K-monoid

Exterior algebra

The exterior algebra of a vector space is the freëst alternating K-monoid containing , falg as formalized by the Universal property.

The exterior algebra is in a sense generalized by, or rather quantized by, the Clifford algebra. Conceptually similar is the Symmetric algebra.

Universal property

Let be a vector space over the associated exterior algebra is a pair consisting of an alternating K-monoid and a linear map such that given any unital associative algebra , a linear map satisfying the identity factorizes uniqely through

such that is a unital algebra homomorphism. This admits a unique extension to a functor such that becomes a natural transformation.

Construction

The exterior algebra may be constructed as a quotient of the tensor algebra

where the divisor is the algebra ideal generated by tensors of the form , where the wedge product is the quotient algebra product.

Proof of universal property

proof

Graded structure

Like the tensor algebra, the exterior algebra is -graded into exterior powers

such that . If is a basis for , then

is a basis for , hence

Elements of the form where are called -blades, whereas -vectors are in general linear combinations of -blades. The distinction is the same as that of separable and entangled tensors. In particular, if then

• An -vector is a pseudovector ()
• An -vector is a pseudoscalar ()

Geometric interpretation

Geometrically, the magnitude of a -blade represents the -hypervolume of the -hyperparallelotope spanned defined by some vectors. Hence it generalizes the cross product, which can be thought of as resulting from the linear isomorphism from to , which is natural if is taken as an oriented vector space.

As antisymmetric tensors

Let be the ^graded natural projection. If is invertible in the ground field, in particular if [[Characteristic|]], then may be identified as a vector space with the subspace of [[Tensor algebra|]] consisting of antisymmetric tensors via the linear section

or more generally for homogenous vectors

This is just the Antisymmetrization and symmetrization of tensors factored via the Universal property.

Properties

• The -blades satisfy the Plücker relations, enabling the Plücker embedding of the Grassmannian into the Projectivization

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