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Take the graded algebra $A=\mathbb C[a_1,a_2,a_3,\dots]$ with grading given by $\deg a_{i}=2i$. Let $a(t)$ be the generating function for generators ($a(t)=a_1t+a_2t^2+\cdots\in A[[t]]$). What is the quotient of $A$ by relations, generating function of which is given by $a(t)^2$ — i.e. is there some nice description of the graded algebra $$ \mathbb C[a_1,a_2,a_3,\dots]/(a(t)^2=0):=\mathbb C[a_1,a_2,a_3,\dots]/(a_1^2,2a_1a_2,2a_1a_3+a_2^2,\dots,\sum_{i+j=k}a_ia_j,\dots)? $$ What about Hilbert series, at least?

Upd (from off-site communications). Feigin-Stoyanovsky looks relevant: the algebra in question appears in theorem 2.2.1 as some representation of $\widehat{sl}_2$, and formula 2.3.3 tells that Hilbert series is exactly $G(q)$ of Rogers-Ramanujan. Mysterious but interesting.

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    What kind of object is $a_i$? A polynomial? If so, in what variable(s)?2011-06-29
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    @Grigory, I love generating functions, I just don't know what $\deg a_i=2i$ means if $a_i$ is a variable.2011-06-30
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    @Gerry Ah. It's just a formal grading -- I want it to be a graded algebra (note that with this grading all relations are homogeneous).2011-06-30
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    I'm confused: how does $a(t)$ live in $\mathbb{C} \langle a_1,a_2,\ldots \rangle$?2011-06-30
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    @Grigory: okay, thanks. (I've never seen that notation before, and I only just now noticed the $:=$.)2011-06-30
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    @Grigory: well, okay. To be honest, I think it was already clear enough and it was I who was being obtuse.2011-06-30

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