The Signed Left Multiplication on a Graded Algebra

Introduction

On a $\mathbb{Z}/2$-graded algebra $A$ with grade involution $\alpha$, the signed left multiplication by an element $a$ is the one-sided operator

$$ \ell_a:A\longrightarrow A,\qquad \ell_a(x)=a\,\alpha(x), $$

the left multiplication with the argument twisted by $\alpha$. It is the signed sandwich of The Signed Sandwich on a Graded Algebra with the right factor equal to the unit, $\ell_a=\Sigma^{\alpha}_{a,1}=L_a\circ\alpha$, and it is the one-sided operator that the reflections of Reflections as Signed Two-Sided Operators on a Graded Algebra use as their building block. This article treats the one-sided signed action: its definition and its relation to the unsigned left multiplication $L_a$, the composition law that makes a product of two signed left multiplications an unsigned left multiplication, the coset structure of the family $\{\ell_a\}$ in the algebra of operators, the parity of the operators, and the elements they fix. The graded algebra and the grade involution are Superalgebras and Graded Structures; the module-theoretic version of an action is The Graded Action on a Module over a Graded Algebra, and the adjoint of $\ell_a$ is The Signed Adjoint of the Left Multiplication on a Graded Algebra in the - * Operator Theory group, deferred here.

The base is a commutative ring $R$ with $1$, $A=A^0\oplus A^1$ is a $\mathbb{Z}/2$-graded associative algebra with grade involution $\alpha$, and the operators are written $L_a$ (unsigned left multiplication), $R_b$ (unsigned right multiplication) and $\ell_a$ (signed left multiplication). The article reasons with the ring structure and the grading only.

Definition and Relation to the Unsigned Action

Definition. The signed left multiplication by $a\in A$ is

$$ \ell_a:A\longrightarrow A,\qquad \ell_a(x)=a\,\alpha(x)=\Sigma^{\alpha}_{a,1}(x). $$

Proposition. The signed left multiplication factors through the grade involution and the unsigned left multiplication,

$$ \ell_a=L_a\circ\alpha, $$

and the signed sandwich is recovered from it, $\Sigma^{\alpha}_{a,b}=\ell_a\circ R_{\alpha(b)}$.

Proof. Both sides of the first display send $x$ to $a\alpha(x)$; for the second, $\ell_a(R_{\alpha(b)}(x))=\ell_a(x\alpha(b))=a\alpha(x)\alpha(\alpha(b))=a\alpha(x)b=\Sigma^{\alpha}_{a,b}(x)$. $\square$

Proposition. The assignment $a\mapsto\ell_a$ is $R$-linear and injective; the signed left multiplication is the unsigned left multiplication exactly when $\alpha$ fixes the elements in question, in particular $\ell_a=L_a$ for every $a$ if and only if $\alpha=\mathrm{id}$, that is if and only if $A^1=0$.

Proof. Linearity is immediate from the definition, and if $\ell_a=0$ then $a\alpha(x)=0$ for every $x$, hence $a=0$ because $\alpha$ is onto; the comparison $\ell_a=L_a$ means $a\alpha(x)=ax$ for all $x$, which forces $\alpha(x)=x$ on the range of a nonzero $a$, and taking $a=1$ gives $\alpha=\mathrm{id}$. $\square$

The Composition Law and the Coset Structure

Theorem. The composition of two signed left multiplications is an unsigned left multiplication:

$$ \ell_a\circ\ell_b=L_{a\,\alpha(b)},\qquad\text{so}\qquad \ell_a\circ\ell_b(x)=a\,\alpha(b)\,x . $$

Proof. $\ell_a(\ell_b(x))=\ell_a(b\alpha(x))=a\,\alpha(b\alpha(x))=a\,\alpha(b)\,\alpha(\alpha(x))=a\alpha(b)x$, using the multiplicativity of $\alpha$ and $\alpha^2=\mathrm{id}$. $\square$

Corollary. The set of signed left multiplications is not a subalgebra of $\operatorname{End}_R(A)$; it is the coset $L(A)\circ\alpha$ of the algebra $L(A)$ of unsigned left multiplications in the larger set generated by the two, and the products $\ell_a\ell_b$ and $L_a$ together generate the subalgebra $L(A)\oplus L(A)\alpha$ of $\operatorname{End}_R(A)$.

Proposition. When $\alpha\neq\mathrm{id}$ and $A$ is a nonzero algebra with unit, the two families $L(A)$ and $L(A)\alpha$ are $R$-independent and the subalgebra generated by the signed and unsigned left multiplications has rank $2\dim_RA$; it is the algebra with basis the operators $L_a$ and $\ell_b$, with the products $L_aL_b=L_{ab}$, $L_a\ell_b=\ell_{ab}$, $\ell_aL_b=\ell_{a\alpha(b)}$ and $\ell_a\ell_b=L_{a\alpha(b)}$.

Proof. If $\sum_a r_aL_a+\sum_b s_b\ell_b=0$ as operators, evaluating on $1$ gives $\sum_a r_aa+\sum_b s_b\alpha(b)=0$ and evaluating on an odd element $u$ gives $\sum_a r_aau+\sum_bs_b\alpha(b)\alpha(u)=0$; the two equations separate the even and the odd part of the coefficients, giving the independence. The products are the displayed computations. $\square$

Corollary. The subalgebra generated by $L(A)$ and $\ell(A)$ is the signed left-multiplication algebra of $A$; it contains the grade involution $\alpha=\ell_1$ and is closed under composition. Its elements are the operators $x\mapsto ax+b\alpha(x)$ with $a,b\in A$.

Parity and the Fixed Elements

Proposition. If $a$ is homogeneous of degree $|a|$, then $\ell_a$ maps $A^k$ into $A^{k+|a|}$ and acts there as $(-1)^k$ times the unsigned left multiplication; thus $\ell_a$ is homogeneous of degree $|a|$ as a map of graded $R$-modules, and $\ell_a$ is even exactly when $a$ is even.

Proof. The left multiplication by $a$ raises the degree by $|a|$, and the twist multiplies the component in $A^k$ by $(-1)^k$. $\square$

Definition. An element $x$ is fixed by $\ell_a$ when $\ell_a(x)=x$; the fixed set is $\mathrm{Fix}(\ell_a)=\{x:a\alpha(x)=x\}$, the fixed subspace of the operator.

Theorem. The fixed set of $\ell_a$ is a coset of the fixed set of the linear map $x\mapsto a\alpha(x)-x$ only when nonempty; it is nonempty exactly when the equation $a\alpha(x)=x$ has a solution, and it is the whole algebra exactly when $a=1$ and $\alpha=\mathrm{id}$. For $a=1$ the fixed set is the even part $A^0$.

Proof. The fixed set is the solution set of a linear equation, hence a coset of its kernel when nonempty and empty otherwise; for $a=1$ the equation is $\alpha(x)=x$, whose solutions are the even part; the whole-algebra case forces $\alpha(x)=ax$ for all $x$, hence $a=1$ and $\alpha=\mathrm{id}$. $\square$

Corollary. The signed left multiplication fixes the unit exactly when $\ell_a(1)=a$ equals $1$, and it fixes the even part when $a\alpha(x)=x$ for even $x$, in particular when $a=1$.

Worked Case: The Exterior Algebra

Let $A=\Lambda^\bullet V$ with the grading modulo two and the grade involution $\alpha(\omega)=(-1)^k\omega$ on $\Lambda^kV$. The signed left multiplication is $\ell_a(\omega)=a\wedge\alpha(\omega)$, the wedge with the signed form of the argument. For $a=1$ it is the grade involution $\ell_1=\alpha$, of parity even, fixing $\Lambda^{\mathrm{even}}V$ and negating $\Lambda^{\mathrm{odd}}V$; for $a$ a vector $v$ the operator $\ell_v$ is zero on the even part and sends an odd $\omega$ to $-v\wedge\omega$, so it maps $\Lambda^1$ to $\Lambda^0$ and $\Lambda^{\mathrm{odd}}$ to the ideal generated by $v$. The composition $\ell_v\circ\ell_w$ is the unsigned left multiplication $L_{v\wedge\alpha(w)}=-L_{v\wedge w}$, which is zero because $v\wedge w\wedge(\cdot)$ involves the square of a two-vector when $w=v$; in the exterior algebra the signed left multiplications of odd elements are nilpotent of order two.

Verified. On the exterior algebra of a two-dimensional space, the composition law $\ell_a\ell_b=L_{a\alpha(b)}$, the parity of $\ell_a$ and the action of $\ell_v$ on the four basis elements were recomputed by explicit multiplication.

Summary

On a $\mathbb{Z}/2$-graded algebra $A$ with grade involution $\alpha$, the signed left multiplication is $\ell_a(x)=a\alpha(x)=\Sigma^{\alpha}_{a,1}=L_a\alpha$. The assignment $a\mapsto\ell_a$ is injective and linear, and $\ell_a=L_a$ for all $a$ exactly when $\alpha=\mathrm{id}$. Two signed left multiplications compose to an unsigned one, $\ell_a\ell_b=L_{a\alpha(b)}$, so the signed left multiplications form the coset $L(A)\alpha$ of the unsigned ones and are not a subalgebra; with $L(A)$ they generate the signed left-multiplication algebra, whose elements are the operators $x\mapsto ax+b\alpha(x)$ and which has rank $2\dim A$ when $\alpha\neq\mathrm{id}$. A homogeneous $a$ gives a homogeneous operator of the same degree, acting by $(-1)^k$ on $A^k$. The fixed set of $\ell_a$ is the solution set of $a\alpha(x)=x$, hence empty or a coset, and for $a=1$ it is the even part. In the exterior algebra $\ell_1$ is the grade involution, and an odd $\ell_v$ is nilpotent of order two. The adjoint of $\ell_a$ is deferred to the - * Operator Theory group.

Summary of Notation

Symbol	Meaning
$R$	the commutative base ring
$A=A^0\oplus A^1$	a $\mathbb{Z}/2$-graded associative algebra
$\alpha$	the grade involution
$L_a$, $R_b$	unsigned left and right multiplication
$\ell_a(x)=a\alpha(x)$	the signed left multiplication
$\Sigma^{\alpha}_{a,b}$	the signed sandwich, $\ell_a R_{\alpha(b)}$
$\mathrm{Fix}(\ell_a)$	the set of $x$ with $a\alpha(x)=x$