# 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. 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). 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. 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? Aug 11 '18 at 11:42

## 1 Answer

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

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 $$D_2(1,1,1)$$.

• I'm sorry Vladimir, I hadn't seen your answer until now. I'll have a look. Thank you. Jul 13 '20 at 6:35