2-cycles invariant exterior algebra

This article is about a quadratic algebra, or a type of quadratic algebra

History

Origin

This term was introduced by: Majid

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 $1\leq i,j\leq n$ 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

Further information: 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.

References

• Noncommutative differentials and Yang-Mills on S_n by Shahn Majid, Hopf algebras in noncommutative geometry and physics