Is it possible ?
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3
Rather obviously yes.
Let $A$ be the algebra over the field $K$ generated by elements $a_1,a_2,\ldots,$ with $a_i$ in dimension $i$ and with $a_ia_j=0$ for all $i$ and $j$. This is an incredibly uninteresting example, but since each graded piece is one-dimensional, its Poincare series is $\sum_{n=0}^\infty t^n=1/(1-t)$.
answered Sep 29, 2010 at 20:37
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$\begingroup$ I agree but I am interested with an algebra with non-trivial multiplication. $\endgroup$– MelaniaCommented Sep 29, 2010 at 20:47
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3$\begingroup$ @Melania: how about start as Robin did, and then kill all $a_ia_ja_k$? $\endgroup$ Commented Sep 29, 2010 at 21:23
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$\begingroup$ Yes..The Poincare series will be $\dfrac{1}{(1-t)(1-t^2)}.$ Thanks! $\endgroup$– MelaniaCommented Sep 29, 2010 at 21:54