Name and properties of this combination of group algebra and semidirect product?

Question

Given a field $k$, a group $G$, and a homomorphism $\phi : G \to \mathrm {Aut} (k)$, we can define a ring $\widehat {k [G]}_\phi$ as follows: As an abelian group it is isomorphic to the group algebra $k [G]$, so every element can be written as $\sum \alpha_i g_i$ for $\alpha_i \in k$ and $g_i \in G$. However, rather than the ordinary group algebra product we can define this "twisted" product

$$ (\alpha g) (\beta h) = (\alpha \phi (g) (\beta)) (g h) \quad (\alpha, \beta \in k, g, h \in G) $$

In the case where $\phi$ is an inclusion modules of this ring correspond to $k$-vector space that are also invariant under the change-of-base-field transformations associated to $G \leq \mathrm {Aut} (k)$, which seems like a pretty natural concept. For instance (this was the inspiration for my question), in Theorem 3.4 of Milne's textbook on Galois Theory, the crucial structure is the null space of the matrix $(\sigma_i (\alpha_j))_{ij}$ where $(\sigma_i)$ enumerates a subgroup $G \leq \mathrm {Aut} (k)$. This null space is a $\widehat {k [G]}$-module, and it seems as though the proof for this theorem can be reformulated in this more abstract algebraic setting.

Has this ring been studied before, and what is its standard name and notation (possibly restricting to $\phi$ an inclusion)? What other interesting properties and results are associated to this ring?

Answer 1

I think that what you are talking about is the skew-group ring $k\rtimes_\phi G$ or the skew-group algebra, in the sense of def. 2.1 of these notes.

This is a special case of the skew-group algebra $A\rtimes_\phi G$, (here $A$ is a $k$-algebra) with multiplication defined by $$ ag\cdot bh=a\phi(g)\big(b\big)gh $$ given a group homomorphism $\phi:G\to Aut(A)$. (See Case I, example 1 of https://mathoverflow.net/a/265402/85967 for some more details).

Furthermore, the skew-group algebra $A\rtimes_\phi G$, can be seen as a special case of the crossed product algebra $A\sharp_\sigma kG$ between a $k$-algebra $A$ and a group $G$ (see Case II of https://mathoverflow.net/a/265402/85967 for more details).