The Symmetric Algebra with an Involution

Introduction

The symmetric algebra $\operatorname{Sym}(M)$ of a module $M$ over a commutative ring $R$ is the free commutative $R$-algebra on $M$, the quotient of the tensor algebra by the two-sided ideal generated by the differences $x\otimes y-y\otimes x$; an involution of $M$ — a map of order two in the sense of Involutive Linear Spaces — therefore extends to an involution of $\operatorname{Sym}(M)$ by the universal property, because on a commutative algebra the difference between an automorphism and an anti-automorphism disappears. The induced involution is this extension; unlike the general case of an involution of an associative algebra, no anti-multiplicativity is imposed or needed, and the induced map is an $R$-algebra automorphism of order two. The purpose of this article is to compute its fixed subalgebra and to situate it with respect to the divided-power pairing, the perfect pairing through which $\operatorname{Sym}(M)$ is dual to the divided-power algebra of the dual module.

The article defines the induced involution, proves that it is the unique algebra automorphism of order two extending a given involution of $M$, and identifies its fixed subalgebra: in degree $k$ the fixed elements are the monomials with an even number of anti-fixed factors, so the fixed subalgebra is generated by the fixed part $M^+$ together with the pairwise products $xy$ of elements of the anti-fixed part $M^-$. It then derives the compatibility of the induced involution with the divided-power pairing: the transpose of the induced involution of $\operatorname{Sym}(M)$ is the induced involution of $\operatorname{Sym}(M^\vee)$, and the pairing is invariant. The two worked cases are the free module with a basis permuted or negated by the involution, where the fixed subalgebra is the even-degree part, and the polynomial ring with the induced involution, whose fixed symmetric part is the object of Involutions of a Polynomial Ring and the Symmetric Part in the linear-space reading.

The article assumes The Symmetric Algebra of a Module for $\operatorname{Sym}(M)$ and its universal property, Tensor Algebras and Free Algebras for the tensor algebra, The Symmetric and Exterior Powers for the graded pieces, Involutive Linear Spaces for the involution of a module, Involutive Linear Algebras for the involution of an associative algebra, The Dual Module and the Transpose for the transpose, and Divided Powers and the Symmetric Pairing for the divided-power algebra. The symmetric powers of an involutive module are the object of The Symmetric Powers of an Involutive Module, later in this group; the involution-invariant ideals are Involution-Invariant Ideals of the Symmetric Algebra. Throughout, $R$ is a commutative ring with identity, $M$ is an $R$-module, $\sigma_M$ is an $R$-linear involution of $M$ with $\sigma_M^2 = \mathrm{id}$, and $\sigma$ is the induced involution of $\operatorname{Sym}(M)$; no form, norm or distance occurs.

The Induced Involution

Definition and Universal Property

Definition. Let $\sigma_M$ be an $R$-linear involution of $M$, and let $\iota : M \to \operatorname{Sym}(M)$ be the canonical map. The induced involution of the symmetric algebra is the unique $R$-algebra homomorphism

$$ \sigma : \operatorname{Sym}(M) \to \operatorname{Sym}(M) \quad \text{with} \quad \sigma\circ\iota = \iota\circ\sigma_M . $$

Theorem. The induced map $\sigma$ exists and is unique, and it is an $R$-algebra automorphism of $\operatorname{Sym}(M)$ of order two; on $\operatorname{Sym}^k(M)$ it maps the product $v_1\cdots v_k$ to $\sigma_M(v_1)\cdots\sigma_M(v_k)$.

Proof. The universal property of the symmetric algebra says that an $R$-linear map $M\to B$ into a commutative $R$-algebra $B$ extends uniquely to an $R$-algebra homomorphism $\operatorname{Sym}(M)\to B$. Applying it with $B = \operatorname{Sym}(M)$ and the map $\iota\circ\sigma_M$ gives existence and uniqueness of $\sigma$. Since $\sigma^2\circ\iota = \iota\circ\sigma_M^2 = \iota$, uniqueness forces $\sigma^2 = \mathrm{id}$. The formula on the degree-$k$ piece is multiplicativity read on the spanning monomials. $\square$

Corollary. The correspondence $\sigma_M\mapsto\sigma$ is a homomorphism from the group of the $R$-linear involutions of $M$ to the group of the $R$-algebra involutions of $\operatorname{Sym}(M)$; it is injective because $\sigma$ restricts to $\sigma_M$ on the degree-one piece, and it is an isomorphism onto the involutions of $\operatorname{Sym}(M)$ that preserve the canonical copy of $M$.

Naturality

Proposition. For involutive modules $(M,\sigma_M)$, $(N,\sigma_N)$ and an $R$-linear map $f : M \to N$ with $f\sigma_M = \sigma_Nf$, the induced map $\operatorname{Sym}(f) : \operatorname{Sym}(M)\to\operatorname{Sym}(N)$ satisfies $\operatorname{Sym}(f)\circ\sigma = \sigma'\circ\operatorname{Sym}(f)$, where $\sigma,\sigma'$ are the induced involutions. Hence the symmetric algebra is a functor of the involutive module.

Proof. Both sides are $R$-algebra homomorphisms agreeing on $\iota(M)$: $\operatorname{Sym}(f)(\sigma(\iota v)) = \operatorname{Sym}(f)(\iota\sigma_Mv) = \iota f\sigma_Mv = \iota\sigma_Nfv = \sigma'(\iota fv) = \sigma'(\operatorname{Sym}(f)(\iota v))$. Uniqueness of the extension gives equality. $\square$

The Fixed Subalgebra

Degree Decomposition

Definition. The symmetric algebra is graded, $\operatorname{Sym}(M) = \bigoplus_{k\ge0}\operatorname{Sym}^k(M)$, and $\sigma$ preserves the degree, $\sigma(\operatorname{Sym}^k(M))\subseteq\operatorname{Sym}^k(M)$, because it is induced by a degree-preserving linear map.

Theorem. The fixed subalgebra of the induced involution is

$$ \operatorname{Sym}(M)^\sigma = \bigoplus_{k\ge 0}\operatorname{Sym}^k(M)^\sigma , \qquad \operatorname{Sym}^k(M)^\sigma = \operatorname{span}_R\{v_1\cdots v_k : \#\{i : v_i \in M^-\} \ \text{is even}\}, $$

where $M^+ = \{v : \sigma_M(v) = v\}$ and $M^- = \{v : \sigma_M(v) = -v\}$ are the fixed and the anti-fixed parts of $M$; equivalently, $\operatorname{Sym}(M)^\sigma$ is the $R$-subalgebra generated by $M^+$ and by the pairwise products $xy$ with $x, y \in M^-$.

Proof. If the involution is linear over $R$ with $2 = 1+1$ not a zero divisor, $M = M^+\oplus M^-$ and a monomial in homogeneous factors is fixed exactly when the number of factors from $M^-$ is even, which is the displayed condition; the subalgebra it generates is spanned by the even monomials, and an even monomial is a product of fixed elements, namely the factors from $M^+$ and the pairs of factors from $M^-$. In the general case the displayed span is still the fixed set, because the condition is linear and the monomials span. $\square$

Corollary (the case $M^- = 0$). If the involution of $M$ is trivial then the induced involution is trivial and $\operatorname{Sym}(M)^\sigma = \operatorname{Sym}(M)$, the whole algebra. If $M^+ = 0$ and $2 \neq 0$ is invertible, then $M = M^-$, the induced involution is the parity operator of the grading, and the fixed subalgebra is the even part $\bigoplus_k\operatorname{Sym}^{2k}(M)$.

Corollary (generation). The fixed subalgebra is generated as an $R$-algebra by $M^+$ together with the set $\{xy : x, y \in M^-\}$; it is the image of the symmetric algebra of the fixed module under the map that adds the squares.

The Divided-Power Pairing

The Pairing

Let $M^\vee = \operatorname{Hom}_R(M,R)$ be the dual module and let $\Gamma(M^\vee)$ be the divided-power algebra of $M^\vee$, the free polynomial-like algebra in which the divided powers $\gamma_n(\varphi)$ satisfy $\gamma_m(\varphi)\gamma_n(\varphi) = \binom{m+n}{m}\gamma_{m+n}(\varphi)$. The divided-power pairing, or symmetric pairing, is the $R$-bilinear map

$$ \langle\cdot,\cdot\rangle : \operatorname{Sym}(M)\times\Gamma(M^\vee) \longrightarrow R , \qquad \langle v_1\cdots v_k, \gamma_k(\varphi)\rangle = \varphi(v_1)\cdots\varphi(v_k), $$

which is a perfect pairing when $M$ is a free module of finite rank and is compatible with the gradings of the two sides.

The Transpose of the Induced Involution

Definition. The transpose of $\sigma_M$ is the unique $R$-linear map $\sigma_M^{\mathsf{T}} : M^\vee \to M^\vee$ with $\langle\sigma_M v, \varphi\rangle = \langle v, \sigma_M^{\mathsf{T}}\varphi\rangle$ for all $v \in M$, $\varphi \in M^\vee$; it is an involution of $M^\vee$ when $\sigma_M$ is an involution.

Theorem. The transpose of the induced involution $\sigma$ of $\operatorname{Sym}(M)$ with respect to the divided-power pairing is the induced involution $\sigma^\vee$ of the divided-power algebra $\Gamma(M^\vee)$ generated by $\sigma_M^{\mathsf{T}}$; the pairing satisfies

$$ \langle \sigma u, \sigma^{\vee}\gamma\rangle = \langle u, \gamma\rangle $$

for all $u \in \operatorname{Sym}(M)$ and $\gamma \in \Gamma(M^\vee)$, and $\sigma$ is the transpose of $\sigma^\vee$ in the same sense.

Proof. On the generators, $\langle\sigma(v_1\cdots v_k),\gamma_k(\varphi)\rangle = \prod_i \langle\sigma_Mv_i,\varphi\rangle = \prod_i \langle v_i,\sigma_M^{\mathsf{T}}\varphi\rangle = \langle v_1\cdots v_k,\gamma_k(\sigma_M^{\mathsf{T}}\varphi)\rangle$, and the extension to $\Gamma(M^\vee)$ is multiplicative, so the displayed invariance holds on the spanning elements and hence everywhere by bilinearity. $\square$

Corollary. The induced involution of the symmetric algebra is the transpose of the induced involution of the divided-power algebra, and the two are intertwined by the pairing; in particular the fixed subalgebra of one side pairs dually with the cofixed quotient of the other, and the invariant functionals on $\operatorname{Sym}(M)$ are the self-adjoint elements of $\Gamma(M^\vee)$ under the induced involution.

Examples

Example (the free module with a permuted basis). Let $M$ be free with basis $e_1,\dots,e_n$ and let $\sigma_M$ exchange $e_1\leftrightarrow e_2$ and fix the remaining basis elements. Then $\operatorname{Sym}(M)$ is the polynomial ring $R[e_1,\dots,e_n]$ and $\sigma$ is the involution exchanging the two variables $e_1,e_2$; the fixed subalgebra is $R[e_1+e_2,\ e_1e_2,\ e_3,\dots,e_n]$, generated by the fixed $M^+ = \langle e_1+e_2,e_3,\dots,e_n\rangle$ together with the product $e_1e_2$ of the two anti-fixed generators.

Example (the negated basis). Let $M$ be free with basis $e_1,\dots,e_n$ and let $\sigma_M(e_i) = -e_i$. Then $M^+ = 0$, $M = M^-$ under the assumption $2 \ne 0$, and the induced involution is the determinant of the sign of a monomial; the fixed subalgebra is the even-degree part of the polynomial ring, generated by the products $e_ie_j$.

Summary

An $R$-linear involution $\sigma_M$ of a module $M$ extends uniquely to an $R$-algebra automorphism $\sigma$ of order two of the symmetric algebra, the induced involution, which preserves the degree and is natural in the involutive module. Its fixed subalgebra is generated by the fixed part $M^+$ together with the pairwise products $xy$ with $x,y \in M^-$; in degree $k$ the fixed elements are the monomials with an even number of anti-fixed factors, so the fixed subalgebra is the even-degree part when the involution has no fixed vectors. With respect to the divided-power pairing $\operatorname{Sym}(M)\times\Gamma(M^\vee)\to R$, the induced involution is the transpose of the induced involution generated by the transpose $\sigma_M^{\mathsf{T}}$ of the involution of $M$, and the pairing is invariant; the invariant functionals correspond to the self-adjoint elements of the divided-power algebra. The polynomial-ring case with the exchanged and the negated generators is worked out. No form, norm or distance occurs.

Summary of Notation

Symbol	Meaning
$M$	$R$-module over a commutative ring $R$
$\sigma_M$	$R$-linear involution of $M$
$M^+, M^-$	Fixed and anti-fixed parts of $M$
$\operatorname{Sym}(M) = \bigoplus_k\operatorname{Sym}^k(M)$	Symmetric algebra, graded
$\sigma$	Induced involution of $\operatorname{Sym}(M)$
$\operatorname{Sym}(M)^\sigma$	Fixed subalgebra
$\langle v_1\cdots v_k,\gamma_k(\varphi)\rangle = \prod\varphi(v_i)$	Divided-power pairing
$\Gamma(M^\vee)$	Divided-power algebra of the dual module
$\sigma_M^{\mathsf{T}}$	Transpose involution of $M^\vee$
$\sigma^\vee$	Induced involution of $\Gamma(M^\vee)$
$\langle\sigma u,\sigma^\vee\gamma\rangle = \langle u,\gamma\rangle$	Invariance of the pairing