# 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 $$H$$ .

Definitions: 1

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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