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Suppose $(O, G, \alpha)$ is a triple where $O$ is some mathematical object, $G$ is a group and $\alpha : G \rightarrow Aut(O)$. Many different areas of mathematics study such triples. However, I only know of a couple of examples of a new object that encodes this system.

1) If $G$ is a group, $N$ a normal subgroup of $G$, $H$ a subgroup of $G$ and $\alpha(h)(n) = hnh^{-1}$, then one can encode $(N, H, \alpha)$ into the (outer) semidirect product $N \rtimes_\alpha H$.

2) If $O = \mathcal A$, a C$^*$-algebra, $G$ is a locally compact group and $\alpha : G \rightarrow Aut(\mathcal A)$ acts by $*$-automorphisms then $(\mathcal A, G, \alpha)$ is encoded as the crossed product algebra $\mathcal A \rtimes_\alpha G$. In very simple terms, this is accomplished by considering unitary representations of $G$. There is also a related construction in Statistical Mechanics called the covariance algebra.

My question then: are there any other semidirect/crossed product type constructions for such triples $(O, G, \alpha)$?

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    $\begingroup$ You can do grothendieck construction when a category acts on a category. $\endgroup$ – Benjamin Steinberg Mar 23 '17 at 19:02
  • $\begingroup$ There are wreath products for other algebraic structures. Joel Vanderwerf did his dissertation under John Rhodes on this. Gerhard "If You Ignore Function Signature" Paseman, 2017.03.23. $\endgroup$ – Gerhard Paseman Mar 23 '17 at 19:06
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    $\begingroup$ Semidirect product of Lie algebras (one being endowed with an action by derivation on the other). $\endgroup$ – YCor Mar 23 '17 at 19:14
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    $\begingroup$ A fairly general semidirect product notion exists in semiabelian categories, see the paper by Bourn and Janelidze "Protomodularity, Descent and Semidirect Products" in TAC (1998). Basically, it works for those varieties of algebras where surjective homomorphisms have identifiable element inverses; then any split surjective homomorphism can be described as a semidirect product. This includes groups, Lie algebras, associative algebras (with an algebra with unit acting on another one without unit via bimultiplications), and many more. $\endgroup$ – მამუკა ჯიბლაძე Mar 23 '17 at 20:30
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    $\begingroup$ @BenjaminSteinberg Strictly speaking, not on a category but on a bunch of categories :D $\endgroup$ – მამუკა ჯიბლაძე Mar 23 '17 at 20:31
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Such triples, involving two objects -or even categories- and some kind of an "action" of one of them on the other (respecting some or all of its structure maps), are quite general and are met -as has already been indicated in the other answers and the comments as well- in various different branches of mathematics and mathematical physics. A fair coverage might deserve a suitably sized expository article.

I will try to sketch some of these ideas, and the corresponding objects, but it is important to note that there are lots of others as well (just to mention a few: $C^*$-algebras, graded algebras, etc, provide somewhat similar examples in different directions).

$\blacksquare$ Case $I$: The smash product $A\sharp H$, between a hopf algebra $H$ and a $H$-module algebra $A$

One notion which is similar and in some sense generalises the first construction discussed in the OP, is the smash product algebra, between a Hopf algebra $H$ and an algebra $A$, which also a $H$-module algebra:

Definition: Let $Η$ be a $k$-Hopf algebra and $Α$ a $k$-algebra, which is also a $H$-module algebra. The smash product of $Α$ and $Η$, denoted by $Α \sharp Η$ will be the algebraic structure determined by the following data:
1. As $k$-vector spaces $Α \sharp Η = Α \otimes_{k} Η$. We will denote the element $a \otimes h$ as $a \sharp h$.
2. For any $a, b \in A$ and for any $h, g \in H$ define a bilinear multiplication by: $$ (a \sharp h)(b \sharp g) = \sum a(h_{(1)} \cdot b) \sharp h_{(2)}g $$ where $\Delta(h)=\sum h_{(1)}\otimes h_{(2)}$ the comultiplication in $H$.
It is relatively easy to show that the $k$-v.s. $Α \sharp Η$ equipped with the above defined multiplication becomes an associative $k$-algebra with unity. This algebra is called the smash product algebra or crossed product algebra between the hopf algebra $H$ and the $H$-module algebra $A$.
It can furtherore be shown that -under suitable circumstances- $Α \sharp Η$ also becomes a $k$-Hopf algebra called the crossed product hopf algebra.
(in the above $k$ is the field).

Now what might be particularly interesting, is that the above smash product algebra, generalizes some similar but well known constructions. So let us study closer some specific cases of smash product algebras:

  • Example 1: The skew group algebra
    Let $Α$ be a (left) $kG$-module algebra. Using the definition above, we get that for all $g \in G$ we have: $$ g\cdot (ab) = (g \cdot a)(g \cdot b) $$ i.e.: the elements of $G$ act as automorphisms of $Α$. We thus get a group homomorphism: $$ \sigma: G \longrightarrow Aut_{k}A $$ and conversely: given any automorphism $\sigma: G \rightarrow Aut_{k}A$, $Α$ becomes a (left) $kG$-module algebra. By the definition of the multiplication in $A \sharp kG$ : $$ (a \sharp g)(b \sharp h) = a(g \cdot b) \sharp gh = a \sigma(g)(b) \sharp gh $$ which proves that, in this case the smash product algebra is isomorphic to the skew group algebra: $$A \sharp kG \cong A \star G$$ as $k$-algebras.

The next example, may be seen as a special case of the above:

  • Example 2: The semidirect product of groups
    Let $Κ$ and $Η$ two groups and let us assume that the elements of $K$ act as automorphisms of $H$. (We thus have a group homomorphism: $\phi: Κ \rightarrow Aut(H)$). Thus, the elements of $Κ$ act as automorphisms of the group algebra $kH$, which becomes a $kK$-module algebra. From the definition of the multiplication in $kH \sharp kK$ we get that for all $h, h' \in H$ και $g, g' \in K$: $$ (h \sharp g)(h' \sharp g') = h(g \cdot h') \sharp gg' = h \phi(g)(h') \sharp gg' $$ i.e., in this case: $$ kH \star K \cong kH \sharp kK \cong k(H \rtimes_{\phi} K) $$ where $H \rtimes_{\phi} K$ stands for the semidirect product of the groups $Η$, $Κ$ (which is a group itself) and $k(H \rtimes_{\phi} K)$ is its group algebra.
    A remarkable consequence, is that, if the action $\phi$ is trivial, in the sense that, for all $g \in K$ and for all $h \in H$: $$ g \cdot h = \phi(g)(h) = h $$ then the semidirect product is identified with the direct product (as groups) and the smash product is identified with the tensor product (as algebras). In that case, the former relation reproduces the well known (from group representation theory): $$ kH \otimes kK \cong k(H \times K) \equiv k(H \oplus K) $$

On the other hand, the smash product algebra described above, plays a much more important role in the structure theory of hopf algebras than simply generalizing or mimicking other known similar constructions: For example, it is well known that any cocommutative hopf algebra $H$ over an algebraically closed field of characteristic zero is isomorphic to the smash product hopf algebra between the group hopf algebra of its grouplikes $G(H)$ and the UEA of the Lie algebra of its primitives $U(P(H))$: $$ H\cong U(P(H))\sharp kG(H) $$ This is commonly refered to as the Cartier-Kostant-Milnor-Moore theorem.

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$\blacksquare$ Case $II$: The crossed product $A\sharp_\sigma kG$, between an algebra $A$ and a finite group $G$

Another similar construction of crossed products, not -in principle- related with the Hopf algebra case and its module algebra described in the former case, is the following:

Let $A$ be a finite dimensional algebra over an algebraically closed field and $G$ a finite group.
Let also, the following additional data:
$\bullet$ a convolution-invertible map $\sigma:G\times G\rightarrow A$ (Note that: $\sigma\in Hom_k(kG\otimes kG,A)$)
$\bullet$ a $k$ linear map $kG\otimes A\rightarrow A$, denoted $g\otimes a\mapsto g\cdot a$
satisfying: $$ g\cdot(h\cdot a)= \sigma(g,h)(gh\cdot a)\sigma^{-1}(g,h) \rightsquigarrow twisted \ module \ condition \\ \big(g\cdot\sigma(h,l)\big)\sigma(g,hl)=\sigma(g,h)\sigma(gh,l) \rightsquigarrow cocycle \ condition \\ g\cdot(ab)=(g\cdot a)(g\cdot b) \rightsquigarrow kG \ measures \ A \\ \sigma(1,g)=\sigma(g,1)=1, \ \ g\cdot1=1, \ \ 1\cdot a=a $$ for all $g,h,l\in G$ and for all $a,b\in A$. Thus, $\sigma$ is a $2$-cocycle.

If we denote the element $a\otimes g\in A\otimes kG$ by $a\sharp g$, then we can define a product on this vector space by: $$ (a\sharp g)(b\sharp h)=a(g\cdot b)\sigma(g,h)\sharp gh $$ This definition results in an associative algebra called crossed product algebra and denoted by $A\sharp_\sigma kG$.

It is a fully $G$-graded algebra, with $g$-component: $(A\sharp_\sigma kG)_g=A\sharp_\sigma kg$ for all $g\in G$ and with the algebra $A$ embedding in it as the identity component. You can see some discussion about it in the article Irreducible representations of crossed products, J. of Pure and applied algebra, v.129, 3, p. 315-326, 1998, by S. Montgomery and S.J. Witherspoon.

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$\blacksquare$ Generalization of Cases $I$ and $II$: The crossed product $A\sharp_\sigma H$, between an algebra $A$ and a hopf algebra $H$, which "measures" $A$

We will say that a hopf algebra $H$ measures the algebra $A$, if there is a $k$-linear map $H\otimes A\to A$, given by $h\otimes a \mapsto h\cdot a$ and such that $h\cdot 1=\epsilon(h)1$ and $h\cdot (ab)=\Sigma(h_{(1)}\cdot a)(h_{(2)}\cdot b)$ for all $h\in H$, $a,b\in A$.

Definition: Assume a triple $(A,H, \sigma)$ consisting of the following data:

  • An algebra $A$,
  • A hopf algebra $H$ measuring $A$ and
  • A convolution invertible map $\sigma\in Hom_k(H\otimes H, A)$

The crossed product $A\sharp_\sigma H$, of $A$ with $H$, is the vector space $A\otimes H$ equipped with the product $$ (a\sharp h)(b\sharp k)=\sum a(h_{(1)} \cdot b)\sigma(h_{(2)},k_{(1)}) \sharp h_{(3)}k_{(2)} $$
for all $a,b\in A$ and for all $h,k\in H$.

Note that now, unlike case $I$, we do not demand $A$ to be an $H$-module. Based on the above definition, the next lemma can be shown in a relatively straightforward manner:

Lemma: The crossed product $A\sharp_\sigma H$, of $A$ with $H$, is an associative algebra with identity $1\sharp 1$ if and only if the following two conditions are met:

  1. $A$ is a twisted $H$-module, that is $1\cdot a=a$, for all $a\in A$ and $$ h\cdot(k\cdot a)= \sum \sigma(h_{(1)},k_{(1)})(h_{(2)}k_{(2)}\cdot a)\sigma^{-1}(h_{(3)},k_{(3)}) $$ for all $h,k\in H$, $a\in A$.
  2. $\sigma$ is a cocycle, that is, $\sigma(1,h)=\sigma(h,1)=\epsilon(h)1$ and $$ \sum \big( h_{(1)}\cdot\sigma(k_{(1)},m_{(1)}) \big)\sigma(h_{(2)},k_{(2)}m_{(2)})=\sum\sigma(h_{(1)},k_{(1)})\sigma(h_{(2)}k_{(2)},m) $$ for all $h,k,m\in H$.

  In case $\sigma$ is trivial, in the sense that $\sigma(h,k)=\epsilon(h)\epsilon(k)1$ for all $h,k\in H$, then the above conditions directly imply that $A$ is an $H$-module algebra and that the crossed product becomes the smash product $$ A\sharp_\sigma H=A\sharp H $$ of case $I$, discussed above.

  In case $H=kG$, where $G$ is a finite group, then the crossed product becomes the group crossed product $$ A\sharp_\sigma H=A\sharp_\sigma kG $$ of case $II$, discussed above.

(More details on the properties of the structure and representations of the crossed product algebra $A\sharp_\sigma H$ can be found in ch. 7 of S. Montgomery's book "Hopf algebras and their actions on rings").

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In any higher category $C$, given an object $X$ and an action of a group $G$ on it you can ask for the homotopy quotient $X_{hG}$ of $X$ by the action of $G$, which is defined by a homotopy-coherent version of the universal property of the usual quotient. This recovers the following constructions in the following higher categories:

  • In the 2-category of groups (thinking of groups as one-object categories), this recovers the semidirect product, and actually a generalization of it describing general group extensions.
  • In the 2-category of rings (thinking of rings as one-object linear categories), this recovers the crossed product, and again a generalization of it including e.g. twisted group algebras.
  • In the $\infty$-category of spaces, this recovers the construction described in Mark Grant's answer.
  • In the $\infty$-category of chain complexes, this recovers a general form of group cohomology with nontrivial coefficients.
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Let $G$ be a topological group, and let $O$ be a topological space with an action of $G$, which can be viewed as a continuous homomorphism $\alpha:G\to \operatorname{Homeo}(O)=\operatorname{Aut}(O)$. There is a universal principal $G$-bundle $EG\to BG$, and an associated bundle $EG\times_G O\to BG$ with fibre $O$. Here $G$ acts diagonally on the product $EG\times O$, and $EG\times_G O$ is the orbit space of this action, with the obvious projection to $EG/G=BG$.

The assignment of $EG\times_G O$ to $(O,G,\alpha)$ seems to be an example of what you are asking, in the topological category.

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    $\begingroup$ Yes, this is the same construction; it reduces to the semidirect product when applied to Eilenberg-MacLane spaces, and model-independently is a homotopy quotient. $\endgroup$ – Qiaochu Yuan Mar 24 '17 at 17:01

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