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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: \(e_{(v_{1},\ldots ,v_{i-1},v+w,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{n})}-e_{(v_{1},\ldots ,v_{i-1},w,v_{i+1},\ldots ,v_{n})}\) for arbitrary \(v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n},v,w\in V\).
  • Scalar Multiplication \(e_{(v_{1},\ldots ,v_{i-1},av,v_{i+1},\ldots ,v_{n})}-ae_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{n})}\) for arbitrary \(v_{1},\ldots ,v_{i-1},v_{i+1},\ldots ,v_{n},v\in V\) und \(a\in K\).
  • Alternating property \(e_{(v_{1},\ldots ,v_{i-1},v,v_{i+1},\ldots ,v_{j-1},v,v_{j+1},\ldots ,v_{n})}\) for \(i < j\) and arbitrary \(v_{1},\ldots, v_{i-1},v_{i+1},\ldots,v_{j-1},v_{j+1},\ldots,v_{n},v\in V\).

• Set \(\bigwedge ^{n}V:=H/U\,,\) as the quotient space defined by the equivalence relation induced by \(U\) on \(H\) .

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References

Adapted from CC BY-SA 3.0 Sources:

1. Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück