Consider ass. algebra with 3 generators a1 a2 a3 and relation: a1a2a3 + a2a3a1 +a3a1a2 - a1a3a2 - a2a1a3 -a3a2a1 = 0.

i.e. $$ \sum_{ s \in S_3} (-1)^{sgn (s)} a_{s(1)} a_{s(2)} a_{s(3 )} = 0.$$

Informal questions: how far this algebra is from commutative polynomial algebra k[a1 a2 a3] ? What is known about this algebra ?

Formal question: what is the Hilbert series of this algebra ?

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Some reformulations of the defining condition:

Consider 3 Grassman variables $\psi_i$ i = 1,2,3. Define $$ \psi = a_1 \psi_1 + a_2 \psi_2 + a_3 \psi_3$$

The condition above is equivalent to $$ \psi ^3 =0 $$.

For commutative algebras we clearly have $\psi ^2=0$. So this algebra extends the commutative ones in certain sense. So I wonder how far it is away from commutative one ?

The other way to say this that M =

a1 a1 a1

a2 a2 a2

a3 a3 a3

The condition above is the same as $det^{column} M =0$ where column-determinant is used i.e. first take elements from the first column, second from the second and so forth.

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It seems Roland Berger discusses similar alegbras at section 3 of

http://arxiv.org/abs/0801.3383

as far as I can understand he proves that such algebra is N-Koszul (i.e. generalization of Koszul duality to non-quadratic algebras). But I cannot get far in his theory.