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.