Some examples of graded algebras

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds to the sum of vectors. More typically we can define the degree of a monomial to be its total degree (the sum of the components of the above vector); this degree is also multiplicative.

Algebras with this additional property are called graded algebras, and they show up surprisingly often in mathematics. As Alexandre Borovik notes, when schoolchildren work with units such as “apples” and “people” they are really working in a $\mathbb{Z}^n$ -graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in $\mathbb{Z}^n$ -graded algebras is the entire content of dimensional analysis.

At this point, I should give some definitions.

To define a graded algebra, we want to generalize the definition of a monomial. To say that a polynomial in one variable can be uniquely written as a sum of monomials is equivalent to giving a direct sum decomposition

$\mathbb{C}[x] \simeq \mathbb{C} \oplus \mathbb{C} x \oplus \mathbb{C} x^2 \oplus ...$

where $\mathbb{C} x^k$ denotes the monomials of the form a_k x^k . Since the product of a monomial of degree with a monomial of degree is a monomial of degree n+m , and ranges over the non-negative integers, we call this a $\mathbb{Z}_{\ge 0}$ -graded algebra.

In general, given a semigroup , a -graded algebra is an algebra with a direct sum decomposition

$\displaystyle A = \bigoplus_{g \in G} A_g$

with the property that the multiplication sends the product of an element of A_g and an element of A_h to an element of $A_{gh}$ . The elements of the factors A_g are called the homogeneous elements, and the value of is called the degree. If you’re unfamiliar with direct sums, just remember that it means that any element of can be written uniquely as a sum of homogeneous elements. Because polynomial rings are the prototypical example, the case $G = \mathbb{Z}_{\ge 0}$ is referred to as “graded.”

Note that a “polynomial” (a sum of homogeneous elements of different degree) doesn’t necessarily have a well-defined degree; we aren’t requiring that have an ordering.

Graded algebras seem to appear whenever symmetry or homogeneity are important, although I don’t have much experience with their more sophisticated uses. Below are a few examples.

Fourier transforms

Every function $f : \mathbb{C} \to \mathbb{C}$ can be uniquely written as the sum of an even function and an odd function. Generically, this takes the form

$\displaystyle f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ .

This gives the set of functions $\mathbb{C} \to \mathbb{C}$ the structure of a $\mathbb{Z}/2\mathbb{Z}$ -graded algebra; the direct sum decomposition is into the even and odd functions.

More generally, let $\omega$ be a primitive $n^{th}$ root of unity and say that a function $f : \mathbb{C} \to \mathbb{C}$ has weight if $f (\omega x) = \omega^k f(x)$ . This gives the set of functions $\mathbb{C} \to \mathbb{C}$ the structure of a $\mathbb{Z}/n\mathbb{Z}$ -graded algebra; the direct sum decomposition is into the functions of weight k, k = 0, 1, ... n-1 . You might know this as the discrete Fourier transform or as the decomposition of a representation of $\mathbb{Z}/n\mathbb{Z}$ into its irreducible one-dimensional representations. For example, for n = 3 the decomposition into functions of weight 0, 1, 2 takes the form

$\displaystyle f(x) = \frac{f(x) + f(\omega x) + f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega^2 f(\omega x) + \omega f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega f(\omega x) + \omega^2 f(\omega^2 x)}{3}$ .

Even more generally, let be a locally compact abelian group and let $\hat{G}$ denote its Pontryagin dual, i.e. the continuous homomorphisms $G \to \mathbb{C}$ . Let be a space on which acts continuously, and given a character $\chi : G \to \mathbb{C}$ , say that a function $f : X \to \mathbb{C}$ has weight $\chi$ if $f(gx) = \chi(g) f(x)$ for every $x \in X$ . Subject to technical assumptions, this gives the space of functions $X \to \mathbb{C}$ the structure of a $\hat{G}$ -graded algebra; the direct sum decomposition is into the functions of weight $\chi, \chi \in \hat{G}$ . (If $\hat{G}$ isn’t discrete; the direct sum is replaced by an integral.) For more details, see Terence Tao’s notes on the Fourier transform. With $G = \mathbb{T}$ (the circle group), X = S^1 (the circle), and $\hat{G} = \mathbb{Z}$ we recover the usual gradation on the space of Fourier series of functions on the circle (equivalently, the space of Fourier series of periodic functions on the real line).

Commutative algebra

A polynomial ring F[x_1, ... x_n] is a graded algebra under total degree. This allows us to focus our attention on homogeneous polynomials, since those are the important ones in algebraic geometry. Given a graded algebra $A = \bigoplus_{n \ge 0} A_n$ , define $H(A, n) = \dim A_n$ and define the Hilbert series

$\displaystyle H_A(t) = \sum_{n \ge 0} H(A, n) t^n$ .

One can verify that when A = F[x_1, ... x_n] the Hilbert series is $\frac{1}{(1 - t)^n}$ , and this should be familiar if you did the exercise about symmetric functions awhile back. Hilbert series behave well under the obvious operations: they are additive under direct sum and multiplicative under tensor product, provided the degree of a tensor product is defined appropriately. One can think of this as a “linearization” of the properties of combinatorial generating functions under disjoint union and Cartesian product. It is therefore reasonable to expect that the Hilbert series of a graded algebra encodes information about its structure.

Hilbert series can be used to study algebraic varieties, as follows: given a projective variety defined over $\mathbb{C}^n$ , the ring of polynomial functions $V \to \mathbb{C}$ is a quotient of $\mathbb{C}[x_1, ... x_n]$ by the ideal I(V) of functions in vanishing on , hence inherits a gradation. The Hilbert series of a variety can be used to define its dimension, as follows.

Theorem: There exists a polynomial (the Hilbert polynomial) P(V, n) such that H(V, n) = P(V, n) for all sufficiently large . The degree of this polynomial is the dimension of .

Intuitively, the degree of the Hilbert polynomial measures the number of “degrees of freedom” that polynomial functions on have. For $V = \mathbb{C}^d$ the ring of functions is $\mathbb{C}[x_1, ... x_d]$ and the Hilbert polynomial is $\displaystyle {n+d-1 \choose d}$ . For $V \subset \mathbb{C}^4$ the Segre variety xy - zw = 0 , the ring of functions is $\mathbb{C}[x, y, z, w]/(xy - zw)$ . Its Hilbert series begins

$\displaystyle 1 + 4t + 9t^2 + ...$

and we can compute its Hilbert polynomial as follows: after replacing the factor by in every monomial, the space of monomials of degree consists of

Monomials in ; there are of these.
Monomials in with a factor of ; there are ${n+2 \choose 2} - (n+1)$ of these.
Monomials in with a factor of ; there are ${n+2 \choose 2} - (n+1)$ of these.

This gives the Hilbert polynomial $2 {n+2 \choose 2} - (n+1) = (n+1)^2$ , hence has dimension ; in fact, it’s a doubly ruled surface.

I haven’t checked, but I believe this generalizes: the Segre embedding might correspond to the Hadamard product of Hilbert series in general.

Supersymmetry

Now we enter the realm of things I don’t understand. A $\mathbb{Z}/2\mathbb{Z}$ -graded algebra is called a superalgebra. Superalgebras have an even part and an odd part, as we have seen. A good example of a superalgebra is the ring of invariants of the alternating group A_n acting on $\mathbb{C}[x_1, ... x_n]$ by permutation of the variables. The even part consists of the polynomials invariant under S_n , the symmetric polynomials, and the odd part consists of the polynomials that gain the sign of a permutation under permutation, the alternating polynomials.

Supersymmetry is an idea from physics relating bosons to fermions. According to Masoud Khalkhali, if is the Hilbert space of states of a single boson, then the Hilbert space of states of bosons is the symmetric tensor power S^n H . If is instead the Hilbert space of states of a single fermion, then the Hilbert space of states of fermions is the exterior power $\wedge^n H$ ; this is the Pauli exclusion principle.

The exterior algebra of a -dimensional vector space has Hilbert series (1 + t)^d , since by Pauli exclusion a monomial of degree corresponds to a subset of size of basis vectors. In other words, fermion = subset. The symmetric algebra of can be identified with the space of polynomials in variables, so as we saw before it has Hilbert series $\frac{1}{(1 - t)^d}$ . In other words, boson = multiset. The “supersymmetry” relating bosons and fermions is hinted at by the following: