The Graded Adjoint Action on a Module over a Jordan Algebra

Introduction

A module over a Jordan algebra $J$ is a module over the multiplication algebra $\operatorname{Mult}(J)$ generated by the one-sided multiplications $L_a$, and when $J$ carries a grade involution $\alpha$ and the module carries a compatible grade involution $\alpha_M$, the action acquires the graded layer $\rho^{\alpha}_a = \rho_{L_a}\circ\alpha_M$ of The Graded Action on a Module over a Jordan Algebra. The present article computes the adjoint of that action with respect to the pairing of the module, and shows that the adjoint action is again a graded action, at the image of the parameter under the grade involution: this is the module-level form of the signed adjoint computations of the group.

The pairing of the module makes the endomorphism algebra $E = \operatorname{End}_{\operatorname{Mult}(J)}(M)$ an involutive algebra with the adjoint ${}^{*}$; the grade involution of the module is an isometry of the pairing, $\alpha_M^{*} = \alpha_M$, and the multiplication operators are self-adjoint, $\rho_{L_a}^{*} = \rho_{L_a}$. The graded action is the composite of the two,

$$ \rho^{\alpha}_a = \rho_{L_a}\circ\alpha_M , \qquad \bigl(\rho^{\alpha}_a\bigr)^{*} = \alpha_M^{*}\circ\rho_{L_a}^{*} = \alpha_M\circ\rho_{L_a} = \rho_{L_{\alpha(a)}}\circ\alpha_M = \rho^{\alpha}_{\alpha(a)} , $$

so the adjoint of the graded action is the graded action at the image of the parameter, and the even elements act by self-adjoint operators while the odd elements act by skew-adjoint operators, which is the sign rule the grading imposes on the adjoint layer. The product of two graded actions composes by the twisted product $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$, and the adjoint of the product is the product of the adjoints in the reverse order, $\rho^{\alpha}_{\alpha(b)}\rho^{\alpha}_{\alpha(a)}$.

The article assumes The Graded Action on a Module over a Jordan Algebra for the module, the grading, the graded action and the sign rule; The Jordan Multiplication Operators for the multiplication algebra $\operatorname{Mult}(J)$; The Adjoint of the Left Multiplication on a Jordan Algebra for the trace form and the self-adjointness of the multiplications; The Signed Adjoint of the Left Multiplication on a Jordan Algebra for the algebra case $M = J$; Module Operators with an Involution for the adjoint involution of the endomorphism ring of a module; and Involutive Linear Algebras for the involution of an endomorphism algebra. Throughout, $R$ is a commutative ring with identity in which $2$ is invertible, $J = A^+$ is a special unital Jordan algebra with grade involution $\alpha$, $M$ is a graded module over $\operatorname{Mult}(J)$ with a compatible grade involution $\alpha_M$ and a non-degenerate reflexive pairing, $E = \operatorname{End}_{\operatorname{Mult}(J)}(M)$ carries the adjoint involution ${}^{*}$, and $\rho^{\alpha}_a = \rho_{L_a}\circ\alpha_M$ is the graded action; no norm, form, distance or positivity occurs.

The Graded Action and Its Adjoint

The Action and the Module Pairing

Definition. The module $M$ over $J$ is a module over $\operatorname{Mult}(J)$; the graded action is

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

the unsigned action of $L_a$ precomposed with the grade involution of the module.

Definition. The adjoint of an operator $F \in E$ with respect to the pairing of $M$ is the unique $F^{*}$ with $\langle Fm,n\rangle = \langle m,F^{*}n\rangle$.

Theorem (the building blocks are self-adjoint). The grade involution of the module is an isometry of the pairing, $\alpha_M^{*} = \alpha_M$, and the multiplication operators are self-adjoint, $\rho_{L_a}^{*} = \rho_{L_a}$, when the pairing is invariant under the action in the twisted sense.

Proof. The first is the invariance $\langle\alpha_Mm,\alpha_Mn\rangle = \langle m,n\rangle$ read as the defining relation of the adjoint; the second is the associativity of the invariant pairing with the action, $T(L_ax,y) = T(x,L_ay)$, transported to the module. $\square$

The Adjoint of the Graded Action

Theorem. The adjoint of the graded action is the graded action at the image of the parameter:

$$ \bigl(\rho^{\alpha}_a\bigr)^{*} = \rho^{\alpha}_{\alpha(a)} . $$

Proof. The adjoint is anti-multiplicative, $(\Phi\circ\Psi)^{*} = \Psi^{*}\circ\Phi^{*}$, so $(\rho^{\alpha}_a)^{*} = (\rho_{L_a}\circ\alpha_M)^{*} = \alpha_M^{*}\circ\rho_{L_a}^{*} = \alpha_M\circ\rho_{L_a}$. Since the grade involution intertwines the multiplication operators by $\alpha_M\rho_{L_a}\alpha_M^{-1} = \rho_{L_{\alpha(a)}}$, one has $\alpha_M\circ\rho_{L_a} = \rho_{L_{\alpha(a)}}\circ\alpha_M = \rho^{\alpha}_{\alpha(a)}$. $\square$

Corollary (the sign rule). The even elements of $J$ act by self-adjoint operators and the odd elements by skew-adjoint operators:

$$ a \in J^+ \Longrightarrow \bigl(\rho^{\alpha}_a\bigr)^{*} = \rho^{\alpha}_a, \qquad a \in J^- \Longrightarrow \bigl(\rho^{\alpha}_a\bigr)^{*} = -\rho^{\alpha}_a , $$

which is the sign rule of the grading on the adjoint layer, the module-level form of $\alpha^{*} = \alpha$ and $L_a^{*} = L_a$.

The Product and the Grading

Theorem. The product of two graded actions is the twisted product $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$, and its adjoint is

$$ \bigl(\rho^{\alpha}_a\rho^{\alpha}_b\bigr)^{*} = \rho^{\alpha}_{\alpha(b)}\rho^{\alpha}_{\alpha(a)} . $$

Proof. $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_a}\alpha_M\rho_{L_b}\alpha_M = \rho_{L_a}\rho_{L_{\alpha(b)}}\alpha_M^2 = \rho_{L_aL_{\alpha(b)}}$, using the intertwining $\alpha_M\rho_{L_b} = \rho_{L_{\alpha(b)}}\alpha_M$; the adjoint is anti-multiplicative and each factor's adjoint is known. $\square$

Corollary (compatibility with the grading). A homogeneous action of degree $|a|$ shifts the module degree by $|a|$, and the adjoint action shifts it back by $|a|$; the representing map $J\to E$ is a graded map and the adjoint is a graded map of the opposite degree, so the adjoint operation reverses the parity of the action.

Corollary (the algebra case). When $M = J$ with the regular action and the trace form, the graded action is the signed left multiplication $\rho^{\alpha}_a = \ell^{\alpha}_a$ of The Signed Adjoint of the Left Multiplication on a Jordan Algebra, and the module-level formula $(\rho^{\alpha}_a)^{*} = \rho^{\alpha}_{\alpha(a)}$ specialises to $(\ell^{\alpha}_a)^{\dagger} = \ell^{\alpha}_{\alpha(a)}$.

Examples

Example (the regular module). For $M = J$ with the regular action the pairing is the trace form, the graded action is the signed left multiplication, and the adjoint action is the signed left multiplication at the image of the parameter; the even elements act self-adjointly and the odd elements skew-adjointly.

Example (the identity grade involutions). For $\alpha = \mathrm{id}$ and $\alpha_M = \mathrm{id}$ the graded action is the unsigned action $\rho_{L_a}$, which is self-adjoint for every $a$; the adjoint action is the same action, and the sign rule is vacuous.

Example (the transpose involution on a matrix module). Let $J = H_n(F)$ and let $M$ be a module of matrices with the transpose grade involution; the graded action $\rho^{\alpha}_a(m) = a\bullet m^{\mathsf{T}}$ has the adjoint $\rho^{\alpha}_{a^{\mathsf{T}}}$, and it is self-adjoint exactly when $a$ is symmetric.

Summary

A module over a graded Jordan algebra carries the graded action $\rho^{\alpha}_a = \rho_{L_a}\circ\alpha_M$ of the multiplication operator precomposed with the grade involution of the module. The pairing of the module makes the endomorphism algebra an involutive algebra; the grade involution of the module is an isometry and the multiplication operators are self-adjoint, and hence the adjoint of the graded action is the graded action at the image of the parameter,

$$ \bigl(\rho^{\alpha}_a\bigr)^{*} = \rho^{\alpha}_{\alpha(a)} . $$

The even elements act by self-adjoint operators and the odd elements by skew-adjoint operators, which is the sign rule of the grading on the adjoint layer; the product of two graded actions is the twisted product $\rho^{\alpha}_a\rho^{\alpha}_b = \rho_{L_aL_{\alpha(b)}}$, whose adjoint is $\rho^{\alpha}_{\alpha(b)}\rho^{\alpha}_{\alpha(a)}$, and the adjoint operation reverses the parity of the action. The regular module recovers the signed left multiplication. No norm, form, distance or positivity occurs.

Summary of Notation

Further Reading

• Nathan Jacobson, Structure and Representations of Jordan Algebras (American Mathematical Society, 1968), for the modules over a Jordan algebra, the multiplication algebra and the involutions.
• Kevin McCrimmon, A Taste of Jordan Algebras (Springer, 2004), for the multiplication algebra, the modules and the graded structures.
• Max-Albert Knus, Alexander Merkurjev, Markus Rost and Jean-Pierre Tignol, The Book of Involutions (American Mathematical Society Colloquium Publications 44, 1998), for the graded modules, the adjoint actions and the descent.
• Richard D. Schafer, An Introduction to Nonassociative Algebras (Academic Press, 1966), for the multiplication algebras, the modules and the operators.
• Nicolas Bourbaki, Algebra II (Springer, 2003), for the modules, the graded algebras and the pairings.