2-cycles invariant exterior algebra

Origin
The notion of the invariant exterior algebra corresponding to the 2-cycles in the symmetric group, was first explored systematically by majod, in his paper Noncommutative differentials and Yang-Mills on S_n.

Definition
The 2-cycles invariant exterior algebra of order $$n$$, denoted $$\Lambda_n$$, is the invariant exterior algebra defined for the symmetric group with respect to 2-cycles (or transpositions). Explicitly, it is the quotient of the free algebra on generators $$e_{(ij)}$$ for $$1 \le i,j \le n$$ by the following relations:


 * $$e_{(ij)} \wedge e_{(ij)} = 0$$
 * $$e_{(ij)} \wedge e_{(km)} + e_{(km)} \wedge e_{(ij)} = 0$$
 * $$e_{(ij)} \wedge e_{(jk)} + e_{(jk)} \wedge e_{(ki)} + e_{(ki)} \wedge e_{(ij)} = 0$$

where $$i,j,k,m$$ are distinct.

Parallel with the Fomin-Kirillov algebra
There is a close relationship between the 2-cycles invariant exterior algebra $$E_n$$ and the Fomin-Kirillov algebra $$\Lambda_n$$. In the Fomin-Kirillov algebra, the generators $$[ij]$$ are antisymmetric but they commute. In the 2-cycles invariant exterior algebra, the generators are symmetric and they anticommute.