Algebra with two generators and relations

Question

I have a $\mathbb{C}$-algebra defined by two generators $a$ and $b$ with the relations that $a^2 = b^2 = 1$.

I would like to know of ways to think about this algebra (commutative or not).

Answer 1

If the algebra is noncommutative, ie $ab\neq ba$ then you have the group algebra over $\mathbb{C}$ of $\mathbb{Z}/(2) *\mathbb{Z}/(2)$, the free product of $\mathbb{Z}/(2)$s. If the algebra is commutative, ie $ab=ba$ then you have the group algebra over $\mathbb{C}$ of $\mathbb{Z}/(2)\times \mathbb{Z}/(2)$, the Klein 4-group.

Answer 2

If you require $a$ and $b$ to commute, then it looks like a graded algebra that one might see in algebraic geometry. It is a quotient of the symmetric algebra of a two-dimensional vector space.

If you require $a$ and $b$ to anticommute ($ab=-ba$) then you are looking at a complex Clifford algebra.