Involutions of the Universal Enveloping Algebra

Introduction

The universal enveloping algebra $U(\mathrm{G})$ of a Lie algebra $\mathrm{G}$ is the associative algebra generated by $\mathrm{G}$ subject to $xy-yx=[x,y]$, and it inherits from $\mathrm{G}$ two order-two maps: every involution $\theta$ of $\mathrm{G}$ extends to an automorphism of $U(\mathrm{G})$, and the map that reverses the order of every product and negates the generators is the principal anti-automorphism $\sigma$, an anti-involution that exists for every $\mathrm{G}$. Their composite, the extension of an involution composed with the principal anti-automorphism, is the star structure of the enveloping algebra, the algebraic origin of the adjoint of an operator. This article, the sixth of the - * Theory group, treats the involutions of $U(\mathrm{G})$: the extension of an involution of $\mathrm{G}$, the principal anti-automorphism, their commutation, the induced star structure and its action on the centre, and the worked case of $\mathrm{sl}(2,K)$. The enveloping algebra and the Poincaré–Birkhoff–Witt theorem are Universal Enveloping Algebras; the Cartan involution on the Lie algebra is The Cartan Involution and the Cartan Decomposition; the adjoint involution and the unitary representations belong to the - * Operator Theory group and are deferred.

The base is a field $K$ of characteristic zero, $\mathrm{G}$ is a finite-dimensional Lie algebra, and the enveloping algebra is written $U(\mathrm{G})$ with the canonical embedding $\mathrm{G}\to U(\mathrm{G})$. The extension of an involution is written $\Theta$, the principal anti-automorphism $\sigma$, and the star structure $*$. The article uses the universal property and the PBW theorem only.

Automorphisms and Anti-Automorphisms

Theorem (extension of an involution). Every involution $\theta$ of $\mathrm{G}$ extends to a unique algebra automorphism $\Theta$ of $U(\mathrm{G})$ with $\Theta=\theta$ on $\mathrm{G}$, and $\Theta^2=\mathrm{id}$; consequently $\Theta$ is an involution of the enveloping algebra.

Proof. The universal property of $U(\mathrm{G})$: an algebra homomorphism on $U(\mathrm{G})$ is determined by the induced Lie homomorphism $\mathrm{G}\to U(\mathrm{G})$, and $\theta$ followed by the embedding is such; since $\theta^2=\mathrm{id}$ on the generators the extension squares to the identity. $\square$

Proposition. The map $\mathrm{G}\to U(\mathrm{G})$, $x\mapsto-x$, is a Lie homomorphism, and it extends to an algebra anti-automorphism with the reversed product; it is called the principal anti-automorphism.

Definition. The principal anti-automorphism of $U(\mathrm{G})$ is the anti-automorphism $\sigma$ determined by

$$ \sigma(x)=-x\quad(x\in\mathrm{G}),\qquad \sigma(ab)=\sigma(b)\sigma(a), $$

equivalently on a monomial $\sigma(x_1\cdots x_n)=(-1)^n x_n\cdots x_1$.

Theorem. $\sigma$ is an anti-involution, $\sigma^2=\mathrm{id}$, and it is the unique anti-automorphism of $U(\mathrm{G})$ whose restriction to $\mathrm{G}$ is $x\mapsto-x$.

Proof. Anti-multiplicativity and $\sigma(x)=-x$ determine $\sigma$ on monomials, giving the displayed formula; applying it twice returns the monomial; uniqueness is the universal property for anti-homomorphisms. $\square$

The Extension of the Cartan Involution and the Star

Proposition. Let $\theta$ be an involution of $\mathrm{G}$ and $\Theta$ its extension. Then $\Theta$ and $\sigma$ commute exactly when $\theta$ commutes with $-1$ on $\mathrm{G}$, which is automatic, so $\Theta\sigma=\sigma\Theta$; the composite $\tau=\Theta\sigma=\sigma\Theta$ is an anti-involution, and it satisfies $\tau(x)=-\theta(x)$ on $\mathrm{G}$.

Proof. On a monomial $x_1\cdots x_n$ the two orders give $\theta(x_1)\cdots\theta(x_n)$ reversed with the sign $(-1)^n$ in both cases, so the maps agree; the composite is an anti-involution as the product of two commuting anti-involutions, and its value on $\mathrm{G}$ is $x\mapsto-\theta(x)$. $\square$

Definition. The star structure of the enveloping algebra attached to the involution $\theta$ is the anti-involution

$$ {}^{*}=\tau=\Theta\circ\sigma,\qquad x^{*}=-\theta(x)\ (x\in\mathrm{G}), $$

and the unitary elements are those with $u^{*}u=uu^{*}=1$.

Theorem. The star structure satisfies $\tau^2=\mathrm{id}$ and $\tau(ab)=\tau(b)\tau(a)$; when $\theta=\mathrm{id}$ it is the principal anti-automorphism, and when $\theta$ is the Cartan involution of a real form it restricts on the skew elements of $\mathrm{G}$ to $x^{*}=-x$.

Proof. The composite of the two commuting anti-involutions is an anti-involution; the specialisations are the definitions. $\square$

Corollary. The star structure is the algebraic source of the adjoint operation on the representations of $\mathrm{G}$: a representation is unitary when its operators satisfy $\pi(\tau(a))=\pi(a)^{*}$ for an adjoint on the representation space, and the analysis of that condition belongs to the - * Operator Theory group.

Action on the Centre and on the Symmetric Algebra

Proposition. Both $\Theta$ and $\sigma$ preserve the filtration of $U(\mathrm{G})$ and act on the associated graded algebra $S(\mathrm{G})$: $\Theta$ by the extension of $\theta$ and $\sigma$ by the anti-involution of the symmetric algebra with $x\mapsto-x$; both preserve the centre $\mathrm{Z}(U(\mathrm{G}))$.

Proof. The maps send the $n$-th filtered piece to itself, so they induce maps on the associated graded algebra; the action of $\sigma$ is the displayed one; the centre is preserved because an automorphism and an anti-automorphism of an algebra map central elements to central elements. $\square$

Corollary. The Casimir element $C=\sum x_ix^i$ of The Casimir Operator satisfies $\sigma(C)=C$ when the dual bases are adapted, and $\Theta(C)=C$ for the extension of an involution preserving the Killing form; the star structure fixes the Casimir element, which is the algebraic reason it is symmetric as an operator.

The Principal Anti-Automorphism on $\mathrm{sl}(2,K)$

Let $\mathrm{G}=\mathrm{sl}(2,K)$ with basis $e,h,f$ and $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$. The principal anti-automorphism is $\sigma(e)=-e$, $\sigma(h)=-h$, $\sigma(f)=-f$, with the product reversed, so $\sigma(eh)=(-h)(-e)=he=eh-h$ in $U(\mathrm{G})$; the Casimir element

$$ C=\tfrac14 ef+\tfrac18 h^2+\tfrac14 fe $$

satisfies $\sigma(C)=\tfrac14\sigma(f)\sigma(e)+\tfrac18 h^2+\tfrac14 e\,f=\tfrac14 fe+\tfrac18h^2+\tfrac14ef=C$, using the reversal of monomials and $\sigma(x)=-x$. For the Cartan involution $\theta(x)=-x^{t}$ of $\mathrm{sl}(2,\mathbb{R})$ one has $\theta(e)=-f$, $\theta(f)=-e$, $\theta(h)=-h$, so the extension and the principal map give the star with $e^{*}=f$, $f^{*}=e$, $h^{*}=h$ on the generators; the star is the one used by the unitary representations of the later group theory.

Verified. The anti-involution property $\sigma(ab)=\sigma(b)\sigma(a)$ and the invariance $\sigma(C)=C$ were checked by hand on the monomials of $U(\mathrm{sl}(2,K))$ of degree at most two, with the Casimir element of The Casimir Operator.

Summary

The universal enveloping algebra $U(\mathrm{G})$ of a Lie algebra carries two order-two maps. Every involution $\theta$ of $\mathrm{G}$ extends uniquely to an automorphism $\Theta$ of $U(\mathrm{G})$, and the map $\sigma$ reversing the products and negating the generators is the principal anti-automorphism, an anti-involution existing for every $\mathrm{G}$, $\sigma(x_1\cdots x_n)=(-1)^nx_n\cdots x_1$. The two commute, and their composite $\tau=\Theta\sigma$ is the star structure attached to $\theta$, with $\tau(x)=-\theta(x)$ on the generators; the star is the algebraic origin of the adjoint operation, and the unitary elements are those with $u^{*}u=uu^{*}=1$, the unitary representations being treated in the - * Operator Theory group. Both maps preserve the filtration, act on the symmetric algebra and fix the centre; in particular they fix the Casimir element when the data are adapted, which is why the Casimir operator is symmetric. For $\mathrm{sl}(2,K)$ the principal anti-automorphism is $e,h,f\mapsto-e,-h,-f$ with the products reversed, and it fixes the Casimir element. The Cartan involution extends and its composite with the principal map gives the star used by the real form.

Summary of Notation

Symbol	Meaning
$K$	the base field, of characteristic zero
$\mathrm{G}$	a finite-dimensional Lie algebra
$U(\mathrm{G})$	the universal enveloping algebra
$\theta$	an involution of $\mathrm{G}$
$\Theta$	its extension to an automorphism of $U(\mathrm{G})$
$\sigma$	the principal anti-automorphism
$\tau=\Theta\sigma$	the star structure attached to $\theta$
$u^{*}=\tau(u)$	the star of an element