# User:Fropuff/Exterior algebra

In mathematics, the **exterior algebra**, denoted Λ^{•}*V* or Λ(*V*), on a vector space *V* is the associative algebra of **alternating tensors** on *V*. The exterior algebra is a graded algebra where the grade is given by the tensor rank:

The subspaces Λ^{k}*V*, consisting of all rank *k* alternating tensors, are called the *k*th **exterior powers** of *V*. Exterior algebras are often called **Grassmann algebras** after their inventor Hermann Grassmann.

The product in the exterior algebra is called the **exterior product** or **wedge product** and is denoted *v* ∧ *w* (read *v wedge w*) for *v*, *w* ∈ Λ^{•}*V*.

## Alternating tensors

Alt : *T*^{k}(*V*) → Λ^{k}(*V*) is the *alternating projection* or *antisymmetrization operation* defined as follows:

That is, Alt(ω) is just the signed average of all permutations (σ in *S*_{k}) of ω. If ω is already antisymmetric then Alt(ω) = ω.

## Exterior product

The exterior product of two alternating tensors is essentially just the tensor product composed with a projection onto the subspace of alternating tensors. That is, let ω and η by homogeneous alternating tensors of rank *k* and *m* respectively. The wedge product is defined as follows:

The wedge product for nonhomogeneous elements is defined by linearity.

**Note**: The funny normalization factor in the front of the definition of the wedge product is included for convenience as it simplifies a number of expressions. Note, however, that many authors prefer to leave it out and simply define

The first convention is sometimes called the *determinant convention* and the latter the *Alt convention*. In this article we will stick to the determinant convention.

The exterior product has the following properties:

- bilinear: for scalars
*a*,*b*and tensors ω, η, and ξ

- (
*a*ω +*b*η) ∧ ξ =*a*(ω ∧ ξ) +*b*(η ∧ ξ) - ξ ∧ (
*a*ω +*b*η) =*a*(ξ ∧ ω) +*b*(ξ ∧ η)

- (

- associative: (ω ∧ η) ∧ ξ = ω ∧ (η ∧ ξ)
- anticommutative: ω ∧ η = (−1)
^{km}η ∧ ω for ω and η homogeneous of degrees*k*and*m*respectively.