◀ ▲ ▶Branches / Algebra / Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
Let \(V\) be a vector space over a field \(F\) and let \(n\in \mathbb {N} \) be a natural number. The exterior algebra or the \(n\)th alternating product of \(V\), denoted by \(\bigwedge ^{n}V\), is a vector space over \(F\) constructed as follows:
 Consider the set \(S\) of all \(n\)tuples \({(v_{1},\ldots ,v_{n})}{\text{ mit }}v_{i}\in V\) and the corresponding basis vectors \(e_{(v_{1},\ldots ,v_{n})}\).
 Build a vector space \(H=K^{(S)}\,,\) generated by the generating system \(a_{1}e_{s_{1}}+\cdots +a_{k}e_{s_{k}}\) with \(a_{i}\in F\) and \(s_{i}\in S,\).

In \(H\), consider the subspace \(U\subseteq H\) generated by the following elements:
 Additivity: \(e_{(v_{1},\ldots ,v_{i1},v+w,v_{i+1},\ldots ,v_{n})}e_{(v_{1},\ldots ,v_{i1},v,v_{i+1},\ldots ,v_{n})}e_{(v_{1},\ldots ,v_{i1},w,v_{i+1},\ldots ,v_{n})}\) for arbitrary \(v_{1},\ldots ,v_{i1},v_{i+1},\ldots ,v_{n},v,w\in V\).
 Scalar Multiplication \(e_{(v_{1},\ldots ,v_{i1},av,v_{i+1},\ldots ,v_{n})}ae_{(v_{1},\ldots ,v_{i1},v,v_{i+1},\ldots ,v_{n})}\) for arbitrary \(v_{1},\ldots ,v_{i1},v_{i+1},\ldots ,v_{n},v\in V\) und \(a\in K\).
 Alternating property \(e_{(v_{1},\ldots ,v_{i1},v,v_{i+1},\ldots ,v_{j1},v,v_{j+1},\ldots ,v_{n})}\) for \(i < j\) and arbitrary \(v_{1},\ldots, v_{i1},v_{i+1},\ldots,v_{j1},v_{j+1},\ldots,v_{n},v\in V\).

Set \(\bigwedge ^{n}V:=H/U\,,\) as the quotient space defined by the equivalence relation induced by \(U\) on \(H\) .
Mentioned in:
Definitions: 1
Thank you to the contributors under CC BYSA 4.0!
 Github:

 nonGithub:
 @Brenner
References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück