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First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qK]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to Q\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $\omega: Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K,\omega)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $\omega: Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times,\omega)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qK]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to Q\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qK]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to G\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qK]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to Q\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qN]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to G\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Q\to\operatorname{Out}(K)$ (which turns $Z(K)$ into a $\mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=\bigoplus_{q\in Q} A_q$ such that $1\in A_1$, $A_{q_1} \cdot A_{q_2} \subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Q\to\operatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^\times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^\times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k[G]$ is $Q$-graded and crossed with $A_q:=k[qK]$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1\to A_1^\times \to (A^\times)_{homog.} \to G\to 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $\operatorname{Out}(A_1)$ may be different from $\operatorname{Out}(A_1^\times)$ and $Z(A_1)^\times$ may have little to do with $Z(A_1^\times)$.

Conversely, the algebra version does not give you the group version, $A_1=k[K]$, again because $\operatorname{Out}(K)$ and $\operatorname{Out}(k[K])$ can be very different.