Exterior product

A fundamental operation in the exterior algebra of tensors defined on an $ n $- dimensional vector space $ V $ over a field $ K $.

Let $ e _ {1} \dots e _ {n} $ be a basis of $ V $, and let $ a $ and $ b $ be $ p $- and $ q $- forms:

$$ a = a ^ {i _ {1} \dots {i _ {p} } } e _ {i _ {1} } \otimes \dots \otimes e _ {i _ {p} } , $$

$$ b = b ^ {j _ {1} \dots {j _ {q} } } e _ {j _ {1} } \otimes \dots \otimes e _ {j _ {q} } . $$

The exterior product of the forms $ a $ and $ b $ is the $ ( p + q) $- form $ c $ obtained by alternation of the tensor product $ a \otimes b $. The form $ c $ is denoted by $ a \wedge b $; its coordinates are skew-symmetric:

$$ c ^ {k _ {1} \dots k _ {p+ q } } = \ \frac{1}{p! q! } \delta _ {i _ {1} \dots i _ {p} j _ {1} \dots j _ {q} } ^ {k _ {1} \dots \dots \dots k _ {p+ q } } a ^ {i _ {1} \dots i _ {p} } b ^ {j _ {1} \dots j _ {q} } , $$

where $ \delta _ {i _ {1} \dots j _ {q} } ^ {k _ {1} \dots k _ {p+} q } $ are the components of the generalized Kronecker symbol. The exterior product of covariant tensors is defined in a similar manner.

The basic properties of the exterior product are listed below:

1) $ ( ka) \wedge b = a \wedge ( kb) = k( a \wedge b) $, $ k \in K $( homogeneity);

2) $ ( a+ b) \wedge c = a \wedge c + b \wedge c $( distributivity);

3) $ ( a \wedge b ) \wedge c = a \wedge ( b \wedge c) $( associativity).

4) $ a \wedge b = (- 1) ^ {pq} b \wedge a $; if the characteristic of $ K $ is distinct from two, the equation $ a \wedge a = 0 $ is valid for any form $ a $ of odd valency.

The exterior product of $ s $ vectors is said to be a decomposable $ s $- vector. Any poly-vector of dimension $ s $ is a linear combination of decomposable $ s $- vectors. The components of this combination are the ( $ s \times s $)- minors of the ( $ n \times s $)- matrix $ ( a _ {j} ^ {i} ) $, $ 1 \leq i \leq n $, $ 1 \leq j \leq s $, of the coefficients of the vectors $ a _ {1} \dots a _ {s} $. If $ s = n $ their exterior product has the form

$$ \alpha _ {n} = a _ {1} \wedge \dots \wedge a _ {n} = \ \mathop{\rm det} ( a _ {j} ^ {i} ) e _ {1} \wedge \dots \wedge e _ {n} . $$

Over fields of characteristic distinct from two, the equation $ a _ {1} \wedge \dots \wedge a _ {n} = 0 $ is necessary and sufficient for vectors $ a _ {1} \dots a _ {n} $ to be linearly dependent. A non-zero decomposable $ s $- vector $ \alpha _ {s} $ defines in $ V $ an $ s $- dimensional oriented subspace $ A $, parallel to the vectors $ a _ {1} \dots a _ {s} $, and the parallelotope in $ A $ formed by the vectors $ a _ {1} \dots a _ {s} $ issuing from one point, denoted by $ [ a _ {1} \dots a _ {s} ] $. The conditions $ a \in A $ and $ \alpha _ {s} \wedge a = 0 $ are equivalent.

For references see Exterior algebra.

Comments

Instead of exterior product the phrase "outer product" is sometimes used. The condition $ a \wedge b = (- 1) ^ {pq} b \wedge a $ for $ a $ of degree $ p $ and $ b $ of degree $ q $ is sometimes called graded commutativity.