Exterior product

In mathematics, the exterior algebra of a vector space (or, more generally, of a module over a commutative ring) is an associative algebra that contains the vector space (or module), and such that the square of any element of the vector space (or module) is zero. The exterior algebra is universal in the sense that every embedding of the vector space or module in an algebra that has these properties may be factored through the exterior algebra.

The multiplication operation of the exterior algebra is called the exterior product or wedge product, and is denoted with the symbol ${\displaystyle \wedge }$. The term "exterior" comes from the exterior product of two vectors not being a vector, while the term "wedge" comes from the shape of the multiplication symbol. The exterior algebra is also named Grassmann algebra after Hermann Grassmann, who introduced it as extended algebras. The exterior product should not be confused with the outer product, which is the tensor product of vectors.

The exterior product of two vectors is sometimes called a 2-blade, which is in turn a bivector. More generally, the exterior product of any number k of vectors is sometimes called a k-blade. Given a vector space (or a module) V, its exterior algebra is denoted ${\displaystyle \Lambda (V)}$. The vector subspace generated by the k-blades is known as the kth exterior power of V, and denoted ${\displaystyle \Lambda ^{k}(V)}$. The exterior algebra ${\displaystyle \Lambda (V)}$ is the direct sum of the ${\displaystyle \Lambda ^{k}(V)}$ as modules with the exterior product as additional structure. The exterior product makes the exterior algebra a graded algebra, and is alternating.

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