Tag: conditions

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1. The usual group 2 cocycle condition:

Let us remember the usual so-called homogeneous group 2 cocycle $ mu (a, b, c) $ the 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 everyone $ a, b, c, d in G $,

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

See references to group cohomology:

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

$$
frac { mu (bd ^ {- 1}, cd ^ {- 1}, 1) mu (ad ^ {- 1}, bd ^ {- 1}, 1)} { mu (ad ^ {- 1}, cd ^ {- 1}, 1) mu (ac ^ {- 1}, bc ^ {- 1}, 1)} = frac { mu (B, C, 1) mu (A, B, 1)} { mu (A, C, 1) mu (AC ^ {- 1}, BC ^ {- 1}, 1)} =
frac { omega (B, C) omega (A, B)} { omega (A, C) omega (A C ^ {- 1}, BC ^ {- 1})} = 1.
$$
or equivalent is the 2-Cocycle condition:
$$
frac { mu (A, B, 1)} { mu (AC ^ {- 1}, BC ^ {- 1}, 1)}
= frac { mu (A, C, 1)} { mu (B, C, 1)}
Leftrightarrow frac { omega (A, B)} { omega (AC ^ {- 1}, BC ^ {- 1})}
= frac { omega (A, C)} { omega (B, C)}.
$$

2. Extended double-two-cycle condition: mathematical structure behind it?

Let's define a new object call $ F $ which is related to the usual homogeneous group 2 cocycles $ mu_1 $ and $ mu_2 $ (also inhomogeneous group 2 cocycles $ omega_1 $ and $ omega_2 $ ) with two tensor product inputs:
$$
F (A, B, alpha, beta): = mu_1 (A otimes alpha, B otimes beta, 1)
= omega_1 (A otimes alpha, B otimes beta)
$$
Likewise
$$
F (A, B,?,?) = Μ 2 (A × B, α × β, 1) = & ohgr;
$$

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

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

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

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

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

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

Here everyone $ A, B, C, alpha, beta, gamma in G $,

My puzzle for you: Are there any known mathematical structures behind these two extended double-two-cycle conditions in Equation (1) and Equation (2)? If so, what does the corresponding co-cycle class solution look like? (Are there certain modified terms in group cohomology?

(Many thanks in advance for your reply.)