-monoid

Exterior algebra

The exterior algebra of a vector space is the freëst alternating -monoid containing , #m/def/ralg 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 -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.

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 , then may be identified as a vector space with the subspace of 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

#state/tidy | #lang/en | #SemBr