The Graded Adjoint Action on a Module over an Ordered Algebra

Introduction

Let $M = M_{\bar 0}\oplus M_{\bar 1}$ be a graded module over a graded ordered involutive algebra with grade involution $\alpha$, with the module grading operator $\alpha_M$, the $\pm1$ operator on the even and the odd parts. The action satisfies the sign rule $\alpha_M\rho(a) = \rho(\alpha(a))\alpha_M$, and the signed action is

$$ \rho^{\alpha}(a) = \rho(a)\,\alpha_M , $$

the action composed with the module grading, the module analogue of the signed left multiplication $L^{\alpha}_a = L_a\Gamma$. When the module carries a positive definite form for which the module grading is self-adjoint and the action is a *-representation, $\alpha_M^{*} = \alpha_M$ and $\rho(a)^{*} = \rho(a^{*})$, the article computes the adjoint action:

$$ \bigl(\rho^{\alpha}(a)\bigr)^{*} = \rho^{\alpha}(\alpha(a^{*})) , $$

so the adjoint of the signed action is the signed action with the $\alpha$-twisted adjoint parameter, the module version of $(L^{\alpha}_a)^{*} = L^{\alpha}_{\alpha(a^{*})}$. The computation is: $(\rho(a)\alpha_M)^{*} = \alpha_M^{*}\rho(a)^{*} = \alpha_M\rho(a^{*}) = \rho(\alpha(a^{*}))\alpha_M = \rho^{\alpha}(\alpha(a^{*}))$, the third equality being the sign rule. The regular module $M = A$ with $\alpha_M = \alpha$ recovers the signed left multiplication of the previous article.

The article states the adjoint formula, identifies the self-adjoint and skew-adjoint signed actions (the parameters with $a = \alpha(a^{*})$ and $a = -\alpha(a^{*})$), shows the compatibility with the twisted composition law $\rho^{\alpha}(a)\rho^{\alpha}(b) = \rho(a\alpha(b))$, and describes the interaction with the order: if the module grading is positive and the action of $a$ is positive with $a\geq0$, then the signed action is positive and so is its adjoint, and the order detects the parity because the signed action agrees with the action on the even part and is its negative on the odd part, $\rho^{\alpha}(a)m = \varepsilon_m\rho(a)m$. The Clifford module is the instance in which the whole structure is generated by the odd part.

The graded module and the sign rule are The Graded Action on a Module over an Ordered Algebra; the signed left multiplication, which is the regular case, is The Signed Left Multiplication on an Ordered Algebra and its adjoint The Signed Adjoint of the Left Multiplication on an Ordered Algebra; the signed sandwich and the reflection are The Signed Sandwich on an Ordered Algebra and Reflections as Signed Two-Sided Operators on an Ordered Algebra; the adjoint and the positivity are The Adjoint of a Positive Operator; the order of the elements is Self-Adjoint Elements and the Order; the two-sided multiplications are The Adjoint of the Left Multiplication on an Ordered Algebra; the order is Ordered Vector Spaces and the Order Unit and Positive Operators on an Ordered Space; the modules and the representations are Modules over a Ring and The Operators on an Algebra of Part I; the grading is Superalgebras and Graded Structures of Part I; the Clifford modules are Clifford Modules of Part I; and the algebra is Ordered Involutive Algebras.

The Graded Adjoint Action

Definition. Let $M$ be a graded module over the graded algebra $A$ with the module grading operator $\alpha_M$ ($\alpha_M = +1$ on $M_{\bar 0}$, $\alpha_M = -1$ on $M_{\bar 1}$), carrying a positive definite form with $\alpha_M^{*} = \alpha_M$ for which the action is a *-representation of the involutive algebra, $\rho(a)^{*} = \rho(a^{*})$; the signed action is

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

Theorem (the adjoint of the signed action). The adjoint of the signed action is the signed action with the $\alpha$-twisted adjoint parameter,

$$ \bigl(\rho^{\alpha}(a)\bigr)^{*} = \rho^{\alpha}(\alpha(a^{*})) , $$

so the family of the signed actions is closed under the adjoint, and the adjoint action is the module version of the one-sided formula $(L^{\alpha}_a)^{*} = L^{\alpha}_{\alpha(a^{*})}$.

Proof. Compute $$ \bigl(\rho^{\alpha}(a)\bigr)^{*} = \bigl(\rho(a)\alpha_M\bigr)^{*} = \alpha_M^{*}\rho(a)^{*} = \alpha_M\,\rho(a^{*}) = \rho(\alpha(a^{*}))\,\alpha_M = \rho^{\alpha}(\alpha(a^{*})) , $$ using the self-adjointness of the module grading, the *-representation property and the sign rule $\alpha_M\rho(b) = \rho(\alpha(b))\alpha_M$ with $b = a^{*}$.

Corollary (the regular module). For $M = A$ with $\alpha_M = \alpha$ the signed action is the signed left multiplication $L^{\alpha}_a$, and the adjoint formula specialises to $(L^{\alpha}_a)^{*} = L^{\alpha}_{\alpha(a^{*})}$ of The Signed Adjoint of the Left Multiplication on an Ordered Algebra; the module formula therefore generalises the one-sided one.

Proof. The identification of the signed action with the signed left multiplication on the regular module is the proposition of The Graded Action on a Module over an Ordered Algebra; the adjoint formula is the theorem, which then reads as the one-sided formula.

Corollary (the twisted composition and its adjoint). The signed actions compose by $\rho^{\alpha}(a)\rho^{\alpha}(b) = \rho(a\alpha(b))$, and the adjoint of a product is the product of the adjoints in the reverse order,

$$ \bigl(\rho^{\alpha}(a)\rho^{\alpha}(b)\bigr)^{*} = \bigl(\rho^{\alpha}(b)\bigr)^{*}\bigl(\rho^{\alpha}(a)\bigr)^{*} = \rho^{\alpha}(\alpha(b^{*}))\rho^{\alpha}(\alpha(a^{*})) = \rho(\alpha(b^{*})a^{*}) , $$

which agrees with the direct computation $\bigl(\rho(a\alpha(b))\bigr)^{*} = \rho(\alpha(b^{*})a^{*})$, so the adjoint is compatible with the twisted composition law.

Proof. The composition law and the adjoint property are the same computations as in The Signed Adjoint of the Left Multiplication on an Ordered Algebra, transported to the module by the functoriality of $\rho$.

Self-Adjointness, Skew-Adjointness and the Order

Proposition (the self-adjoint signed actions). The signed action is self-adjoint,

$$ \bigl(\rho^{\alpha}(a)\bigr)^{*} = \rho^{\alpha}(a) , $$

exactly when the parameter is $\alpha$-Hermitian, $a = \alpha(a^{*})$; it is skew-adjoint, $\bigl(\rho^{\alpha}(a)\bigr)^{*} = -\rho^{\alpha}(a)$, exactly when $a = -\alpha(a^{*})$; and every signed action decomposes into the self-adjoint and the skew-adjoint parts, being normal exactly when these commute. For a faithful action, the self-adjoint signed action is an involution exactly on the reflection elements, by $(\rho^{\alpha}(a))^{2} = \rho(a\alpha(a))$.

Proof. The self-adjointness criterion is the theorem read as the equation $\rho^{\alpha}(a) = \rho^{\alpha}(\alpha(a^{*}))$, which for a faithful action is the equality of the parameters; the skew case is the minus-sign equation; the decomposition and the normality are standard; the involution statement is the square formula of the graded action.

Theorem (the order of the signed actions). If the module grading is positive, $\alpha_M(M_+)\subseteq M_+$, and the action of $a$ is positive with $a\geq0$, then the signed action and its adjoint are positive,

$$ \rho^{\alpha}(a)\geq0 \quad \text{and} \quad \bigl(\rho^{\alpha}(a)\bigr)^{*}\geq0 , $$

so the adjoint is an order isomorphism of the family of the signed actions; on the homogeneous elements the signed action acts by the parity sign,

$$ \rho^{\alpha}(a)m = \varepsilon_m\,\rho(a)m \qquad (m\ \text{homogeneous}) , $$

agrees with the action on the even part and is its negative on the odd part, so the order is the invariant that detects the sign of the module grading.

Proof. The positivity of $\rho^{\alpha}(a)$ is the positivity proposition of the graded action; the positivity of the adjoint is the general positivity preservation of the adjoint of The Adjoint of a Positive Operator; the parity formula is the definition of $\alpha_M$ in the graded case; the order-isomorphism statement is the two-sided preservation.

Corollary (the order interval and the fixed elements). The signed action of a reflection element is an involution whose fixed set is the sum of the fixed space of $\rho(a)$ on the even part and the $(-1)$-eigenspace on the odd part,

$$ \operatorname{Fix}\rho^{\alpha}(a) = \ker(\rho(a) - I)\cap M_{\bar 0}\ \oplus\ \ker(\rho(a) + I)\cap M_{\bar 1} , $$

and the adjoint of the signed action preserves this fixed set because it preserves the parity and the order.

Proof. The fixed-set description is the proposition of The Graded Action on a Module over an Ordered Algebra; the adjoint statement is the order-isomorphism and parity-preservation property.

Worked Cases

The Regular Module

For $M = A$ with the module grading the grade involution, the signed action is the signed left multiplication, and the adjoint action is $(L^{\alpha}_a)^{*} = L^{\alpha}_{\alpha(a^{*})}$; the self-adjoint parameters are the $\alpha$-Hermitian ones, and the positivity is the positivity of the grade involution and of the parameter. The regular module is the base point of the article and reduces it to The Signed Adjoint of the Left Multiplication on an Ordered Algebra.

The Natural Module of the Biquaternions

Let $A = \mathbb{B} = M_2(\mathbb{C})$ with the $\operatorname{diag}(1,-1)$ grading and $M = \mathbb{C}^{2}$ with the grading in which the first coordinate is even and the second odd, so that $\alpha_M = \operatorname{diag}(1,-1)$ is self-adjoint for the standard Hermitian form. The signed action of the odd element $a$ with $a^{2} = -1$ is an involution, and its adjoint is the signed action of $\alpha(a^{*}) = -a^{*}$, which is again odd; the fixed set is the diagonal on the even coordinate and the off-diagonal on the odd one. This is the smallest non-regular module in which the adjoint action is computed.

The Clifford Module

Let $A = \mathrm{Cl}(V,q)$ with the parity grading and $M$ a Clifford module with the induced grading. The vectors act by odd operators, the sign rule is the statement that the Clifford relations involve only the even part of the contraction, and the signed action of a vector $e$ with $e^{2} = -1$ is an involution; its adjoint is the signed action of $\alpha(e^{*}) = -e^{*}$, the twist of the adjoint vector, and the whole computation reduces to the parity of $e$. The Clifford module is the instance in which the graded action and its adjoint are the entire content of the structure.

Summary

The graded module of a graded ordered involutive algebra carries the module grading operator $\alpha_M$ and the signed action

$$ \rho^{\alpha}(a) = \rho(a)\,\alpha_M , $$

the module version of the signed left multiplication; when the module form makes $\alpha_M$ self-adjoint and the action a *-representation, the adjoint action is

$$ \bigl(\rho^{\alpha}(a)\bigr)^{*} = \rho^{\alpha}(\alpha(a^{*})) , $$

so the family of the signed actions is closed under the adjoint, the regular module recovers the one-sided formula, and the adjoint is compatible with the twisted composition law $\rho^{\alpha}(a)\rho^{\alpha}(b) = \rho(a\alpha(b))$. The signed action is self-adjoint exactly for the $\alpha$-Hermitian parameters, $a = \alpha(a^{*})$, skew-adjoint for the skew ones, and it decomposes into the self-adjoint and the skew-adjoint parts; for a faithful action it is an involution exactly on the reflection elements, with the fixed set the fixed space of the action on the even part plus the $(-1)$-eigenspace on the odd part. If the module grading is positive and the action of a positive parameter is positive, then the signed action and its adjoint are positive and the adjoint is an order isomorphism of the family; the order detects the parity, since $\rho^{\alpha}(a)m = \varepsilon_m\rho(a)m$. The graded module is The Graded Action on a Module over an Ordered Algebra; the signed left multiplication and its adjoint are The Signed Left Multiplication on an Ordered Algebra and The Signed Adjoint of the Left Multiplication on an Ordered Algebra; the sandwich and the reflection are The Signed Sandwich on an Ordered Algebra and Reflections as Signed Two-Sided Operators on an Ordered Algebra; the adjoint and the positivity are The Adjoint of a Positive Operator; the order is Self-Adjoint Elements and the Order and Ordered Vector Spaces and the Order Unit; the modules and the grading are Modules over a Ring, The Operators on an Algebra and Superalgebras and Graded Structures; and the Clifford instance is Clifford Modules.

Summary of Notation

Further Reading

• Richard Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, vol. 1 (Academic Press, 1983), for the *-representations, the adjoints and the positivity.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 2001), for the graded modules, the sign rule and the Clifford actions.
• F. Reese Harvey, Spinors and Calibrations (Academic Press, 1990), for the graded modules, the parity and the Clifford operators.
• Gert K. Pedersen, C*-Algebras and their Automorphism Groups (Academic Press, 1979), for the representations, the automorphisms and the order.
• Erik M. Alfsen and Frederik W. Shultz, State Spaces of Operator Algebras (Birkhäuser, 2001), for the order, the positivity and the operator adjoints.