# The Unapologetic Mathematician

## Mathematics for the interested outsider

We’re about to talk about certain kinds of algebras that have the added structure of a “grading”. It’s not horribly important at the moment , but we might as well talk about it now so we don’t forget later.

Given a monoid $G$, a $G$-graded algebra is one that, as a vector space, we can write as a direct sum

$\displaystyle A=\bigoplus\limits_{g\in G}A_g$

so that the product of elements contained in two grades lands in the grade given by their product in the monoid. That is, we can write the algebra multiplication by

$\displaystyle\mu:A_g\otimes A_h\rightarrow A_{gh}$

for each pair of grades $g$ and $h$. As usual, we handle elements that are the sum of two elements with different grades by linearity.

By far the most common grading is by the natural numbers under addition, in which case we often just say “graded”. For example, the algebra of polynomials is graded, where the grading is given by the total degree. That is, if $A=R[X_1,\dots,X_k]$ is the algebra of polynomials in $k$ variables, then the $n$ grade consists of sums of products of $n$ of the variables at a time. This is a grading because the product of two such homogeneous polynomials is itself homogeneous, and the total degree of each term in the product is the sum of the degrees of the factors. For instance, the product of $xy+yz$ in grade $2$ and $x^3+xyz+yz^2$ in grade $3$ is

$\displaystyle (xy+yz)(x^3+xyz+yz^2)=x^4y+x^3yz+x^2y^2z+2xy^2z^2+y^2z^3$

in grade $5=2+3$.

Other common gradings include $\mathbb{Z}$-grading and $\mathbb{Z}_2$-grading. The latter algebras are often called “superalgebras”, related to their use in studying supersymmetry in physics. “Superalgebra” sounds a lot more big and impressive than “$\mathbb{Z}_2$-graded algebra”, and physicists like that sort of thing.

In the context of graded algebras we also have graded modules. A $G$-graded module $M$ over the $G$-graded algebra $A$ can also be written down as a direct sum

$\displaystyle M=\bigoplus\limits_{g\in G}M_g$

But now it’s the action of $A$ on $M$ that involves the grading:

$\displaystyle\alpha:A_g\otimes M_h\rightarrow M_{gh}$

We can even talk about grading in the absence of a multiplicative structure, like a graded vector space. Now we don’t even really need the grades to form a monoid. Indeed, for any index set $I$ we might have the graded vector space

$\displaystyle V=\bigoplus\limits_{i\in I}V_i$

This doesn’t seem to be very useful, but it can serve to recognize natural direct summands in a vector space and keep track of them. For instance, we may want to consider a linear map $T$ between graded vector spaces $V$ and $W$ that only acts on one grade of $V$ and with an image contained in only one grade of $W$:

\displaystyle\begin{aligned}T(V_i)&\subseteq W_j\\T(V_k)&=0\qquad k\neq i\end{aligned}

We’ll say that such a map is graded $(i,j)$. Any linear map from $V$ to $W$ can be decomposed uniquely into such graded components

$\displaystyle\hom(V,W)=\bigoplus\limits_{(i,j)\in I\otimes J}\hom(V_i,W_j)$

giving a grading on the space of linear maps.

October 23, 2009