graded algebra in nLab

Contents

Contents

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 GG be a group. (Often GG will be abelian, and, in fact, one usually takes by default G=ℤG = \mathbb{Z} the additive group of integers, in which case the actual group being used is omitted from the terminology and notation.)

A graded ring is a ring RR equipped with a decomposition of the underlying abelian group as a direct sum R=⊕ g∈GR gR = \oplus_{g \in G} R_g such that the product takes R g×R g′→R gg′R_{g} \times R_{g'} \to R_{g g'}.

Analogously there is the notion of graded kk-associative algebra over any commutative ring kk.

Specifically for kk a field a graded algebra is a monoid in graded vector spaces over kk.

An ℕ\mathbb{N}-graded algebra is called connected if in degree-0 it is just the ground ring.

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

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

Properties

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

Examples

Group ring

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

k[G]=⨁ g∈GL gk[G] = \bigoplus_{g\in G}L_g

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

ℓ g 1⋅ℓ g 2=ℓ g 1g 2,\ell_{g_1}\cdot \ell_{g_2} = \ell_{g_1g_2},

given on basis elements, by kk-linearity.

Lazard ring

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

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