ExteriorAlgebra

ExteriorAlgebra, often called the exterior or Grassmann algebra, of a vector space V over a field F, is the graded algebra obtained by adjoining a wedge product that enforces antisymmetry. It can be constructed as the quotient T(V)/I of the tensor algebra T(V) by the two-sided ideal I generated by v⊗w + w⊗v for all v,w in V (in characteristic not 2 this implies v∧v = 0). The wedge product is the multiplication in the resulting algebra, and Λ^k V denotes the degree-k component.

If dim V = n, then ExteriorAlgebra Λ(V) decomposes as Λ(V) = ⊕_{k=0}^n Λ^k V, with dim Λ^k V = C(n,k) and total dimension 2^n. A basis is given by wedge products of an ordered basis e1,...,en: e_{i1} ∧ ... ∧ e_{ik} for i1 < ... < ik.

The product is graded-commutative: for α ∈ Λ^p V and β ∈ Λ^q V, α ∧ β = (-1)^{pq} β ∧ α. The space Λ^k V is canonically isomorphic to the space of alternating k-linear maps V^k → F.

Universal property: any linear map f: V → A into an algebra A that sends pairs of vectors to anti-commuting elements extends uniquely to an algebra homomorphism Λ(V) → A. Equivalently, Λ(V) ≅ T(V)/I.

In differential geometry, ExteriorAlgebra generalizes to the exterior algebra of differential forms on a manifold. The differential d increases degree by 1, satisfies d^2 = 0, and obeys the graded Leibniz rule d(α ∧ β) = dα ∧ β + (-1)^{deg α} α ∧ dβ. The top exterior power Λ^n V relates to volumes and determinants, and the exterior algebra forms the core of the de Rham complex.