nLab graded algebra

Idea

A graded algebra is an associative algebra which is with a labelling on its elements by elements of some monoid or group, and such that the multiplication in the algebra is reflected in the multiplication in the labelling group.

Definition

Let $M$ be a monoid. (Often $M$ will be an abelian group, and often by default $M = \mathbb{Z}$, the additive group of integers. However, another common choice is $M = \mathbb{N}$, the additive monoid of natural numbers.)

An $M$-graded ring or simply a graded ring is a ring $R$ equipped with a decomposition of the underlying abelian group as a direct sum $R = \oplus_{m \in M} R_m$ such that the product maps $R_{m} \times R_{m'}$ to $R_{m m'}$.

Analogously there is the notion of graded associative algebra over any field $k$. We can express it as follows: an $M$-graded algebra or simply graded algebra is a monoid in a monoidal category of $M$-graded vector spaces over $k$.

Both graded rings and graded algebras over a field are, in turn, special cases of graded algebras over a commutative ring.

An $\mathbb{N}$-graded ring is often called a nonnegatively graded ring. An $\mathbb{N}$-graded ring is called connected if in degree zero it is just the ground ring.

A differential graded algebra is a graded algebra $A$ equipped with a derivation $d : A\to A$ of degree +1 (or -1, depending on conventions) and such that $d \circ d = 0$. This is the same as a monoid in the category of chain complexes.

A $\mathbb{N}$-graded algebra is called strongly $\mathbb{N}$-graded (in Ardizzoni & Menini (2007), Def. 3.2) if for every $n,p \ge 0$, the multiplication $A_{n} \otimes A_{p} \rightarrow A_{n+p}$ is an epimorphism.

Properties

Proposition

For $R$ a commutative ring write $Spec R \in Ring^{op}$ for the corresponding object in the opposite category. Write $\mathbb{G}_m$ for the multiplicative group underlying the affine line.

There is a natural isomorphism between

• $\mathbb{Z}$-gradings on $R$;

• $\mathbb{G}_m$-actions on $Spec R$.

The proof is spelled out at affine line in the section Properties.

Examples

Group ring

Let $G$ be any (discrete) group and $k[G]$, its group algebra. This has a direct sum decomposition as a $k$-module,

$$k[G] = \bigoplus_{g\in G}L_g$$

where each $L_g$ is a one dimensional free $k$-module, for which it is convenient, here, to give a basis $\{\ell_g\}$. The graded algebra structure is obtained by extending the multiplication rule,

$$\ell_{g_1}\cdot \ell_{g_2} = \ell_{g_1g_2},$$

given on basis elements, by $k$-linearity.

Lazard ring

The Lazard ring, carrying the universal (1-dimensional, commutative) formal group law is naturally an $\mathbb{N}$-graded ring.

References

Textbook account:

• Gregory Karpilovsky, Chapter 2 of: The Algebraic Structure of Crossed Products, Mathematics Studies 142, North Holland 1987 (ISBN:9780080872537)

For Hopf algebras:

• Ken Brown, Paul Gilmartin, James J. Zhang, Connected (graded) Hopf algebras (arXiv:1601.06687)

The notion of strongly $\mathbb{N}$-graded algebra is defined in:

• Alessandro Ardizzoni, Claudia Menini, Associated graded algebras and coalgebras (arXiv:0704.2106)