = a\,x\,b , $$

the product being taken in the associative algebra $A$ whose underlying module is $J$. It is additive, it is $R$-linear in each of $a$, $b$ and $x$, and $S_{a,b} = L^A_aR^A_b$ is the product of a left and a right multiplication of $A$.

The sandwich is an operator on the module $J$; it is not in general an operator of the Jordan structure. It need not commute with the Jordan product, and it need not lie in the multiplication algebra $\operatorname{Mult}(J)$ of The Jordan Multiplication Operators.

The Symmetrisation is the Quadratic Representation

Theorem. For all $a, b \in J$,

$$ \tfrac12\bigl(S_{a,b} + S_{b,a}\bigr) = U_{a,b} , $$

the quadratic representation of The Left and Right Multiplication Operators on a Jordan Algebra; in particular the symmetrisation of the sandwiches is an operator of the Jordan structure, and it lies in $\operatorname{Mult}(J)$.

Proof. For $x \in J$ one has $\tfrac12(S_{a,b}+S_{b,a})(x) = \tfrac12(axb+bxa)$, and the special form of the quadratic representation is $U_{a,b}(x) = \tfrac12(axb+bxa)$. That $U_{a,b}$ lies in $\operatorname{Mult}(J)$ is its definition, $U_{a,b} = L_aL_b+L_bL_a-L_{a\bullet b}$. $\square$

Corollary. The antisymmetrisation $\tfrac12(S_{a,b}-S_{b,a})$ is nonzero exactly when the sandwich does not commute with the transposition of its parameters; it equals $\tfrac12(axb-bxa)$ and is the operator that the Jordan structure discards. The Jordan two-sided operator is the part of $S_{a,b}$ that is symmetric in $a$ and $b$, and the discarded part measures the failure of the Jordan product to see the order of the factors.

Remark. The unsigned sandwich and the quadratic representation agree in the associative case: if $A$ is commutative, $S_{a,b} = U_{a,b} = L_{ab}$. They differ already for $A = M_2(k)$, where $S_{a,b}(x)$ need not be symmetric in the parameters while $U_{a,b}(x) = \tfrac12(axb+bxa)$ always is.

The Graded Structure and the Twist

The Grade Involution

Definition. A grade involution of the associative algebra $A$ is an automorphism $\alpha \in \operatorname{Aut}(A)$ with $\alpha^2 = \mathrm{id}$; it is the trivial one when $\alpha = \mathrm{id}$. The decomposition

$$ A = A_{\bar 0} \oplus A_{\bar 1}, \qquad A_{\bar 0} = \{x : \alpha(x) = x\}, \qquad A_{\bar 1} = \{x : \alpha(x) = -x\}, $$

makes $A$ a $\mathbb{Z}/2$-graded algebra, with $A_iA_j \subseteq A_{i+j}$; this is the graded structure of The Signed Sandwich on a Ring, and $\alpha$ is an automorphism of the special Jordan algebra $J = A^+$, since $\alpha(x\bullet y) = \alpha(x)\bullet\alpha(y)$ for the halved product.

Proposition. $\alpha$ is a grade involution of the Jordan algebra $J$, that is, an automorphism of $J$ with $\alpha^2 = \mathrm{id}$, and the Jordan product respects the induced grading, $J_i\bullet J_j \subseteq J_{i+j}$ with $J_i = A_i$; the commutativity of $\bullet$ is not disturbed, and the only new datum is the sign rule that an odd element carries.

Proof. $\alpha$ preserves the associative product, hence the halved product, hence is an automorphism of $J$; $\alpha^2 = \mathrm{id}$ is assumed; the grading statement is the multiplicativity of $\alpha$ read on the decomposition. $\square$

The Signed Sandwich

Definition. For $a, b \in J$ the signed sandwich is the operator

$$ S^{\alpha}_{a,b} : J \longrightarrow J, \qquad S^{\alpha}_{a,b}(x) = a\,\alpha(x)\,b . $$

It is additive, $R$-linear in $a$ and $b$, and it is the unsigned sandwich precomposed with the grade involution,

$$ S^{\alpha}_{a,b} = S_{a,b}\circ\alpha . $$

Theorem (the signed sandwich is the unsigned sandwich at $\alpha$). For all $a, b \in J$,

$$ S^{\alpha}_{a,b} = S_{a,b}\circ\alpha, \qquad S^{\alpha}_{a,b}(x) = S_{a,b}(\alpha(x)) \ \text{ for all } x , $$

and the two coincide exactly when $\alpha = \mathrm{id}$.

Proof. $S_{a,b}(\alpha(x)) = a\alpha(x)b = S^{\alpha}_{a,b}(x)$, by the definitions. If $\alpha = \mathrm{id}$ the two formulas are identical; conversely if $S^{\alpha}_{a,b} = S_{a,b}$ for all $a,b$ then $\alpha(x) = x$ for all $x$ by choosing $a, b$ with $a\alpha(x)b \ne axb$ unless $\alpha(x) = x$. $\square$

Proposition (the symmetrisation of the signed sandwich). For all $a, b \in J$,

$$ \tfrac12\bigl(S^{\alpha}_{a,b} + S^{\alpha}_{b,a}\bigr) = U_{a,b}\circ\alpha , $$

and the operator on the right lies in $\operatorname{Mult}(J)$; write it $\Sigma^{\alpha}_{a,b} = U_{a,b}\circ\alpha$ and call it the symmetrised signed sandwich. The symmetrisation is exactly the twist of the quadratic representation by the grade involution, and it is the signed two-sided operator of the Jordan structure.

Proof. $\tfrac12(S^{\alpha}_{a,b}+S^{\alpha}_{b,a})(x) = \tfrac12(a\alpha(x)b+b\alpha(x)a) = U_{a,b}(\alpha(x)) = (U_{a,b}\circ\alpha)(x)$, using the special form of $U_{a,b}$. $\square$

Composition and the Failure of the Ring Law

The Unsymmetrised Case

The unsigned and signed sandwiches themselves obey the ring laws, since they are products of one-sided multiplications of $A$: by The Signed Sandwich on a Ring, $S_{a,b}S_{c,d} = S_{ac,db}$ and

$$ S^{\alpha}_{a,b}S^{\alpha}_{c,d} = S^{\alpha}_{a\alpha(c),\,b\alpha(d)} . $$

These two laws are used here and not reproved. They are laws of operators on the module $J$, and the second inserts the grade involution into the parameters, which is the sign rule of the grading.

The Symmetrised Case

Theorem (the failed law). For all $a, b, c, d \in J$,

$$ \Sigma^{\alpha}_{a,b}\,\Sigma^{\alpha}_{c,d} = U_{a,b}\,U_{\alpha(c),\alpha(d)} , $$

a product of two quadratic representations. This product is not in general of the form $\Sigma^{\alpha}_{e,f}$; indeed it lies in the span of the $U_{e,f}$ only when $U_{a,b}U_{\alpha(c),\alpha(d)}$ is itself a quadratic representation, which is the exception and not the rule. Hence the monoid law of the ring case does not survive the symmetrisation.

Proof. From $\Sigma^{\alpha}_{a,b} = U_{a,b}\alpha$ and $\alpha U_{c,d} = U_{\alpha(c),\alpha(d)}\alpha$ (the equivariance of the quadratic representation under the automorphism $\alpha$), the composite is $U_{a,b}\alpha U_{c,d}\alpha = U_{a,b}U_{\alpha(c),\alpha(d)}\alpha^2 = U_{a,b}U_{\alpha(c),\alpha(d)}$. For the failure of the Jordan form, take $\alpha = \mathrm{id}$, $A = M_2(k)$, $a = b = E_{12}$ and $c = d = E_{21}$. Then $U_{a,a}(x) = E_{12}xE_{12} = r\,E_{12}$ and $U_{c,c}(x) = E_{21}xE_{21} = q\,E_{21}$, where $x = \bigl(\begin{smallmatrix}p&q\\ r&s\end{smallmatrix}\bigr)$, so

$$ U_{a,a}U_{c,c}(x) = E_{12}\,(qE_{21})\,E_{12} = q\,E_{12} . $$

This operator is the $(1,2)$-coordinate functional times the matrix unit $E_{12}$. Suppose it were a quadratic representation $U_{e,f}$ with $e = \bigl(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\bigr)$, $f = \bigl(\begin{smallmatrix}p&q_1\\ r&s\end{smallmatrix}\bigr)$. Evaluating at $x = E_{12}$ gives $U_{e,f}(E_{12}) = E_{12}$, whose $(1,2)$-entry is the $(1,2)$-entry of $\tfrac12(eE_{12}f+fE_{12}e)$, namely $\tfrac12(bq_1+q_1b) = bq_1$; hence $bq_1 = 1$. Evaluating at $x = E_{21}$ gives $U_{e,f}(E_{21}) = 0$, whose $(1,2)$-entry is $\tfrac12(bq_1+q_1b) = bq_1$ again; hence $bq_1 = 0$, a contradiction. So the composite is not a quadratic representation. $\square$

Corollary. The set $\{\Sigma^{\alpha}_{a,b} : a, b \in J\}$ generates an algebra under composition, but it is not a monoid, and the composition law of the ring case is replaced by the multiplication in the algebra of quadratic representations. The one-sided signed operator $x\mapsto a\,\alpha(x)$ of The Signed Left Multiplication on a Jordan Algebra, which is linear in a single parameter, is the element from which the symmetrised signed sandwiches are rebuilt in the next articles of this group.

Reflections Realised by the Signed Sandwich

The Diagonal Signed Sandwich

Definition. For a unit $u \in A^\times$ the diagonal signed sandwich is

$$ r_u = S^{\alpha}_{u,u^{-1}} : J \to J, \qquad r_u(x) = u\,\alpha(x)\,u^{-1} . $$

It is the signed conjugation by $u$, and it is an operator of the Jordan structure when it is the twist of the conjugation $U_{u,u^{-1}}$.

Theorem. Let $u \in A^\times$. Then

$$ r_u^2 = \operatorname{conj}_{u\alpha(u)} , \qquad r_u^2(x) = u\alpha(u)\,x\,(u\alpha(u))^{-1} . $$

Hence $r_u$ is an involution, a reflection, exactly when $u\,\alpha(u)$ is central in $A$; it is then the inner automorphism of $J$ by $u\alpha(u)$, of order two.

Proof. The square is computed from the ring law $S^{\alpha}_{a,b}S^{\alpha}_{c,d} = S^{\alpha}_{a\alpha(c),b\alpha(d)}$ with $a = c = u$ and $b = d = u^{-1}$: $r_u^2 = S^{\alpha}_{u\alpha(u),\,u^{-1}\alpha(u^{-1})}$, and $u^{-1}\alpha(u^{-1}) = (u\alpha(u))^{-1}$ because $\alpha(u^{-1}) = \alpha(u)^{-1}$. The conjugation by an element is the identity exactly when that element is central. $\square$

Corollary (parity). If $u$ is fixed by $\alpha$ then $u\alpha(u) = u^2$, and $r_u$ is an involution exactly when $u^2$ is central; if $u$ is negated by $\alpha$ then $u\alpha(u) = -u^2$, central exactly when $u^2$ is central. In particular $u^2 \in Z(A)$ makes $r_u$ an involution whatever the parity of $u$, and this is the criterion inherited from The Signed Sandwich on a Ring.

The Reflection and the Quadratic Representation

Proposition. The a symmetrised diagonal signed sandwich is

$$ \Sigma^{\alpha}_{u,u^{-1}} = U_{u,u^{-1}}\circ\alpha , \qquad \Sigma^{\alpha}_{u,u^{-1}}(x) = \tfrac12\bigl(u\alpha(x)u^{-1} + u^{-1}\alpha(x)u\bigr) , $$

and it is an element of $\operatorname{Mult}(J)$. It is an involution when $U_{u,u^{-1}}$ commutes with the twist in the appropriate sense; the exact statement and the correspondence between reflections and the elements acting by an involution — with its failure in the degenerate case of a non-central $u\alpha(u)$ — is the subject of Reflections as Signed Two-Sided Operators on a Jordan Algebra, next in this group.

Proof. The formula is the symmetrisation of $r_u$, together with the identification $\tfrac12(S^{\alpha}_{u,u^{-1}}+S^{\alpha}_{u^{-1},u}) = U_{u,u^{-1}}\alpha$ proved above. $\square$

A Worked Example

Example (the matrix Jordan algebra). Let $A = M_2(k)$ over a field $k$ of characteristic not two, let $\alpha$ be the conjugation $X \mapsto JXJ^{-1}$ with $J = \operatorname{diag}(1,-1)$, a grade involution whose even part is the diagonal matrices and whose odd part the off-diagonal ones, and let $J = M_2(k)^+$ be the special Jordan algebra with the halved product. Take the unit $u = E_{12}+E_{21}$, so that $\alpha(u) = -u$ and $u^{-1} = u$. Then $u\alpha(u) = -u^2 = -I$ is central, so $r_u$ is an involution:

$$ r_u(x) = u\,\alpha(x)\,u^{-1} = u\,\alpha(x)\,u , \qquad r_u^2(x) = (-I)\,x\,(-I)^{-1} = x . $$

The reflection $r_u$ acts on the Jordan algebra $J$ by the signed conjugation, and its symmetrisation $\Sigma^{\alpha}_{u,u} = U_{u,u}\circ\alpha$ lies in the multiplication algebra. The operator $r_u$ itself does not lie in the multiplication algebra, since $U_{u,u}(x) = uxu$ while $r_u(x) = u\alpha(x)u$ and $\alpha \ne \mathrm{id}$.

Example (a degenerate sandwich). With $A = M_2(k)$, $\alpha = \mathrm{id}$ and $u = I + E_{12}$, one has $\alpha(u) = u$ and $u^2 = I + 2E_{12}$, which is not central; $r_u = \operatorname{conj}_u$ has $r_u^2 = \operatorname{conj}_{u^2} \ne \mathrm{id}$, so the sandwich is an automorphism of infinite order and not a reflection. The failure is exactly the non-centrality of $u\alpha(u)$, and it is the degenerate case that the reflection correspondence must exclude.

Summary

On a special Jordan algebra $J = A^+$ the unsigned sandwich is $S_{a,b}(x) = axb$ and the signed sandwich is $S^{\alpha}_{a,b}(x) = a\alpha(x)b$, with $S^{\alpha}_{a,b} = S_{a,b}\circ\alpha$; both are operators on the underlying module and obey the ring laws $S_{a,b}S_{c,d} = S_{ac,db}$ and $S^{\alpha}_{a,b}S^{\alpha}_{c,d} = S^{\alpha}_{a\alpha(c),b\alpha(d)}$. The Jordan two-sided operator is the symmetrisation: $\tfrac12(S_{a,b}+S_{b,a}) = U_{a,b}$ is the quadratic representation, and $\tfrac12(S^{\alpha}_{a,b}+S^{\alpha}_{b,a}) = U_{a,b}\circ\alpha = \Sigma^{\alpha}_{a,b}$ is the symmetrised signed sandwich, the two being elements of $\operatorname{Mult}(J)$. The ring composition law does not survive the symmetrisation: $\Sigma^{\alpha}_{a,b}\Sigma^{\alpha}_{c,d} = U_{a,b}U_{\alpha(c),\alpha(d)}$, a product of quadratic representations that is not itself one in general. The diagonal signed sandwich $r_u = S^{\alpha}_{u,u^{-1}}$ is the signed conjugation $x\mapsto u\alpha(x)u^{-1}$; its square is the conjugation by $u\alpha(u)$, so it is a reflection exactly when $u\alpha(u)$ is central, with the parity corollary that $u^2$ central suffices. The reflection correspondence, with its degenerate cases, is Reflections as Signed Two-Sided Operators on a Jordan Algebra; the unsymmetrised signed one-sided operator is The Signed Left Multiplication on a Jordan Algebra.

Summary of Notation

Symbol	Meaning
$J = A^+$	Special unital Jordan algebra with halved product
$A$	The associative algebra whose symmetrisation is $J$
$\alpha$	Grade involution, an automorphism with $\alpha^2 = \mathrm{id}$
$A_{\bar 0}, A_{\bar 1}$	Even and odd parts, $J_i = A_i$
$S_{a,b}(x) = axb$	Unsigned sandwich
$U_{a,b} = \tfrac12(S_{a,b}+S_{b,a})$	Quadratic representation as symmetrised sandwich
$S^{\alpha}_{a,b}(x) = a\alpha(x)b$	Signed sandwich
$S^{\alpha}_{a,b} = S_{a,b}\circ\alpha$	Signed sandwich is the unsigned at $\alpha$
$\Sigma^{\alpha}_{a,b} = U_{a,b}\circ\alpha$	Symmetrised signed sandwich
$\Sigma^{\alpha}_{a,b}\Sigma^{\alpha}_{c,d} = U_{a,b}U_{\alpha(c),\alpha(d)}$	Failed composition law
$r_u = S^{\alpha}_{u,u^{-1}}(x) = u\alpha(x)u^{-1}$	Diagonal signed sandwich
$r_u^2 = \operatorname{conj}_{u\alpha(u)}$	Reflection iff $u\alpha(u) \in Z(A)$