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:
- <math>\Lambda^\bullet V = \bigoplus_{k=0}^{\infty} \Lambda^k V<math>
The subspaces ΛkV, consisting of all rank k alternating tensors, are called the kth 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 : Tk(V) → Λk(V) is the alternating projection or antisymmetrization operation defined as follows:
- <math>{\rm Alt}(\omega)=\frac{1}{k!}\sum_{\sigma\in S_k}{\rm sgn}(\sigma)\,\omega(x_{\sigma(1)}, \cdots , x_{\sigma(k)})<math>
That is, Alt(ω) is just the signed average of all permutations (σ in Sk) 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:
- <math>\omega\wedge\eta=\frac{(k+m)!}{k!\,m!}{\rm Alt}(\omega\otimes\eta)<math>
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
- <math>\omega\wedge\eta={\rm Alt}(\omega\otimes\eta)<math>
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
- (aω + bη) ∧ ξ = a(ω ∧ ξ) + b(η ∧ ξ)
- ξ ∧ (aω + bη) = 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.