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 Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $\mathbb{Z}^n$ -graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in Image may be NSFW.
Clik here to view. $\mathbb{Z}^n$ -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

Image may be NSFW.
Clik here to view. $\mathbb{C}[x] \simeq \mathbb{C} \oplus \mathbb{C} x \oplus \mathbb{C} x^2 \oplus ...$

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

In general, given a semigroup Image may be NSFW.
Clik here to view., a Image may be NSFW.
Clik here to view.-graded algebra is an algebra Image may be NSFW.
Clik here to view. with a direct sum decomposition

Image may be NSFW.
Clik here to view. $\displaystyle A = \bigoplus_{g \in G} A_g$

with the property that the multiplication sends the product of an element of Image may be NSFW.
Clik here to view. A_g and an element of Image may be NSFW.
Clik here to view. A_h to an element of Image may be NSFW.
Clik here to view. $A_{gh}$ . The elements of the factors Image may be NSFW.
Clik here to view. A_g are called the homogeneous elements, and the value of Image may be NSFW.
Clik here to view. is called the degree. If you’re unfamiliar with direct sums, just remember that it means that any element of Image may be NSFW.
Clik here to view. can be written uniquely as a sum of homogeneous elements. Because polynomial rings are the prototypical example, the case Image may be NSFW.
Clik here to view. $G = \mathbb{Z}_{\ge 0}$ 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 Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $f : \mathbb{C} \to \mathbb{C}$ can be uniquely written as the sum of an even function and an odd function. Generically, this takes the form

Image may be NSFW.
Clik here to view. $\displaystyle f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$ .

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

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

Image may be NSFW.
Clik here to view. $\displaystyle f(x) = \frac{f(x) + f(\omega x) + f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega^2 f(\omega x) + \omega f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega f(\omega x) + \omega^2 f(\omega^2 x)}{3}$ .

Even more generally, let Image may be NSFW.
Clik here to view. be a locally compact abelian group and let Image may be NSFW.
Clik here to view. $\hat{G}$ denote its Pontryagin dual, i.e. the continuous homomorphisms Image may be NSFW.
Clik here to view. $G \to \mathbb{C}$ . Let Image may be NSFW.
Clik here to view. be a space on which Image may be NSFW.
Clik here to view. acts continuously, and given a character Image may be NSFW.
Clik here to view. $\chi : G \to \mathbb{C}$ , say that a function Image may be NSFW.
Clik here to view. $f : X \to \mathbb{C}$ has weight Image may be NSFW.
Clik here to view. $\chi$ if Image may be NSFW.
Clik here to view. $f(gx) = \chi(g) f(x)$ for every Image may be NSFW.
Clik here to view. $x \in X$ . Subject to technical assumptions, this gives the space of functions Image may be NSFW.
Clik here to view. $X \to \mathbb{C}$ the structure of a Image may be NSFW.
Clik here to view. $\hat{G}$ -graded algebra; the direct sum decomposition is into the functions of weight Image may be NSFW.
Clik here to view. $\chi, \chi \in \hat{G}$ . (If Image may be NSFW.
Clik here to view. $\hat{G}$ isn’t discrete; the direct sum is replaced by an integral.) For more details, see Terence Tao’s notes on the Fourier transform. With Image may be NSFW.
Clik here to view. $G = \mathbb{T}$ (the circle group), Image may be NSFW.
Clik here to view. X = S^1 (the circle), and Image may be NSFW.
Clik here to view. $\hat{G} = \mathbb{Z}$ 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 Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. $A = \bigoplus_{n \ge 0} A_n$ , define Image may be NSFW.
Clik here to view. $H(A, n) = \dim A_n$ and define the Hilbert series

Image may be NSFW.
Clik here to view. $\displaystyle H_A(t) = \sum_{n \ge 0} H(A, n) t^n$ .

One can verify that when Image may be NSFW.
Clik here to view. A = F[x_1, ... x_n] the Hilbert series is Image may be NSFW.
Clik here to view. $\frac{1}{(1 - t)^n}$ , 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 Image may be NSFW.
Clik here to view. defined over Image may be NSFW.
Clik here to view. $\mathbb{C}^n$ , the ring of polynomial functions Image may be NSFW.
Clik here to view. $V \to \mathbb{C}$ is a quotient of Image may be NSFW.
Clik here to view. $\mathbb{C}[x_1, ... x_n]$ by the ideal Image may be NSFW.
Clik here to view. I(V) of functions in Image may be NSFW.
Clik here to view. vanishing on Image may be NSFW.
Clik here to view., hence inherits a gradation. The Hilbert series of a variety Image may be NSFW.
Clik here to view. can be used to define its dimension, as follows.

Theorem: There exists a polynomial (the Hilbert polynomial) Image may be NSFW.
Clik here to view. P(V, n) such that Image may be NSFW.
Clik here to view. H(V, n) = P(V, n) for all sufficiently large Image may be NSFW.
Clik here to view.. The degree of this polynomial is the dimension of Image may be NSFW.
Clik here to view..

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

Image may be NSFW.
Clik here to view. $\displaystyle 1 + 4t + 9t^2 + ...$

and we can compute its Hilbert polynomial as follows: after replacing the factor Image may be NSFW.
Clik here to view. by Image may be NSFW.
Clik here to view. in every monomial, the space of monomials of degree Image may be NSFW.
Clik here to view. consists of

Monomials in Image may be NSFW.
Clik here to view.; there are Image may be NSFW.
Clik here to view. of these.
Monomials in Image may be NSFW.
Clik here to view. with a factor of Image may be NSFW.
Clik here to view.; there are Image may be NSFW.
Clik here to view. ${n+2 \choose 2} - (n+1)$ of these.
Monomials in Image may be NSFW.
Clik here to view. with a factor of Image may be NSFW.
Clik here to view.; there are Image may be NSFW.
Clik here to view. ${n+2 \choose 2} - (n+1)$ of these.

This gives the Hilbert polynomial Image may be NSFW.
Clik here to view. $2 {n+2 \choose 2} - (n+1) = (n+1)^2$ , hence Image may be NSFW.
Clik here to view. has dimension Image may be NSFW.
Clik here to view.; 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 Image may be NSFW.
Clik here to view. $\mathbb{Z}/2\mathbb{Z}$ -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 Image may be NSFW.
Clik here to view. A_n acting on Image may be NSFW.
Clik here to view. $\mathbb{C}[x_1, ... x_n]$ by permutation of the variables. The even part consists of the polynomials invariant under Image may be NSFW.
Clik here to view. 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 Image may be NSFW.
Clik here to view. is the Hilbert space of states of a single boson, then the Hilbert space of states of Image may be NSFW.
Clik here to view. bosons is the symmetric tensor power Image may be NSFW.
Clik here to view. S^n H . If Image may be NSFW.
Clik here to view. is instead the Hilbert space of states of a single fermion, then the Hilbert space of states of Image may be NSFW.
Clik here to view. fermions is the exterior power Image may be NSFW.
Clik here to view. $\wedge^n H$ ; this is the Pauli exclusion principle.

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

Image may be NSFW.
Clik here to view. $\displaystyle [x^k] (1 + t)^d = {d \choose k}$
Image may be NSFW.
Clik here to view. $\displaystyle [x^k] \frac{1}{(1 - t)^d} = (-1)^k {-d \choose k}$ .

The next GILA post will attempt to discuss these issues from a combinatorial perspective.

Some examples of graded algebras

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