The Graded Action on a Module over a Jordan Algebra

Introduction

A module over a Jordan algebra is not an object of the same kind as a module over an associative algebra, because the Jordan product is not associative: there is no intrinsic meaning to the iterated products $a(bm)$ and $(ab)m$ agreeing. The multiplication algebra $\operatorname{Mult}(J)$ of The Jordan Multiplication Operators, the associative algebra generated by the one-sided multiplications $L_a$, repairs this: a module over the Jordan algebra $J$ is a module over $\operatorname{Mult}(J)$, and the action of $J$ is the action of the generators $L_a$ of that associative algebra. When $J$ carries a grade involution $\alpha$ and the module carries a compatible grade involution $\alpha_M$, the action acquires a second, signed layer: the graded action

$$ \rho^{\alpha}_a = \rho_{L_a}\circ\alpha_M , \qquad \rho^{\alpha}_a(m) = a\cdot\alpha_M(m), $$

obtained from the unsigned action of $L_a$ by precomposition with the grade involution of the module.

This article sets up the algebra of that layer. It defines modules over a Jordan algebra through $\operatorname{Mult}(J)$, defines the grading of $\operatorname{Mult}(J)$ induced by the grading of $J$, and shows the compatibility of the action with the grading: a homogeneous multiplication by $a$ of degree $|a|$ shifts the module degree by $|a|$, so the representing map $J\to\operatorname{End}_R(M)$ is a graded map. It then derives the sign rule: the grade involutions of the algebra and of the module are intertwined by the action, $\alpha_M\rho_{L}\alpha_M^{-1} = \rho_{\alpha(L)}$, and the graded action composes by the twisted product $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$, which inserts a parity sign into the multiplication operators. The upshot is that the graded action is not a representation of the Jordan algebra but a representation of a twisted multiplication algebra, and the twist is exactly the sign rule the grading imposes.

The article assumes Modules over an Algebra for modules and the regular module, The Jordan Multiplication Operators for $J$, $L_a$ and $\operatorname{Mult}(J)$, The Left and Right Multiplication Operators on a Jordan Algebra for the structure of the multiplication algebra, The Signed Left Multiplication on a Jordan Algebra for the twist by the grade involution, and Jordan Algebras for the product. The adjoint action, in which the pairing is carried through the graded action, is The Graded Adjoint Action on a Module over a Jordan Algebra, in the * Operator Theory group; the involution of the elements is not used. Throughout, $R$ is a commutative ring with identity, $J$ is a unital Jordan $R$-algebra with grade involution $\alpha$ (an automorphism of $J$ with $\alpha^2 = \mathrm{id}$), $M$ is a graded module over $\operatorname{Mult}(J)$ with grade involution $\alpha_M$, and no form, norm or distance occurs.

Modules over a Jordan Algebra

Definition Through the Multiplication Algebra

Definition. The multiplication algebra of $J$ is the unital associative $R$-algebra $\operatorname{Mult}(J)$ generated inside $\operatorname{End}_R(J)$ by the operators $L_a$, $a \in J$, of The Jordan Multiplication Operators. A module over $J$ is a left $\operatorname{Mult}(J)$-module, that is, an $R$-module $M$ with a unital algebra homomorphism

$$ \rho : \operatorname{Mult}(J) \longrightarrow \operatorname{End}_R(M) . $$

The action of $J$ on $M$ is the composite $J\to\operatorname{Mult}(J)\to\operatorname{End}_R(M)$ sending $a$ to $\rho_{L_a}$, written $\rho_a = \rho(L_a)$.

Proposition. A module over $J$ is the same thing as an $R$-module $M$ with an $R$-linear map $\rho : J \to \operatorname{End}_R(M)$ such that

$$ \rho_{a\bullet b} = \tfrac12\bigl(\rho_a\rho_b + \rho_b\rho_a\bigr) \qquad \text{and} \qquad [\rho_a, \rho_{a^2}] = 0 , $$

the second being the operator form of the Jordan identity.

Proof. A Jordan homomorphism $J \to \operatorname{End}_R(M)^+$ extends uniquely to a homomorphism of the associative algebra generated by the images, which is $\operatorname{Mult}(J)$ mapped onto the image; conversely the restriction of a homomorphism $\operatorname{Mult}(J)\to\operatorname{End}_R(M)$ to the generators satisfies the two identities because they hold in $\operatorname{Mult}(J)$. The uniqueness uses that $\operatorname{Mult}(J)$ is generated by the $L_a$. $\square$

Example. The regular module is $M = J$ with the action $\rho_a = L_a$; a module is a generalisation of this, and the regular module is the one in which the representer is injective when $J$ is unital.

The Grading of the Multiplication Algebra

Definition. Let $J = J_{\bar 0}\oplus J_{\bar 1}$ be the grading by $\alpha$. The multiplication algebra is graded by

$$ \operatorname{Mult}(J)_k = \operatorname{span}_R\bigl\{L_{a_1}\cdots L_{a_r} : a_i \text{ homogeneous}, \ \textstyle\sum_i |a_i| = k \bmod 2\bigr\}, $$

so that $\operatorname{Mult}(J) = \operatorname{Mult}(J)_{\bar 0}\oplus\operatorname{Mult}(J)_{\bar 1}$, and a product of $r$ homogeneous multiplications by elements of total parity $k$ lies in the part $k$.

Proposition. The grading is compatible with the product, $\operatorname{Mult}(J)_i\operatorname{Mult}(J)_j\subseteq\operatorname{Mult}(J)_{i+j}$; the even part $J_{\bar 0}$ acts on $J$ preserving the degree, and a multiplication by an odd element exchanges the two graded parts of $J$.

Proof. The first statement is the multiplicativity of the parity of the sum of degrees. For the second, $L_a(J_{\bar i})\subseteq J_{\overline{i+|a|}}$ by the multiplicativity of the grading of $J$, which is the same statement as for $\ell^{\alpha}$ in The Signed Left Multiplication on a Jordan Algebra. $\square$

The Graded Module and the Graded Action

Graded Modules

Definition. A graded module over the graded algebra $(\operatorname{Mult}(J),\alpha)$ is a module $M$ with a decomposition

$$ M = M_{\bar 0}\oplus M_{\bar 1}, \qquad \operatorname{Mult}(J)_i\,M_j \subseteq M_{i+j} , $$

together with a grade involution of the module, an additive involution $\alpha_M$ of $M$ with $\alpha_M^2 = \mathrm{id}$ and $\alpha_M|_{M_{\bar j}} = (-1)^j$, compatible with the algebra in the sense

$$ \alpha_M\rho_{L}\alpha_M^{-1} = \rho_{\alpha(L)} \ \text{ for } L \in \operatorname{Mult}(J), \qquad \alpha_M(a\cdot m) = \alpha(a)\cdot\alpha_M(m) . $$

Theorem (compatibility of the action with the grading). For homogeneous $a \in J$ and homogeneous $m \in M$,

$$ \rho_a(M_j) \subseteq M_{\overline{j+|a|}} , \qquad \rho_a(m) = a \cdot m \in M_{\overline{j+|a|}} . $$

Hence the representing map $J \to \operatorname{End}_R(M)$ is graded: it maps $J_i$ into the operators that shift the module degree by $i$.

Proof. $L_a$ for $a \in J_i$ lies in $\operatorname{Mult}(J)_i$, and $\operatorname{Mult}(J)_iM_j\subseteq M_{i+j}$. $\square$

The Graded Action

Definition. The graded action (or signed action) of $J$ on the graded module $M$ is

$$ \rho^{\alpha}_a = \rho_a\circ\alpha_M : M \to M, \qquad \rho^{\alpha}_a(m) = a \cdot \alpha_M(m) . $$

It is $R$-linear, additive in $a$, and it inserts the grade involution of the module between the action and the argument.

Proposition (the sign rule on elements). For $a \in J_i$ and $m \in M_j$,

$$ \rho^{\alpha}_a(m) = (-1)^j\,\rho_a(m) = (-1)^j\,a\cdot m . $$

Thus the graded action differs from the unsigned action by the parity sign $(-1)^j$ of the module element, and on the even part $M_{\bar 0}$ the two actions coincide while on the odd part $M_{\bar 1}$ the graded action is the negative of the unsigned one.

Proof. $\alpha_M(m) = (-1)^j m$ for $m \in M_j$, and $\rho_a$ is additive. $\square$

The Sign Rule in the Operators

Intertwining of the Grade Involutions

Theorem (the sign rule). For every $L \in \operatorname{Mult}(J)$,

$$ \alpha_M\,\rho_L\,\alpha_M^{-1} = \rho_{\alpha(L)} , $$

where $\alpha(L)$ is the automorphism of the multiplication algebra induced by $\alpha$ on the generators, $\alpha(L_a) = L_{\alpha(a)}$; consequently, for homogeneous $a, b$,

$$ \rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}} , $$

the twisted product law: the graded actions compose by the multiplication algebra operator with one factor twisted by $\alpha$.

Proof. The first identity is the compatibility assumed in the definition of the graded module, applied to the generator $L_a$ and extended multiplicatively. For the second, $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_a\alpha_M\rho_b\alpha_M = \rho_a(\alpha_M\rho_b\alpha_M^{-1})\alpha_M^2 = \rho_a\rho_{\alpha(b)} = \rho_{L_aL_{\alpha(b)}}$. $\square$

Corollary. The graded action is not a representation of the Jordan algebra: whereas the unsigned actions satisfy $\rho_a\rho_b = \rho_{L_aL_b}$, the graded actions satisfy $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$, and $L_aL_{\alpha(b)}$ is generally neither $L_{a\bullet\alpha(b)}$ nor of the form $L_c$. The set $\{\rho^{\alpha}_a : a \in J\}$ is closed under composition in the multiplication algebra, with the twisted law, but it is a representation of $J$ only when $\alpha = \mathrm{id}$ or when the twist is absorbed, which is the analogue for the action of the phenomenon recorded for the signed sandwich.

Example. The regular graded module is $M = J$ with $\alpha_M = \alpha$ and $\rho_a = L_a$. The graded action is $\rho^{\alpha}_a = L_a\alpha = \ell^{\alpha}_a$ of The Signed Left Multiplication on a Jordan Algebra, and the twisted law reads $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}} = L_aL_{\alpha(b)}$, the composite of the unsigned multiplication by $a$ and the unsigned multiplication by $\alpha(b)$. The sign rule is the parity shift of that article, recovered here as the twisted product law of the graded action.

Compatibility and Consequences

The Graded Representing Map

Proposition. The map $a\mapsto\rho_a$ is a graded homomorphism of the module structure: it carries $J_i$ into $\operatorname{End}_R(M)_{\bar i}$, the operators shifting the degree by $i$; the graded action $a\mapsto\rho^{\alpha}_a$ is graded in the same sense, and it is the composition of the graded homomorphism with the module grade involution.

Proof. The first statement is the compatibility theorem; the second is the definition $\rho^{\alpha}_a = \rho_a\alpha_M$, with $\alpha_M$ of degree zero. $\square$

The Adjoint Action

The action of $J$ on $M$ has an adjoint with respect to the pairing of the category, in which the operators on $M$ are transported by the involutions of the algebra and of the module. The compatibility of that adjoint action with the grading, and the sign rule it obeys, is the subject of The Graded Adjoint Action on a Module over a Jordan Algebra, in the * Operator Theory group; the present article stops at the level of the action and the twist, and no pairing, form or adjoint is used here.

Summary

A module over a Jordan algebra $J$ is a module over the multiplication algebra $\operatorname{Mult}(J)$, equivalently an $R$-module $M$ with $\rho:J\to\operatorname{End}_R(M)$ satisfying $\rho_{a\bullet b}=\tfrac12(\rho_a\rho_b+\rho_b\rho_a)$ and $[\rho_a,\rho_{a^2}]=0$. When $J$ is graded by its grade involution $\alpha$ and $M$ by a compatible $\alpha_M$, the multiplication algebra is graded by the total parity of the multiplications, a homogeneous action by $a$ shifts the module degree by $|a|$, and the representing map is graded. The graded action $\rho^{\alpha}_a=\rho_a\alpha_M$ inserts the module grade involution; on the odd part it equals the negative of the unsigned action, and it composes by the twisted product $\rho^{\alpha}_a\rho^{\alpha}_b=\rho_{L_aL_{\alpha(b)}}$. The twist is the sign rule the grading imposes, and it makes the graded action a representation of the twisted multiplication algebra rather than of the Jordan algebra. No form, norm or distance occurs; the adjoint action is deferred to the * Operator Theory group.

Summary of Notation

Symbol	Meaning
$J$	Unital Jordan $R$-algebra with grade involution $\alpha$
$J_{\bar 0}, J_{\bar 1}$	Even and odd parts of $J$
$\operatorname{Mult}(J)$	Multiplication algebra generated by the $L_a$
$\operatorname{Mult}(J)_k$	Graded part of parity $k$
$M = M_{\bar 0}\oplus M_{\bar 1}$	Graded module over $\operatorname{Mult}(J)$
$\alpha_M$	Grade involution of the module
$\rho_a = \rho(L_a)$	Action of $a \in J$ on $M$
$\rho_a(M_j)\subseteq M_{\overline{j+\|a\|}}$	Grading compatibility
$\rho^{\alpha}_a = \rho_a\alpha_M$	Graded (signed) action
$\rho^{\alpha}_a(m) = (-1)^j\rho_a(m)$, $m\in M_j$	Sign rule on elements
$\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$	Twisted product law
$\alpha_M\rho_L\alpha_M^{-1} = \rho_{\alpha(L)}$	Intertwining of the grade involutions