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 ?

Yet another way to reformulate the defining relation is the following:denote by "M" 3* matrix:

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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Some example.

Consider $E_{ij}$ - "elementary matrices" i.e. n*n matrix with zeros everywhere except position (i,j) where we put 1.

Take for example $E_{11} , E_{21} , E_{31} $,

**Observation:**
they satisfy the above relation.

More generally one can take $E_{i p} , E_{j p} , E_{k p} $ (important the second index is the same).

This means that the algebra above admits a homomorphism to universal enveloping of $E_{11} , E_{21} , E_{31} $. Universal enveloping algebras are very close to commuttive (at least their size is the same). So it suggests that in general such algebra is close to commutative, but probably this is wrong...

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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.