Consider $A = \mathbb{R}[t\_1,\ldots,t\_{k}]$, the ring of real $k$-variate polynomials in the indeterminates $t\_1,\ldots,t\_k$. For $y\in\mathbb{R}$, define the element $p(y) \in A$ through
$$
p(y) = t\_1 y + t\_2 y^2 + \cdots + t\_k y^k \in A\,.
$$


Let $I$ be the ideal in $A$ generated by $\{p(y)^2 \mid y \in \mathbb{R}\}$, and let $S = A/I$ be the quotient algebra of $A$ by $I$.


Has $S$ been studied? Does it have a name?



A commutative variant of the exterior algebra
For $k=\infty$, this algebra appears when studying integrable representations of level 1 of the Lie algebra $\widehat{\mathfrak{sl}}\_2$, see, for example, discussion in Section 2 of 


[A. V. Stoyanovsky, B. L. Feigin. Functional models for representations of current algebras and semi-infinite Schubert cells. Functional analysis and its applications, 28 (1994), 55–72.](https://link.springer.com/article/10.1007%2FBF01079010) 


The case of finite $k$ is used when applying when applying this sort of representation theoretic construction to obtain new combinatorial identities. In particular, in the notation of the paper


[B. L. Feigin, S. A. Loktev. On the finitization of the Gordon identities. Functional analysis and its applications, 35 (2001), 44–51](https://link.springer.com/article/10.1023%2FA%3A1004172516908)


your algebra has the name $D\_2(1,1,1)$.


