Q: A commutative variant of the exterior algebra 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: 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. 
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
your algebra has the name $D_2(1,1,1)$.
