Ring graded by a non-Abelian monoid

Question

I'm looking for interesting examples of a $G$-graded ring where $G$ is a non-Abelian semigroup, monoid or group. Obvious examples are the semigroup algebra $kG$, but I haven't come across any others.

I could come up with some contrived ones of course, but I'm looking for ones that arise naturally.

Answer 1

A classical example would be a crossed-product: if $A$ is a central simple algebra of dimension $n^2$ over a field $K$, and $L/K$ is a Galois extension of degree $n$ with Galois group $G$ such that $L\subset A$ as $K$-algebras, then there is a canonical decomposition $$A = \bigoplus_{g\in G}A_g$$ giving a $G$-graded algebra structure, where $a\in A_g$ if for all $\lambda\in L$, $a\lambda=g(\lambda)a$. As a $K$-vector space, $A_g\simeq L$.

It is an important construction since any central simple algebra is Brauer-equivalent to such an algebra, for any $L$ which splits $A$ (ie $A\otimes_K L\simeq M_n(L)$).

This can be seen as a twisted (actually, twice twisted) version of your example $L[G]$: in both cases, any element is a sum $\sum_{g\in G}\lambda_g u_g$ for some fixed elements $u_g$ (and we can take $u_1=1$), and $\lambda_g\in L$.

• First twist, by an action of $G$ on $L$: in $L[G]$, the $u_g$ commute with the $\lambda\in L$, while in $A$ we have $u_g\lambda=g(\lambda)u_g$;
• Second twist, by a $2$-cocycle $\alpha\in Z^2(G,L^\times)$: in $L[G]$, $u_gu_h=u_{gh}$, while in $A$ we have $u_gu_h=\alpha(g,h)u_{gh}$ for some $\alpha(g,h)\in L^\times$.