4
$\begingroup$

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 $F(A,B,\alpha ,\beta)$?

(Thanks in advance for your answer.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.