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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.

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.

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

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

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).

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:

Such triples, involving two objects -or even categories- and an 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.

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.