2-cycles invariant exterior algebra

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Template:Symmetric groups

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 \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

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