# Extended double 2-cocycle conditions: Mathematical structure behind?

Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly.

1. The ordinary group 2-cocycle condition:

Let us remind the usual so-called homogeneous group 2-cocycle $$\mu(a,b,c)$$ of cohomology group $$H^2(G,U(1))$$ where $$U(1)=\mathbb{R}/\mathbb{Z}$$ is given by

$$\frac{\mu(b,c,d)\mu(a,b,d)}{\mu(a,c,d)\mu(a,b,c)}=1.$$ where all $$a,b,c,d \in G$$.

• We can focus on the case $$G$$ is a finite group (or even finite Abelian group if you want to simplify further.)

See References on group cohomogy:

The homogeneous group 2-cocycle $$\mu(a,b,c)$$ can be coverted to a homogeneous group 2-cocycle via $$\omega(A,B):=\omega(a d^{-1},b d^{-1})= \mu(a d^{-1},b d^{-1},1).$$ so if we define $$a d^{-1}=A$$ and $$b d^{-1}=B$$, $$c d^{-1}=C$$, then

$$\frac{\mu(bd^{-1},cd^{-1},1)\mu(ad^{-1} , b d^{-1},1)}{\mu(a d^{-1},c d^{-1},1)\mu(a c^{-1},b c^{-1},1)}=\frac{\mu(B,C,1)\mu(A , B,1)}{\mu(A,C,1)\mu(AC^{-1}, BC^{-1},1)}= \frac{\omega(B,C)\omega(A,B)}{\omega(A,C)\omega(A C^{-1},BC^{-1})}=1.$$ or equivalently the 2-cocycle condition is: $$\frac{\mu(A,B,1)}{\mu(AC^{-1}, BC^{-1},1)} =\frac{\mu(A,C,1)}{\mu(B,C,1)} \Leftrightarrow\frac{\omega(A,B)}{\omega(AC^{-1}, BC^{-1})} =\frac{\omega(A,C)}{\omega(B,C)}.$$

1. Extended double 2-cocycle condition: Mathematical structure behind?

Let us define a new object call $$F$$ which is related to the ordinary homogeneous group 2-cocycles $$\mu_1$$ and $$\mu_2$$ (also inhomogeneous group 2-cocycles $$\omega_1$$ and $$\omega_2$$ ) with two tensor product inputs: $$F(A,B,\alpha ,\beta) :=\mu_1(A \otimes \alpha,B \otimes \beta,1) =\omega_1(A \otimes \alpha,B \otimes \beta)$$ also $$F(A,B,\alpha ,\beta) :=\mu_2(A \otimes B,\alpha \otimes \beta,1)=\omega_2(A \otimes B,\alpha \otimes \beta).$$

The 2-cocycle condition for a homogeneous group 2-cocycle $$\mu_1$$ (also an inhomogeneous group 2-cocycle $$\omega_1$$ ) becomes:

$$\frac{\omega_1(A \otimes \alpha,B \otimes \beta)}{\omega_1(AC^{-1} \otimes \alpha \gamma^{-1}, BC^{-1} \otimes \beta \gamma^{-1})} =\frac{\omega_1(A \otimes \alpha,C \otimes \gamma)}{\omega_1(B \otimes \beta,C \otimes \gamma)}$$

$$\Rightarrow\boxed{ \frac{F(A,B,\alpha ,\beta)}{ F(AC^{-1},BC^{-1},\alpha\gamma^{-1} ,\beta \gamma^{-1}) } = \frac{F(A,C,\alpha ,\gamma)}{ F(B,C,\beta, \gamma) }} \tag{1}$$

The 2-cocycle condition for a homogeneous group 2-cocycle $$\mu_2$$ (also an inhomogeneous group 2-cocycle $$\omega_2$$ ) becomes:

$$\frac{\omega_2(A \otimes B, \alpha \otimes \beta)}{\omega_2(AC^{-1} \otimes B \gamma^{-1} , \alpha C^{-1} \otimes \beta \gamma^{-1})} =\frac{\omega_2(A \otimes B,C \otimes \gamma)}{\omega_2(\alpha \otimes \beta,C \otimes \gamma)}$$

$$\Rightarrow\boxed{ \frac{F(A,B,\alpha ,\beta)}{ F(AC^{-1},B\gamma^{-1},\alpha C^{-1} ,\beta \gamma^{-1}) } = \frac{F(A , B,C ,\gamma)}{ F(\alpha , \beta,C ,\gamma) }} \tag{2}$$

Here all $$A,B,C,\alpha,\beta,\gamma \in G$$.

My puzzle for you: Is there any known mathematical structures behind these two extended double 2-cocycle conditions in Eq.(1) and Eq.(2)? If so, what does the corresponding cocycle class solution form? Are there certain modified notions of cohomology group counting distinct classes of these cocycles $$F(A,B,\alpha ,\beta)$$?