# 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?

• Your ring $S$ is a fairly simple ring (your description makes it look harder than it is). Is there something specific that you want to know about this ring? I do not know whether this particular ring has been studied exclusively in literature. – Mohan Aug 9 '18 at 18:14
• @Mohan If you see a much simpler description of the ring, please feel free to add it to the question. The description I gave is the one that popped up in my application. I am interested in performing fast arithmetic in the ring, where "fast" means roughly in time $O(d\log^c d)$ for some $c$, where $d$ is the dimension of $S$ (which is easily determined). – Cornelius Brand Aug 10 '18 at 7:35
• Let $u_m=\sum_{i+j=m} t_it_j$ for $2\leq m\leq 2k$. Then $I$ is generated by the $u_m$s. – Mohan Aug 10 '18 at 13:27
• Thank you! I was aware of this (you get the dimension from that very quickly), but wasn't not sure if this was overall a more useful way to look at it. Is there any striking property of ideals of this form, or any well-known special class of rings that this falls into that comes to your mind? – Cornelius Brand Aug 11 '18 at 11:42

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
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
your algebra has the name $$D_2(1,1,1)$$.