Exterior Algebra

The exterior algebra, or Grassmann algebra after Hermann Grassmann, is the algebraic system whose product is the exterior product. The exterior algebra provides an algebraic setting in which to answer geometric questions. For instance, whereas blades have a concrete geometrical interpretation, objects in the exterior algebra can be manipulated according to a set of unambiguous rules. The exterior algebra contains objects that are not just k-blades, but sums of k-blades; such a sum is called a k-vector. The k-blades, because they are simple products of vectors, are called the simple elements of the algebra. The rank of any k-vector is defined to be the smallest number of simple elements of which it is a sum. The exterior product extends to the full exterior algebra, so that it makes sense to multiply any two elements of the algebra. Equipped with this product, the exterior algebra is an associative algebra, which means that α ∧ (β ∧ γ) = (α ∧ β) ∧ γ for any elements α, β, γ. The k-vectors have degree k, meaning that they are sums of products of k vectors. When elements of different degrees are multiplied, the degrees add like multiplication of polynomials. This means that the exterior algebra is a graded algebra.

In a precise sense, given by what is known as a universal construction, the exterior algebra is the largest algebra that supports an alternating product on vectors, and can be easily defined in terms of other known objects such as tensors. The definition of the exterior algebra makes sense for spaces not just of geometric vectors, but of other vector-like objects such as vector fields or functions. In full generality, the exterior algebra can be defined for modules over a commutative ring, and for other structures of interest in abstract algebra. It is one of these more general constructions where the exterior algebra finds one of its most important applications, where it appears as the algebra of differential forms that is fundamental in areas that use differential geometry. Differential forms are mathematical objects that represent infinitesimal areas of infinitesimal parallelograms (and higher-dimensional bodies), and so can be integrated over surfaces and higher dimensional manifolds in a way that generalizes the line integrals from calculus. The exterior algebra also has many algebraic properties that make it a convenient tool in algebra itself. The association of the exterior algebra to a vector space is a type of functor on vector spaces, which means that it is compatible in a certain way with linear transformations of vector spaces. The exterior algebra is one example of a bialgebra, meaning that its dual space also possesses a product, and this dual product is compatible with the exterior product. This dual algebra is precisely the algebra of alternating multilinear forms on V, and the pairing between the exterior algebra and its dual is given by the interior product.

Read more about Exterior Algebra:  Formal Definitions and Algebraic Properties, Functoriality, The Alternating Tensor Algebra, History

Other articles related to "exterior algebra, exterior, algebra":

Exterior Algebra - History
... The exterior algebra was first introduced by Hermann Grassmann in 1844 under the blanket term of Ausdehnungslehre, or Theory of Extension ... Saint-Venant also published similar ideas of exterior calculus for which he claimed priority over Grassmann ... The algebra itself was built from a set of rules, or axioms, capturing the formal aspects of Cayley and Sylvester's theory of multivectors ...
Berezin Integral - Integration On An Exterior Algebra
... Let be the exterior algebra of polynomials in anticommuting elements over the field of complex numbers ... of the generators is fixed and defines the orientation of the exterior algebra.) The Berezin integral on is the linear functional with the following properties for any where means the left or ...
Generalizations Of The Derivative - Derivatives in Geometry - Differential Topology
... On the exterior algebra of differential forms over a smooth manifold, the exterior derivative is the unique linear map which satisfies a graded version of the Leibniz law and squares to zero ... It is a grade 1 derivation on the exterior algebra ... fields, it is an example of a Lie bracket (vector fields form the Lie algebra of the diffeomorphism group of the manifold) ...
Grassmann Number - Exterior Algebra
... The Grassmann algebra is the exterior algebra of the vector space spanned by the generators ... The exterior algebra is defined independent of a choice of basis ...
Hermann Grassmann - Mathematician
... Works of 1894–1911, contains the first known appearance of what are now called linear algebra and the notion of a vector space ... Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows “ The definition of a linear space (vector space).. ... in a way which is astonishingly similar to the presentation one finds in modern linear algebra texts ...

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