In Kitaev's reasoning of constructing the algebra of symmetry group, he said,

"considering the symmetry group G of a fermionic system and a map $\alpha$ $$ \alpha: G \rightarrow \mathbb{Z}_2 $$ where $\alpha$ is the indicator that determines whether the group element is unitary or anti-unitary ,then we consider a element in the center of the $G$ $$ p \in Z(G) $$ where $P$ is the fermionic parity operator $P = (-1)^{N} $, N is the number of particles, we can easily know that $P$ is unitary so that $\alpha(P) = 1$ ,then we can construct the "group algebra"(not sure whether it is called like this?) $$ \mathscr{A} = \mathbb{R}[G]/(P+1) $$ "

First of all, what is $\mathbb{R}[G]$?

It seems like a clifford algebra construction but I have no idea of how these things are organized, especially how to calculate $\mathscr{A}$? Hope that someone could give me a logical picture of how things go.


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