The Graded Adjoint Action on a Module over a Ring

Introduction

When a ring $A$ carries a $\mathbb{Z}/2$-grading and a module $M$ over it is graded as well, the operators that respect the grading acquire a sign: a homogeneous operator $T$ of degree $\lvert T\rvert$ satisfies $T(xa) = (-1)^{\lvert T\rvert\lvert a\rvert}T(x)a$ and $T(ax) = (-1)^{\lvert T\rvert\lvert a\rvert}aT(x)$ according to the side, and the adjoint of a product carries the Koszul sign,

$$ (ST)^{*} = (-1)^{\lvert S\rvert\lvert T\rvert}\,T^{*}S^{*} . $$

The adjoint action is the inner operator $\operatorname{ad}_x(y) = xy-(-1)^{\lvert x\rvert\lvert y\rvert}yx$ of the graded commutator; it is a graded derivation of degree $\lvert x\rvert$, it is compatible with the grading in the sense that it shifts the degree by $\lvert x\rvert$, and it is compatible with an involution $\sigma$ of degree zero exactly when $x$ lies in the appropriate eigenspace of $\sigma$. This article fixes the graded pairing on a graded module, computes the adjoint of a homogeneous operator, states the Koszul sign rule, and reads the adjoint action as a graded derivation with its sign.

It assumes The Graded Action on a Module over a Ring for the graded action, Involutions of the Endomorphism Ring for the adjoint, Star-Derivations and the Skew Derivations for the inner derivations; the superalgebra of the sign rule is Superalgebras and Graded Structures and is named rather than used. Throughout, $A$ is a ring with a $\mathbb{Z}/2$-grading, $M$ is a graded module over $A$, the pairing $\langle-,-\rangle$ is biadditive and homogeneous of degree zero, homogeneous elements are written with their parity $\lvert a\rvert\in\mathbb{Z}/2$, and the sign $(-1)^{\lvert a\rvert\lvert b\rvert}$ is the Koszul sign.

The Graded Pairing and the Adjoint

Definition. The pairing $\langle x,y\rangle$ on the graded module is graded when it is biadditive and homogeneous of degree zero, $\lvert\langle x,y\rangle\rvert = \lvert x\rvert+\lvert y\rvert$, and supersymmetric when

$$ \langle y,x\rangle = (-1)^{\lvert x\rvert\lvert y\rvert}\langle x,y\rangle . $$

The grade involution of the ring acts on homogeneous elements by $a\mapsto (-1)^{\lvert a\rvert}a$; it is the operator $\alpha$ of the signed articles of this group.

Theorem. For a homogeneous operator $T$ the adjoint $T^{*}$ defined by $\langle Tx,y\rangle = \langle x,T^{*}y\rangle$ is homogeneous of degree $\lvert T\rvert$, and for homogeneous $S, T$

$$ (ST)^{*} = (-1)^{\lvert S\rvert\lvert T\rvert}T^{*}S^{*}, \qquad (T^{*})^{*} = T, \qquad \mathrm{id}^{*} = \mathrm{id} . $$

The adjoint is a graded involution: it is anti-multiplicative up to the Koszul sign.

Proof. The existence and uniqueness of $T^{*}$ are those of Involutions of the Endomorphism Ring; the degree is read off from $\lvert\langle Tx,y\rangle\rvert = \lvert T\rvert+\lvert x\rvert+\lvert y\rvert = \lvert x\rvert+\lvert T^{*}y\rvert$, so $\lvert T^{*}y\rvert = \lvert T\rvert+\lvert y\rvert$ and $\lvert T^{*}\rvert = \lvert T\rvert$. For the product, $(-1)^{\lvert S\rvert\lvert T\rvert}T^{*}S^{*}$ is the ordinary adjoint of $ST$ with the sign inserted to make the two factors homogeneous of the correct degree: passing $S$ past $T^{*}$ in the chain of the pairing introduces the sign $(-1)^{\lvert S\rvert\lvert T\rvert}$, which is the Koszul rule; the order-two and the unit statements are as in the ungraded case.

Corollary (compatibility with the grading). The adjoint of a homogeneous operator preserves the parity, and the map $T\mapsto T^{*}$ is compatible with the grading in the sense of the sign rule above; on the even operators it restricts to the adjoint involution of the ungraded article, and on the odd ones it is the twisted (signed) adjoint.

Proof. The degree statement is the theorem; the restriction is the observation that for $\lvert T\rvert = 0$ the Koszul sign is $1$ and the rule is the ungraded one, while for odd operators the sign survives.

The Adjoint Action

Definition. For $x \in A$ the adjoint action of $x$ is

$$ \operatorname{ad}_x(y) = xy-(-1)^{\lvert x\rvert\lvert y\rvert}yx , $$

the graded commutator; it is a linear map on $A$ and, for each $x$, an operator on the module by the action of $A$.

Theorem. For a homogeneous $x$ the operator $\operatorname{ad}_x$ is a graded derivation of degree $\lvert x\rvert$,

$$ \operatorname{ad}_x(yz) = \operatorname{ad}_x(y)z + (-1)^{\lvert x\rvert\lvert y\rvert}y\operatorname{ad}_x(z) , $$

it satisfies the graded Leibniz rule in the variable $x$ with the sign $(-1)^{\lvert y\rvert}$, and it is compatible with the grading by shifting the degree, $\lvert\operatorname{ad}_x(y)\rvert = \lvert x\rvert+\lvert y\rvert$.

Proof. Expand $\operatorname{ad}_x(yz) = xyz-(-1)^{\lvert x\rvert(\lvert y\rvert+\lvert z\rvert)}yzx$ and compare with $\operatorname{ad}_x(y)z + (-1)^{\lvert x\rvert\lvert y\rvert}y\operatorname{ad}_x(z) = xyz-(-1)^{\lvert x\rvert\lvert y\rvert}yxz+(-1)^{\lvert x\rvert\lvert y\rvert}yxz-(-1)^{\lvert x\rvert\lvert y\rvert}(-1)^{\lvert x\rvert\lvert z\rvert}yzx$; the middle terms cancel and the last term is $-(-1)^{\lvert x\rvert(\lvert y\rvert+\lvert z\rvert)}yzx$, matching. The degree is immediate from the homogeneity of the product.

Theorem (the graded Lie structure). The adjoint action satisfies the graded antisymmetry and the graded Jacobi identity,

$$ \operatorname{ad}_x(y) = -(-1)^{\lvert x\rvert\lvert y\rvert}\operatorname{ad}_y(x), \qquad \operatorname{ad}_x\operatorname{ad}_y - (-1)^{\lvert x\rvert\lvert y\rvert}\operatorname{ad}_y\operatorname{ad}_x = \operatorname{ad}_{\operatorname{ad}_x(y)} , $$

so the graded commutator turns $A$ into a graded Lie algebra and $\operatorname{ad}$ is a representation of it by graded derivations, with kernel the graded centre $Z_{\mathrm{gr}}(A)$.

Proof. The antisymmetry is the definition read twice; the Jacobi identity is the expansion of the graded commutator, the same computation as in the ungraded case with the Koszul signs inserted at each transposition; the kernel is the graded centre because $\operatorname{ad}_x = 0$ means $x$ graded-commutes with every element.

Proposition (compatibility with an involution). Let $\sigma$ be an involution of $A$ of degree zero commuting with the grade involution $\alpha$. Then

$$ \operatorname{ad}_x^{*_\sigma} = -\operatorname{ad}_{\delta(x)} , \qquad \delta = \sigma\alpha , $$

with respect to the twisted pairing: the adjoint of the adjoint action is the adjoint action of $-\delta(x)$, so the adjoint action is skew-adjoint exactly when $\delta(x) = -x$.

Proof. $\operatorname{ad}_x = L_x-(-1)^{\lvert x\rvert\lvert\cdot\rvert}R_x$ is the difference of a signed left and a signed right multiplication, and the signed adjoints of the two are $T_{\delta(x)}$ and the corresponding signed right multiplication by $\delta(x)$, by The Signed Adjoint of the Left Multiplication on a Ring; the difference is $-\operatorname{ad}_{\delta(x)}$.

Examples

(a) The exterior algebra. For the exterior algebra of a module with the sign $(-1)^{\lvert x\rvert\lvert y\rvert}$, the adjoint action of a vector $v$ is the contraction-insertion operator $\operatorname{ad}_v(\omega) = v\omega-(-1)^{\lvert\omega\rvert}\omega v$, a graded derivation of degree one; its adjoint is $-\operatorname{ad}_v$ for the natural pairing, so the odd part acts by skew-adjoint operators.

(b) The Clifford algebra. With the Clifford product and the grading by degree, the adjoint action of a vector is the commutator $[v,\omega] = v\omega-\omega v$ on the even part and the anti-commutator on the odd part, the sign rule of Hilbert Algebras; the adjoint is $-\operatorname{ad}_{\delta(v)}$, and for the dagger of that theory $\delta(v) = -v$, so the adjoint action is self-adjoint.

(c) The matrix superalgebra. $A = M_{p|q}$ with the transpose and the grading by blocks: the adjoint action is the graded commutator of matrices, and the Koszul sign appears in the product rule the moment two odd matrices are multiplied.

(d) The sign rule. In every case the essential point is the sign: the adjoint of a product is $(-1)^{\lvert S\rvert\lvert T\rvert}T^{*}S^{*}$, and the graded commutator is $xy-(-1)^{\lvert x\rvert\lvert y\rvert}yx$. Setting the grading trivial, $\lvert a\rvert = 0$ for all $a$, recovers the ungraded adjoint involution and the ordinary commutator of Involutions of the Endomorphism Ring and The Skew Field of a Ring with Involution.

Summary

On a graded module over a graded ring the pairing is graded and supersymmetric, $\langle y,x\rangle = (-1)^{\lvert x\rvert\lvert y\rvert}\langle x,y\rangle$, and the adjoint of a homogeneous operator is homogeneous of the same degree with the Koszul sign rule $(ST)^{*} = (-1)^{\lvert S\rvert\lvert T\rvert}T^{*}S^{*}$ for products. The adjoint action $\operatorname{ad}_x(y) = xy-(-1)^{\lvert x\rvert\lvert y\rvert}yx$ is a graded derivation of degree $\lvert x\rvert$, it satisfies the graded antisymmetry and the graded Jacobi identity, so the graded commutator makes the ring a graded Lie algebra and $\operatorname{ad}$ a representation of it with kernel the graded centre. With respect to the twisted pairing the adjoint of the adjoint action is $-\operatorname{ad}_{\delta(x)}$ for $\delta = \sigma\alpha$, so the adjoint action is skew-adjoint exactly when $\delta(x) = -x$; when the grading is trivial the whole article reduces to the ungraded adjoint involution and the ordinary commutator.

Summary of Notation

Symbol	Meaning
$\lvert a\rvert$	Parity of a homogeneous element
$(-1)^{\lvert a\rvert\lvert b\rvert}$	Koszul sign
$\langle y,x\rangle=(-1)^{\lvert x\rvert\lvert y\rvert}\langle x,y\rangle$	Supersymmetric graded pairing
$(ST)^{}=(-1)^{\lvert S\rvert\lvert T\rvert}T^{}S^{*}$	Koszul sign rule for the adjoint
$\alpha$	Grade involution, $a\mapsto(-1)^{\lvert a\rvert}a$
$\operatorname{ad}_x(y)=xy-(-1)^{\lvert x\rvert\lvert y\rvert}yx$	Adjoint action; graded commutator
$\lvert\operatorname{ad}_x(y)\rvert=\lvert x\rvert+\lvert y\rvert$	Degree shift
graded Jacobi	Graded Lie algebra; $\operatorname{ad}_{\operatorname{ad}_x(y)}$
$\operatorname{ad}_x^{*_\sigma}=-\operatorname{ad}_{\delta(x)}$	Adjoint of the adjoint action
$\lvert a\rvert = 0$	Trivial grading; recovers the ungraded case