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wonderich
wonderich
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The corresponding 3-cocycles of $H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n$ are very simple: $$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{N^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$$$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{n^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$ with $p_I \in \mathbb{Z}_n$ labels the element in $H^3(\mathbb{Z}_n,U(1))$. Also $a,b,c \in \mathbb{Z}_n$. $[b^{}+c^{}] \equiv (b^{}+c^{})$mod $N$$n$. You can check explicitly it satisfies 3-cocycles conditions.

The corresponding 3-cocycles of $H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n$ are very simple: $$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{N^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$ with $p_I \in \mathbb{Z}_n$ labels the element in $H^3(\mathbb{Z}_n,U(1))$. Also $a,b,c \in \mathbb{Z}_n$. $[b^{}+c^{}] \equiv (b^{}+c^{})$mod $N$. You can check explicitly it satisfies 3-cocycles conditions.

The corresponding 3-cocycles of $H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n$ are very simple: $$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{n^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$ with $p_I \in \mathbb{Z}_n$ labels the element in $H^3(\mathbb{Z}_n,U(1))$. Also $a,b,c \in \mathbb{Z}_n$. $[b^{}+c^{}] \equiv (b^{}+c^{})$mod $n$. You can check explicitly it satisfies 3-cocycles conditions.

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wonderich
wonderich
  • 10.5k
  • 3
  • 26
  • 70

The corresponding 3-cocycles of $H^3(\mathbb{Z}_n,U(1))=\mathbb{Z}_n$ are very simple: $$ \omega_{{I}}^{}(a,b,c) = \exp \left( \frac{2 \pi i p^{}_{{I}}}{N^{2}} \; a^{}(b^{} +c^{} -[b^{}+c^{}]) \right) $$ with $p_I \in \mathbb{Z}_n$ labels the element in $H^3(\mathbb{Z}_n,U(1))$. Also $a,b,c \in \mathbb{Z}_n$. $[b^{}+c^{}] \equiv (b^{}+c^{})$mod $N$. You can check explicitly it satisfies 3-cocycles conditions.