Question

Let $S$ be a finite set. Let $R$ be a complex vector space with basis indexed by subsets of $S$. Define a product on $R$ by defining it on the basis elements as $1_A\cdot 1_B=1_{A\Delta B}$, where $A\Delta B$ is the symmetric difference of $A$ and $B$. This gives $R$ the structure of a commutative and associative $C$-algebra.

Is this a well-understood algebra? Does it have a name?

Answer 1

It is the complex group algebra over $((\mathbb{Z}/2)^S,+)$. This may be also described as the tensor product of $S$ copies of $\mathbb{C}[\mathbb{Z}/2] = \mathbb{C} \times \mathbb{C}$.

Answer 2

This is the group algebra $\mathbb{C}G$ where $G$ is an elementary Abelian group of order $2^n$ where $n=|S|$ (the elements of $G$ are your $1_A$). Every group ring over $\mathbb{C}$ is isomorphic to the direct product of matrix algebras over $\mathbb{C}$. In this case as $G$ is Abelian, $\mathbb{C} G$ is isomorphic to the product of $2^n$ copies of $\mathbb{C}$.

Answer 3

This is the group algebra of the additive group $(\mathbb Z/2\mathbb Z)^S$, hence it is the product of $2^{|S|}$ copies of $\mathbb C$.