Anticommutativity

Anticommutativity is a property of a binary operation on a set or algebra in which swapping the order of the operands changes the sign of the result: for all elements a and b, a*b = - b*a. This concept is meaningful over fields whose characteristic is not 2; if the characteristic is 2, the relation trivializes to a*b = b*a.

Common examples occur in structures used in geometry, topology, and physics. The wedge product in exterior or Grassmann algebras is anticommutative: for differential forms, α ∧ β = - β ∧ α (at least when α and β are 1-forms). The cross product in three-dimensional space is anticommutative: a × b = - b × a. In quantum theory, fermionic operators satisfy anticommutation relations, and certain matrix representations exhibit anticommutation among distinct generators; for instance, distinct Pauli matrices anticommute: σ_i σ_j = - σ_j σ_i for i ≠ j.

In many contexts, anticommutativity is a special case of graded or so-called graded-commutative behavior. In Z2-graded algebras, the product of two odd elements anticommutes, while even elements commute with all elements. The Lie bracket is an antisymmetric bilinear operation, [x,y] = -[y,x], which is closely related to the notion of anticommutativity when viewed as a binary operation on a Lie algebra.

Applications of anticommutativity span differential geometry, topology, and mathematical physics. It underpins the algebraic structure of differential forms and exterior calculus, appears in the construction of Grassmann algebras used in supersymmetry and quantum field theory, and informs the behavior of fermionic systems in physics.