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 , denoted , 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 for by the following relations:
where 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 and the Fomin-Kirillov algebra . In the Fomin-Kirillov algebra, the generators 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