Exterior algebra

The Grassmann algebra or exterior algebra of a vector space V is an associative, skew-symmetric - graded algebra with unit element. She is - depending on the definition - subalgebra or a factor algebra of an antisymmetrized tensor algebra of V and is represented by. The multiplication is called the outer product, wedge product, wedge product or wedge product. A special case of this product is related to the cross product. Applies this calculation not only in elementary linear algebra (for example, in the theory of determinants) but mainly in algebraic geometry and differential geometry than algebra of differential forms. In this form, the theory of alternating differential forms on Élie Cartan goes back, so the unified the existing concepts of the theory of surfaces. Antikommutative products of vectors, as well as abstract vector spaces in general were first considered in 1846 by Hermann Grassmann.

• 2.1 The outer product
• 2.2 graduation, base and dimension
• 2.3 Universal property

• 5.1 Relationship to the cross product and triple product ( Hodge duality of vectors ) and concepts of physics
• 5.2 Relationship to the determinant theory; Extent dimension of m- vectors

Definition

Exterior power

It is a vector space over a field. Next was

( with the conventions and ). The subspace is generated by Elementartensoren in which two factors are equal:

The external power is then defined as the ratio of space

Exterior algebra

The direct sum

Is a two-sided homogeneous ideal in the tensor algebra

The exterior algebra is the factor algebra

Construed as a vector space this is isomorphic to

(For, see below. ) The product in the exterior algebra is traditionally written as.

Analogously, one can define the external algebra of modules over commutative rings.

Alternating tensors

In addition to the above-mentioned definition of the outer algebra, there are other equivalent ways to define the outer algebra. For example, you can understand the elements of the external algebra as alternating tensors. The following are the characteristic of the field is equal to 0

In the homogeneous components respectively the symmetric group operates. A tensor is called alternating if

For all permutations applies ( is the sign of the permutation ). The vector space of alternating tensors of the stage was.

You can assign (or " Alternator" ) on canonical way an alternating tensor each tensor using the Antisymmetrisierungsabbildung. It is defined by

With the product

For bilinear and continuation arises in a total space of alternating tensors associative, anticommutative - graded algebra. The canonical map is a Algebrenisomorphismus.

Properties

This section discusses the main characteristics of the external algebra as their graduation and the universal property and on their product. A prerequisite for getting that one -dimensional vector space.

The outer product

The product of the exterior algebra is associative. It is also commutative - graded, which means that it applies

For and. In particular, for all, but in general, for with just.

In the terminology of Super geometry is used instead of commutative - graded the equivalent term superkommutativ and using the Superkommutators can be expressed as the condition of Superkommutativität

For and.

If a shape and a shape that is the explicit formula for the outer product of finite dimensional vector spaces and for arbitrary ( infinite-dimensional Banach spaces and for ):

The symmetric group of order and the sign of the permutation to represent.

Graduation, base and dimension

The exterior algebra

Is a graded algebra. That is, they can be represented as a direct sum of subalgebras. For the exterior algebra, this follows directly from the definition. The external powers are the corresponding subalgebras.

Let now a basis of the vector space. Then

A basis of. The dimension. In particular, is necessary.

The base of the outer algebra is then obtained by combining the bases of all grades. Then apply to the dimension

Where the binomial coefficients called. It follows that each element of the Grassmann algebra can be represented as

Wherein the coefficients characterize the element with respect to a base and.

As an example, one can choose the vector space with the canonical basis. The third level of the outer algebra spanned by:

By counting one sees that is.

Universal property

Is a vector space (or module) and an associative algebra, so there is a bijection between

• The homomorphisms of vector spaces (or modules ), such that for all

And

• The Algebrenhomomorphismen.

Scalar product

Is an inner product of the vector space V, as well as the outer algebra can be provided with such. This subspaces of different degrees defined as orthogonal. Within a sub- area, it is sufficient to define the inner product in the pure products. Be pure and products. You may receive the Gram matrix of scalar products are assigned. Then the scalar product can be defined as the determinant of the Gram matrix:

V is the n-dimensional column vector space, may be defined to be the matrix. From this one can consider the maximal square submatrices. This is a multi- index of

And consists of exactly those rows of A.

There is the following identity for the set of Binet - Cauchy, in the case m = 2 and A = B also called " Flächenpythagoras ":

Differential forms

The main field of application of the external algebra lies in differential geometry. Let be a -dimensional differentiable manifold. How to Choose the cotangent space of this manifold as the underlying vector space and forms the outer algebra. A differential form is a section in the bundle of the vector space, Thus, a mapping that maps each point of an element of the outer manifold via the algebra cotangent at this point. These forms have the great advantage that you can integrate cards independently on a manifold with their help.

Hodge operator

Let ( as above) is a vector space and the external algebra of. Be an oriented basis of. The Hodge operator or Hodge star operator is a natural isomorphism with. So the Hodge operator assigns a to uniquely to ω, the so-called "dual element ". For this

Since scalars are dual to the given unit vector.

Relationship with the cross product and triple product ( Hodge duality of vectors ) and concepts of physics

Be the canonical basis of and be two elements from the exterior algebra (or external power) the real vector space. With the Hodge operator is called. For the outer product of and is using the distributive

The Hodge operator rotates in three-dimensional space the product of the basis vectors to the vector. By cyclically interchanging the indices, the assignments of the other basis vectors arise. This results in the cross product in three-dimensional real space. So you can understand on the external algebra as a generalization of the cross product. Using this generalization can also generalize the well-known from the vector analysis differential operator rotation on the n- dimensional case.

The scalar triple product of three vectors can be conceived as an element corresponding to the third external potency. Note that the star Hodge operator is defined only with respect to a base ( or alternatively, with respect to an inner product, which ensures the existence of an orthonormal basis). The outer product, however, can be independent of such a choice set.

The classical physics that were found sizes that are called in physics pseudo vectors, such as a magnetic field strength or angular momentum, can be used as elements of understand. With a pseudo-scalar is meant in many cases, a size that can be understood as an element of.

Relationship to determinants theory; Extent dimension of m- vectors

Even easier the associated with the Hodge operator concept is the duality in angelfish: These are the dual of the determinant of a matrix. In particular:

It should apply the same conditions as in the previous section; was only now admitted, and it is now if, (ie a sum of elementary m- legs) is given for an m- leg shape, then results as above, the antisymmetrized product, up to an alternating sign, the orientation of the respective dependent ( " handedness " versus " left-handed " ), the Hyperflächenmaß of m- leg dual of the respective " base direction ", ie its m -dimensional " volume " in the same time or that term represents a minor of a matrix with m columns and n rows represent. thus obtained in an elementary way, namely because of the multi -linearity and multi- associativity of the given expression, the known determinants of development rates. In particular, the volume measure thus generated ( = Grundflächenmaß times height ) of each parallelepiped invariant to displacements parallel to the base, because determinants of linearly dependent vectors disappear.

Relationship with the Clifford algebra

Let be a symmetric bilinear form on.

Now is the two-digit, bilinear combination

Defined by

For. The hats on the factors mean here the omission in the product. By introducing this new link as multiplication one obtains the Clifford algebra. In particular, we obtain the Nullbilinearform again the Grassmann algebra, since the additional term is omitted in the above equation and thus applies.