The Adjoint of the Left Multiplication on a Ring

Introduction

On a ring $A$ with a trace $\tau$ the natural pairing of the category is $\langle x,y\rangle = \tau(xy)$, symmetric and nondegenerate under the hypothesis of Involutions of the Endomorphism Ring, and the adjoint of the left multiplication is the right multiplication: the identity $\tau(a x y) = \tau(x y a)$ says that $L_a$ transfers the pairing to $R_a$, and the pair $\{L_a\}$ and $\{R_a\}$ is the regular representation read through its own dual. The formula is the model for all the adjoints of the group, and it is compatible with an involution $\sigma$ of $A$ in a precise sense: the right multiplication is the left multiplication by the image under $\sigma$ composed with the conjugation by $\sigma$, $R_a = \sigma L_{\sigma(a)}\sigma$, and with respect to the $\sigma$-twisted pairing the adjoint of $L_a$ is $L_{\sigma(a)}$.

This article computes the adjoint of the left and the right multiplications with respect to the natural pairing, records the compatibility with the involution, identifies the self-adjoint and the unitary left multiplications, and states the relation with the signed adjoint of The Signed Adjoint of the Left Multiplication on a Ring. It assumes Left and Right Multiplication in a Ring, Involutive Rings, Involutions of the Endomorphism Ring for the natural pairing, and Derivations of a Ring for the inner derivations $L_a-R_a$. Throughout, $A$ is a ring with $1 \neq 0$ and an involution $\sigma$, $\tau$ is a trace on $A$ that is $\sigma$-invariant, the pairing is $\langle x,y\rangle = \tau(xy)$, and $L_a$, $R_a$ are the left and right multiplications.

The Natural Pairing and the Adjoint

Theorem. With respect to the pairing $\langle x,y\rangle = \tau(xy)$,

$$ \langle L_a x, y\rangle = \langle x, R_a y\rangle, \qquad \langle R_b x, y\rangle = \langle x, L_b y\rangle , $$

so $L_a^{*} = R_a$ and $R_b^{*} = L_b$: the adjoint of the left multiplication is the right multiplication by the same element. The adjoint involution of $\operatorname{End}(A)$ switches the two sides of the regular representation.

Proof. $\langle L_a x,y\rangle = \tau(axy) = \tau(xya) = \langle x,R_a y\rangle$ by the cyclicity of the trace, and the second identity is the same computation with the sides exchanged.

Proposition (the adjoint is a ring anti-isomorphism when the pairing is perfect). The assignment $T\mapsto T^{*}$ is the adjoint involution of the endomorphism ring of Involutions of the Endomorphism Ring; on the subring generated by the left and right multiplications it exchanges $L_a$ and $R_a$, hence the subring $\{L_a, R_a\}$ is stable under it and the map $a\mapsto L_a$ composed with the adjoint gives $R_a$.

Proof. The first statement is the theorem of Involutions of the Endomorphism Ring; the exchange is the computation above.

Compatibility with the Involution

Proposition. Let $\sigma$ be an involution of $A$ and $\tau$ a $\sigma$-invariant trace. Then the right multiplication is the conjugate of a left multiplication,

$$ R_a = \sigma\, L_{\sigma(a)}\,\sigma , $$

and the adjoint of $L_a$ is expressed on the left by

$$ L_a^{*} = R_a = \sigma\,L_{\sigma(a)}\,\sigma . $$

With respect to the $\sigma$-twisted pairing $\{x,y\} = \tau(x\sigma(y))$ the adjoint is instead $L_a^{*_\sigma} = L_{\sigma(a)}$ and $R_b^{*_\sigma} = R_{\sigma(b)}$.

Proof. $\sigma L_{\sigma(a)}\sigma(x) = \sigma(\sigma(a)\sigma(x)) = x a = R_a x$. For the twisted pairing, $\{L_a x,y\} = \tau(ax\sigma(y)) = \tau(x\sigma(y)a) = \tau(x\sigma(\sigma(a)y)) = \{x,L_{\sigma(a)}y\}$, using $\sigma$-invariance of $\tau$ and the anti-multiplicativity of $\sigma$; the right-multiplication case is the same computation.

Corollary (self-adjointness and the regular representation). The left multiplication $L_a$ is self-adjoint for the natural pairing exactly when $a$ is central, and then $L_a = R_a$; for the twisted pairing $L_a$ is self-adjoint exactly when $\sigma(a) = a$. The unitary left multiplications, those with $L_a^{*}L_a = L_aL_a^{*} = \mathrm{id}$, are the $a$ that are both central and involutive, $a^2 = 1$ and $a \in Z(A)$; in general only the central involutions give unitary left multiplications, and the signed version adds the twisted case.

Proof. $L_a^{*} = R_a$, so $L_a = R_a$ is the centrality of $a$. For the twisted pairing $L_a^{*_\sigma} = L_{\sigma(a)}$, so self-adjointness is $\sigma(a) = a$. For unitarity, $L_a^{*}L_a = R_aL_a$ is the map $x\mapsto axa$, which is the identity exactly when $a^2 = 1$ (from $x = 1$) and $axa = x$ for all $x$, and the latter with $a^2 = 1$ is the centrality of $a$.

Examples

(a) The matrix ring. $A = M_n(R)$ and $a = X$: $L_X^{*} = R_X$ under the trace pairing, so the adjoint of the left multiplication is the right multiplication, and the self-adjoint ones are the scalar matrices. This is the trace-duality statement of The Transpose as an Adjoint read as an adjoint of the regular representation.

(b) The group ring. $A = K[G]$ with the standard involution and $\tau$ the coefficient of the identity; $\sigma$-invariance of $\tau$ is immediate because $\sigma$ permutes the group basis, and $L_g^{*} = R_g$, $L_g^{*_\sigma} = L_{g^{-1}}$: the twisted adjoint of the left multiplication by a group element is the left multiplication by its inverse.

(c) The commutative case. For a commutative $A$, $L_a = R_a$ and every $L_a$ is self-adjoint; the adjoint involution is the identity on the image of $A$ in $\operatorname{End}(A)$, which is the content of the centrality criterion.

(d) The connection with the derivation. The inner derivation of Derivations of a Ring is $D_a = L_a-R_a$, so the adjoint of the inner derivation is the inner derivation of the image, $(L_a-R_a)^{*} = R_a-L_a = -D_a$: the inner derivations are skew-adjoint for the natural pairing, and the self-adjoint part of $L_a$ is the centre. This is the ring-level face of the skew-derivation calculus of Star-Derivations and the Skew Derivations.

Summary

With respect to the natural pairing $\langle x,y\rangle = \tau(xy)$ of a ring with a trace, the adjoint of the left multiplication is the right multiplication and conversely, $L_a^{*} = R_a$, $R_b^{*} = L_b$; the adjoint is the involution of Involutions of the Endomorphism Ring on the subring of the regular representation, and it switches the sides. The right multiplication is the conjugate of a left multiplication, $R_a = \sigma L_{\sigma(a)}\sigma$ for an involution $\sigma$ and a $\sigma$-invariant trace, and with respect to the twisted pairing $\{x,y\} = \tau(x\sigma(y))$ the adjoint is $L_a^{*_\sigma} = L_{\sigma(a)}$. The left multiplication is self-adjoint for the natural pairing exactly when $a$ is central and for the twisted pairing exactly when $a$ is symmetric; the unitary left multiplications are the central involutions. The inner derivation $D_a = L_a-R_a$ is skew-adjoint, which is the ring-level form of the skew derivations of Star-Derivations and the Skew Derivations.

Summary of Notation

Further Reading

• Nathan Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications 37 (1964), for the left and right multiplications, the trace form and the subring they generate.
• I. N. Herstein, Rings with Involution (University of Chicago Press, 1976), for the compatibility of the involution with the left and right multiplications.
• Nicolas Bourbaki, Algebra I, Chapters 1–3 (Springer, 1998), for the opposite ring, the regular representation and the adjoint under a symmetric pairing.
• Matej Brešar, Introduction to Noncommutative Algebra (Springer, 2014), for the inner derivations $L_a-R_a$ and their adjoint properties.