Exterior algebra

In mathematics, the exterior algebra or Grassmann algebra of a vector space $V$ is an associative algebra that contains $V,$ which has a product, called exterior product or wedge product and denoted with $\backslash wedge$, such that $v\backslash wedge\; v=0$ for every vector $v$ in $V.$ The exterior algebra is named after Hermann Grassmann, introduced these as extended algebras (cf. ). and the names of the product come from the "wedge" symbol $\backslash wedge$ and the fact that the product of two elements of $V$ is "outside" $V.$

The wedge product of $k$ vectors $v\_1\; \backslash wedge\; v\_2\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_k$ is called a blade of degree $k$ or $k$-blade. The wedge product was introduced originally as an algebraic construction used in geometry to study , , and their higher-dimensional analogues: the magnitude of a bivector $v\backslash wedge\; w$ is the area of the parallelogram defined by $v$ and $w,$ and, more generally, the magnitude of a $k$-blade is the (hyper)volume of the parallelotope defined by the constituent vectors. The alternating property that $v\backslash wedge\; v=0$ implies a skew-symmetric property that $v\; \backslash wedge\; w\; =\; -w\; \backslash wedge\; v,$ and more generally any blade flips sign whenever two of its constituent vectors are exchanged, corresponding to a parallelotope of opposite orientation.

The full exterior algebra contains objects that are not themselves blades, but linear combinations of blades; a sum of blades of homogeneous degree $k$ is called a Multivector, while a more general sum of blades of arbitrary degree is called a multivector.The term k-vector is not equivalent to and should not be confused with similar terms such as 4-vector, which in a different context could mean an element of a 4-dimensional vector space. A minority of authors use the term $k$-multivector instead of $k$-vector, which avoids this confusion. The linear span of the $k$-blades is called the $k$- th exterior power of $V.$ The exterior algebra is the direct sum of the $k$-th exterior powers of $V,$ and this makes the exterior algebra a graded algebra.

The exterior algebra is universal in the sense that every equation that relates elements of $V$ in the exterior algebra is also valid in every associative algebra that contains $V$ and in which the square of every element of $V$ is zero.

The definition of the exterior algebra can be extended for spaces built from vector spaces, such as and functions whose domain is a vector space. Moreover, the field of scalars may be any field. More generally, the exterior algebra can be defined for modules over a commutative ring. In particular, the algebra of differential forms in $k$ variables is an exterior algebra over the ring of the in $k$ variables.

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Motivating examples

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Areas in the plane The two-dimensional Euclidean vector space $\backslash mathbf\{R\}^2$ is a Real number vector space equipped with a basis consisting of a pair of orthogonal $$\backslash mathbf\{e\}\_1\; =\; \backslash begin\{bmatrix\}1\backslash \backslash 0\backslash end\{bmatrix\},\backslash quad\; \backslash mathbf\{e\}\_2\; =\; \backslash begin\{bmatrix\}0\backslash \backslash 1\backslash end\{bmatrix\}.$$

Suppose that $$\backslash mathbf\{v\}\; =\; \backslash begin\{bmatrix\}a\backslash \backslash b\backslash end\{bmatrix\}\; =\; a\; \backslash ,\backslash mathbf\{e\}\_1\; +\; b\; \backslash ,\backslash mathbf\{e\}\_2,\; \backslash quad\; \backslash mathbf\{w\}\; =\; \backslash begin\{bmatrix\}c\backslash \backslash d\backslash end\{bmatrix\}\; =\; c\; \backslash ,\backslash mathbf\{e\}\_1\; +\; d\; \backslash ,\backslash mathbf\{e\}\_2$$ are a pair of given vectors in , written in components. There is a unique parallelogram having $\backslash mathbf\{v\}$ and $\backslash mathbf\{w\}$ as two of its sides. The area of this parallelogram is given by the standard determinant formula:

Consider now the exterior product of $\backslash mathbf\{v\}$ and : where the first step uses the distributive law for the exterior product. The second one uses the fact that the exterior product is an alternating map, i.e., $\backslash mathbf\{e\}\_1\; \backslash wedge\; \backslash mathbf\{e\}\_1\; =\; \backslash mathbf\{e\}\_2\; \backslash wedge\; \backslash mathbf\{e\}\_2\; =\; 0.$ Being alternating also implies being anticommutative, $\backslash mathbf\{e\}\_2\; \backslash wedge\; \backslash mathbf\{e\}\_1\; =\; -(\backslash mathbf\{e\}\_1\; \backslash wedge\; \backslash mathbf\{e\}\_2)$, which gives the last line. Note that the coefficient in this last expression is precisely the determinant of the matrix . The fact that this may be positive or negative has the intuitive meaning that v and w may be oriented in a counterclockwise or clockwise sense as the vertices of the parallelogram they define. Such an area is called the signed area of the parallelogram: the absolute value of the signed area is the ordinary area, and the sign determines its orientation.

The fact that this coefficient is the signed area is not an accident. In fact, it is relatively easy to see that the exterior product should be related to the signed area if one tries to axiomatize this area as an algebraic construct. In detail, if denotes the signed area of the parallelogram of which the pair of vectors v and w form two adjacent sides, then A must satisfy the following properties:

1. for any real numbers r and s, since rescaling either of the sides rescales the area by the same amount (and reversing the direction of one of the sides reverses the orientation of the parallelogram).
2. , since the area of the degenerate parallelogram determined by v (i.e., a line segment) is zero.
3. , since interchanging the roles of v and w reverses the orientation of the parallelogram.
4. for any real number r, since adding a multiple of w to v affects neither the base nor the height of the parallelogram and consequently preserves its area.
5. , since the area of the unit square is one.

With the exception of the last property, the exterior product of two vectors satisfies the same properties as the area. In a certain sense, the exterior product generalizes the final property by allowing the area of a parallelogram to be compared to that of any chosen parallelogram in a parallel plane (here, the one with sides e₁ and e₂). In other words, the exterior product provides a basis-independent formulation of area.This axiomatization of areas is due to Leopold Kronecker and Karl Weierstrass; see . For a modern treatment, see . For an elementary treatment, see .

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Cross and triple products For vectors in $\backslash mathbb\{R\}^3$, the exterior algebra is closely related to the cross product and triple product. Using the standard basis $\backslash \{\backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3\backslash \}$, the exterior product of a pair of vectors $$\backslash mathbf\{u\}\; =\; u\_1\; \backslash mathbf\{e\}\_1\; +\; u\_2\; \backslash mathbf\{e\}\_2\; +\; u\_3\; \backslash mathbf\{e\}\_3$$ and $$\backslash mathbf\{v\}\; =\; v\_1\; \backslash mathbf\{e\}\_1\; +\; v\_2\; \backslash mathbf\{e\}\_2\; +\; v\_3\; \backslash mathbf\{e\}\_3$$ is where $\backslash \{\backslash mathbf\{e\}\_1\backslash wedge\backslash mathbf\{e\}\_2,\backslash mathbf\{e\}\_3\backslash wedge\backslash mathbf\{e\}\_1,\backslash mathbf\{e\}\_2\backslash wedge\backslash mathbf\{e\}\_3\backslash \}$ is the natural basis for the three-dimensional space $\backslash Lambda^2(\backslash mathbb\{R\}^3)$. The coefficients above are the same as those in the usual definition of the cross product of vectors in three dimensions, the only difference being that the exterior product is not an ordinary vector, but instead is a bivector.

Bringing in a third vector $$\backslash mathbf\{w\}\; =\; w\_1\; \backslash mathbf\{e\}\_1\; +\; w\_2\; \backslash mathbf\{e\}\_2\; +\; w\_3\; \backslash mathbf\{e\}\_3,$$ the exterior product of three vectors is $$\backslash mathbf\{u\}\; \backslash wedge\; \backslash mathbf\{v\}\; \backslash wedge\; \backslash mathbf\{w\}\; =\; (u\_1\; v\_2\; w\_3\; +\; u\_2\; v\_3\; w\_1\; +\; u\_3\; v\_1\; w\_2\; -\; u\_1\; v\_3\; w\_2\; -\; u\_2\; v\_1\; w\_3\; -\; u\_3\; v\_2\; w\_1)\; (\backslash mathbf\{e\}\_1\; \backslash wedge\; \backslash mathbf\{e\}\_2\; \backslash wedge\; \backslash mathbf\{e\}\_3)$$ where $\backslash mathbf\{e\}\_1\backslash wedge\backslash mathbf\{e\}\_2\backslash wedge\backslash mathbf\{e\}\_3$ is the basis vector for the one-dimensional space $\backslash Lambda^3(\backslash mathbb\{R\}^3)$. The scalar coefficient is the triple product of the three vectors.

The cross product and triple product in three dimensions each admit both geometric and algebraic interpretations. The cross product $\backslash mathbf\{u\}\backslash times\backslash mathbf\{v\}$ can be interpreted as a vector which is perpendicular to both $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$ and whose magnitude is equal to the area of the parallelogram determined by the two vectors. It can also be interpreted as the vector consisting of the minors of the matrix with columns $\backslash mathbf\{u\}$ and $\backslash mathbf\{v\}$. The triple product of $\backslash mathbf\{u\}$, $\backslash mathbf\{v\}$, and $\backslash mathbf\{w\}$ is geometrically a (signed) volume. Algebraically, it is the determinant of the matrix with columns $\backslash mathbf\{u\}$, $\backslash mathbf\{v\}$, and $\backslash mathbf\{w\}$. The exterior product in three dimensions allows for similar interpretations. In fact, in the presence of a positively oriented orthonormal basis, the exterior product generalizes these notions to higher dimensions.

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Formal definition The exterior algebra $\backslash bigwedge\; (V)$ of a vector space $V$ over a field $K$ is defined as the Quotient ring of the tensor algebra T( V), where

$T(V)=\; \backslash bigoplus\_\{k=0\}^\backslash infty\; T^kV\; =\; K\backslash oplus\; V\; \backslash oplus\; (V\backslash otimes\; V)\; \backslash oplus\; (V\backslash otimes\; V\backslash otimes\; V)\; \backslash oplus\; \backslash cdots,$

by the two-sided ideal $I$ generated by all elements of the form $x\; \backslash otimes\; x$ such that $x\backslash in\; V$. This definition is a standard one. See, for instance, . Symbolically,

$\backslash bigwedge\; (V)\; :=\; T(V)/I.\backslash ,$

The exterior product $\backslash wedge$ of two elements of $\backslash bigwedge\; (V)$ is defined by

$\backslash alpha\backslash wedge\backslash beta\; =\; \backslash alpha\backslash otimes\backslash beta\; \backslash pmod\; I.$

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Algebraic properties

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Alternating product The exterior product is by construction alternating on elements of , which means that $x\; \backslash wedge\; x\; =\; 0$ for all $x\; \backslash in\; V,$ by the above construction. It follows that the product is also anticommutative on elements of , for supposing that ,

$$

0 = (x + y) \wedge (x + y) = x \wedge x + x \wedge y + y \wedge x + y \wedge y = x \wedge y + y \wedge x hence

$x\; \backslash wedge\; y\; =\; -(y\; \backslash wedge\; x).$

More generally, if $\backslash sigma$ is a permutation of the integers , and , , ..., are elements of , it follows that

$$

x_{\sigma(1)} \wedge x_{\sigma(2)} \wedge \cdots \wedge x_{\sigma(k)} = \sgn(\sigma)x_1 \wedge x_2 \wedge \cdots \wedge x_k, where $\backslash sgn(\backslash sigma)$ is the signature of the permutation .A proof of this can be found in more generality in .

In particular, if $x\_i\; =\; x\_j$ for some , then the following generalization of the alternating property also holds:

$x\_\{1\}\; \backslash wedge\; x\_\{2\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; x\_\{k\}\; =\; 0.$

Together with the distributive property of the exterior product, one further generalization is that a necessary and sufficient condition for $\backslash \{\; x\_\{1\},\; x\_\{2\},\; \backslash dots,\; x\_\{k\}\; \backslash \}$ to be a linearly dependent set of vectors is that

$x\_\{1\}\; \backslash wedge\; x\_\{2\}\; \backslash wedge\; \backslash cdots\; \backslash wedge\; x\_\{k\}\; =\; 0.$

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Exterior power The th exterior power of , denoted , is the vector subspace of linear span by elements of the form

$x\_1\; \backslash wedge\; x\_2\; \backslash wedge\; \backslash cdots\; \backslash wedge\; x\_k,\backslash quad\; x\_i\; \backslash in\; V,\; i=1,2,\; \backslash dots,\; k\; .$

If , then $\backslash alpha$ is said to be a p-vector. If, furthermore, $\backslash alpha$ can be expressed as an exterior product of $k$ elements of , then $\backslash alpha$ is said to be decomposable (or  simple, by some authors; or a  blade, by others). Although decomposable -vectors span , not every element of $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ is decomposable. For example, given with a basis , the following 2-vector is not decomposable:

$\backslash alpha\; =\; e\_1\; \backslash wedge\; e\_2\; +\; e\_3\; \backslash wedge\; e\_4.$

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Basis and dimension If the dimension of $V$ is $n$ and $\backslash \{e\_1,\backslash dots,e\_n\backslash \}$ is a basis for $V$, then the set

$$

\{\,e_{i_1} \wedge e_{i_2} \wedge \cdots \wedge e_{i_k} ~ \big| ~~ 1 \le i_1 < i_2 < \cdots < i_k \le n \,\} is a basis for . The reason is the following: given any exterior product of the form

$v\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; v\_k\; ,$

every vector $v\_j$ can be written as a linear combination of the basis vectors ; using the bilinearity of the exterior product, this can be expanded to a linear combination of exterior products of those basis vectors. Any exterior product in which the same basis vector appears more than once is zero; any exterior product in which the basis vectors do not appear in the proper order can be reordered, changing the sign whenever two basis vectors change places. In general, the resulting coefficients of the basis -vectors can be computed as the minors of the matrix that describes the vectors $v\_j$ in terms of the basis .

By counting the basis elements, the dimension of $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ is equal to a binomial coefficient:

$\backslash dim\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; =\; \backslash binom\{n\}\{k\}\; ,$

where is the dimension of the vectors, and is the number of vectors in the product. The binomial coefficient produces the correct result, even for exceptional cases; in particular, $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; =\; \backslash \{\; 0\; \backslash \}$ for .

Any element of the exterior algebra can be written as a sum of p-vector. Hence, as a vector space the exterior algebra is a direct sum

$$

{\textstyle\bigwedge}(V) = {\textstyle\bigwedge}^{\!0}(V)

 \oplus {\textstyle\bigwedge}^{\!1}(V)
 \oplus {\textstyle\bigwedge}^{\!2}(V)
 \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V)
     

(where, by convention, , the field underlying , and ), and therefore its dimension is equal to the sum of the binomial coefficients, which is .

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Rank of a k-vector If , then it is possible to express $\backslash alpha$ as a linear combination of decomposable p-vector:

$\backslash alpha\; =\; \backslash alpha^\{(1)\}\; +\; \backslash alpha^\{(2)\}\; +\; \backslash cdots\; +\; \backslash alpha^\{(s)\}$

where each $\backslash alpha^\{(i)\}$ is decomposable, say

$$

\alpha^{(i)} = \alpha^{(i)}_1 \wedge \cdots \wedge \alpha^{(i)}_k,\quad i = 1,2,\ldots, s.

The rank of the -vector $\backslash alpha$ is the minimal number of decomposable -vectors in such an expansion of . This is similar to the notion of tensor rank.

Rank is particularly important in the study of 2-vectors . The rank of a 2-vector $\backslash alpha$ can be identified with half the rank of the matrix of coefficients of $\backslash alpha$ in a basis. Thus if $e\_i$ is a basis for , then $\backslash alpha$ can be expressed uniquely as

$\backslash alpha\; =\; \backslash sum\_\{i,j\}a\_\{ij\}e\_i\; \backslash wedge\; e\_j$

where $a\_\{ij\}\; =\; -a\_\{ji\}$ (the matrix of coefficients is skew-symmetric). The rank of the matrix $a\_\{ij\}$ is therefore even, and is twice the rank of the form $\backslash alpha$.

In characteristic 0, the 2-vector $\backslash alpha$ has rank $p$ if and only if

$$

\underset{p}{\underbrace{\alpha \wedge \cdots \wedge \alpha}} \neq 0 \ and $\backslash \; \backslash underset\{p+1\}\{\backslash underbrace\{\backslash alpha\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash alpha\}\}\; =\; 0.$

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Graded structure The exterior product of a -vector with a -vector is a $(k\; +\; p)$-vector, once again invoking bilinearity. As a consequence, the direct sum decomposition of the preceding section

$$

{\textstyle\bigwedge}(V) = {\textstyle\bigwedge}^{\!0}(V) \oplus {\textstyle\bigwedge}^{\!1}(V) \oplus {\textstyle\bigwedge}^{\!2}(V) \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V) gives the exterior algebra the additional structure of a graded algebra, that is

$$

{\textstyle\bigwedge}^{\!k}(V) \wedge {\textstyle\bigwedge}^{\!p}(V) \sub {\textstyle\bigwedge}^{\!k+p}(V).

Moreover, if is the base field, we have

$$

{\textstyle\bigwedge}^{\!0}(V) = K and $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !1\}(V)\; =\; V.$

The exterior product is graded anticommutative, meaning that if $\backslash alpha\; \backslash in\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ and , then

$\backslash alpha\; \backslash wedge\; \backslash beta\; =\; (-1)^\{kp\}\backslash beta\; \backslash wedge\; \backslash alpha.$

In addition to studying the graded structure on the exterior algebra, studies additional graded structures on exterior algebras, such as those on the exterior algebra of a graded module (a module that already carries its own gradation).

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Universal property Let be a vector space over the field . Informally, multiplication in $\{\backslash textstyle\backslash bigwedge\}(V)$ is performed by manipulating symbols and imposing a distributive law, an associative law, and using the identity $v\; \backslash wedge\; v\; =\; 0$ for . Formally, $\{\backslash textstyle\backslash bigwedge\}(V)$ is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative -algebra containing with alternating multiplication on must contain a homomorphic image of . In other words, the exterior algebra has the following universal property:See , and . More detail on universal properties in general can be found in , and throughout the works of Bourbaki.

To construct the most general algebra that contains and whose multiplication is alternating on , it is natural to start with the most general associative algebra that contains , the tensor algebra , and then enforce the alternating property by taking a suitable quotient ring. We thus take the two-sided ideal in generated by all elements of the form for in , and define $\{\backslash textstyle\backslash bigwedge\}(V)$ as the quotient

$\{\backslash textstyle\backslash bigwedge\}(V)\; =\; T(V)\backslash ,/\backslash ,I$

(and use as the symbol for multiplication in ). It is then straightforward to show that $\{\backslash textstyle\backslash bigwedge\}(V)$ contains and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space its exterior algebra $\{\backslash textstyle\backslash bigwedge\}(V)$ is a functor from the category of vector spaces to the category of algebras.

Rather than defining $\{\backslash textstyle\backslash bigwedge\}(V)$ first and then identifying the exterior powers $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ as certain subspaces, one may alternatively define the spaces $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ first and then combine them to form the algebra . This approach is often used in differential geometry and is described in the next section.

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Generalizations Given a commutative ring $R$ and an $R$-module , we can define the exterior algebra $\{\backslash textstyle\backslash bigwedge\}(M)$ just as above, as a suitable quotient of the tensor algebra . It will satisfy the analogous universal property. Many of the properties of $\{\backslash textstyle\backslash bigwedge\}(M)$ also require that $M$ be a projective module. Where finite dimensionality is used, the properties further require that $M$ be finitely generated and projective. Generalizations to the most common situations can be found in .

Exterior algebras of are frequently considered in geometry and topology. There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the Serre–Swan theorem. More general exterior algebras can be defined for sheaves of modules.

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Alternating tensor algebra For a field of characteristic not 2,See for generalizations. the exterior algebra of a vector space $V$ over $K$ can be canonically identified with the vector subspace of $\backslash mathrm\{T\}(V)$ that consists of antisymmetric tensors. For characteristic 0 (or higher than ), the vector space of $k$-linear antisymmetric tensors is transversal to the ideal , hence, a good choice to represent the quotient. But for nonzero characteristic, the vector space of -linear antisymmetric tensors could be not transversal to the ideal (actually, for , the vector space of $K$-linear antisymmetric tensors is contained in $I$); nevertheless, transversal or not, a product can be defined on this space such that the resulting algebra is isomorphic to the exterior algebra: in the first case the natural choice for the product is just the quotient product (using the available projection), in the second case, this product must be slightly modified as given below (along Arnold setting), but such that the algebra stays isomorphic with the exterior algebra, i.e. the quotient of $\backslash mathrm\{T\}(V)$ by the ideal $I$ generated by elements of the form . Of course, for characteristic (or higher than the dimension of the vector space), one or the other definition of the product could be used, as the two algebras are isomorphic (see V. I. Arnold or Kobayashi-Nomizu).

Let $\backslash mathrm\{T\}^r(V)$ be the space of homogeneous tensors of degree $r$. This is spanned by decomposable tensors

$v\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; v\_r,\backslash quad\; v\_i\; \backslash in\; V.$

The antisymmetrization (or sometimes the skew-symmetrization) of a decomposable tensor is defined by

$$

\operatorname{\mathcal{A}^{(r)}}(v_1 \otimes \cdots \otimes v_r) =\sum_{\sigma \in \mathfrak{S}_r} \operatorname{sgn}(\sigma) v_{\sigma(1)} \otimes \cdots \otimes v_{\sigma(r)} and, when $r!\; \backslash neq\; 0$ (for nonzero characteristic field $r!$ might be 0):

$$

\operatorname{Alt}^{(r)}(v_1 \otimes \cdots \otimes v_r) = \frac{1}{r!}\operatorname{\mathcal{A}^{(r)}}(v_1 \otimes \cdots \otimes v_r) where the sum is taken over the symmetric group of permutations on the symbols . This extends by linearity and homogeneity to an operation, also denoted by $\backslash mathcal\{A\}$ and $\backslash rm\{Alt\}$, on the full tensor algebra .

Note that

$$

\operatorname{\mathcal{A}^{(r)}}\operatorname{\mathcal{A}^{(r)}}=r!\operatorname{\mathcal{A}^{(r)}}.

Such that, when defined, $\backslash operatorname\{Alt\}^\{(r)\}$ is the projection for the exterior (quotient) algebra onto the r-homogeneous alternating tensor subspace. On the other hand, the image $\backslash mathcal\{A\}(\backslash mathrm\{T\}(V))$ is always the alternating tensor graded subspace (not yet an algebra, as product is not yet defined), denoted . This is a vector subspace of , and it inherits the structure of a graded vector space from that on . Moreover, the kernel of $\backslash mathcal\{A\}^\{(r)\}$ is precisely , the homogeneous subset of the ideal , or the kernel of $\backslash mathcal\{A\}$ is . When $\backslash operatorname\{Alt\}$ is defined, $A(V)$ carries an associative graded product $\backslash widehat\{\backslash otimes\}$ defined by (the same as the wedge product)

$t\backslash wedge\; s=t~\backslash widehat\{\backslash otimes\}~s\; =\; \backslash operatorname\{Alt\}(t\; \backslash otimes\; s).$

Assuming $K$ has characteristic 0, as $A(V)$ is a supplement of $I$ in , with the above given product, there is a canonical isomorphism

$A(V)\backslash cong\; \{\backslash textstyle\backslash bigwedge\}(V).$

When the characteristic of the field is nonzero, $\backslash mathcal\{A\}$ will do what $\backslash rm\{Alt\}$ did before, but the product cannot be defined as above. In such a case, isomorphism $A(V)\backslash cong\; \{\backslash textstyle\backslash bigwedge\}(V)$ still holds, in spite of $A(V)$ not being a supplement of the ideal , but then, the product should be modified as given below ($\backslash dot\{\backslash wedge\}$ product, Arnold setting).

Finally, we always get isomorphic with , but the product could (or should) be chosen in two ways (or only one). Actually, the product could be chosen in many ways, rescaling it on homogeneous spaces as $c(r+p)/c(r)c(p)$ for an arbitrary sequence $c(r)$ in the field, as long as the division makes sense (this is such that the redefined product is also associative, i.e. defines an algebra on ). Also note, the interior product definition should be changed accordingly, in order to keep its skew derivation property.

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Index notation Suppose that V has finite dimension n, and that a basis of V is given. Then any alternating tensor can be written in index notation with the Einstein summation convention as

$t\; =\; t^\{i\_1i\_2\backslash cdots\; i\_r\}\backslash ,\; \{\backslash mathbf\; e\}\_\{i\_1\}\; \backslash otimes\; \{\backslash mathbf\; e\}\_\{i\_2\}\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \{\backslash mathbf\; e\}\_\{i\_r\},$

where t ^{i₁⋅⋅⋅ i _r} is completely antisymmetric in its indices.

The exterior product of two alternating tensors t and s of ranks r and p is given by

$$

t~\widehat{\otimes}~s = \frac{1}{(r+p)!}\sum_{\sigma \in {\mathfrak S}_{r+p}}\operatorname{sgn}(\sigma)t^{i_{\sigma(1)} \cdots i_{\sigma(r)}} s^{i_{\sigma(r+1)} \cdots i_{\sigma(r+p)}} {\mathbf e}_{i_1} \otimes {\mathbf e}_{i_2} \otimes \cdots \otimes {\mathbf e}_{i_{r+p}}.

The components of this tensor are precisely the skew part of the components of the tensor product , denoted by square brackets on the indices:

$$

= t^{i_1\cdots}.

The interior product may also be described in index notation as follows. Let $t\; =\; t^\{i\_0i\_1\backslash cdots\; i\_\{r-1\}\}$ be an antisymmetric tensor of rank . Then, for , is an alternating tensor of rank , given by

$$

= r\sum_{j=0}^n\alpha_j t^{ji_1\cdots i_{r-1}}. where n is the dimension of V.

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Duality

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Alternating operators Given two vector spaces V and X and a natural number k, an alternating operator from V ^k to X is a multilinear map

$f\; :\; V^k\; \backslash to\; X$

such that whenever v₁, ..., v _k are linearly dependent vectors in V, then

$f(v\_1,\backslash ldots,\; v\_k)\; =\; 0.$

The map

$w\; :\; V^k\; \backslash to\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V),$

which associates to $k$ vectors from $V$ their exterior product, i.e. their corresponding $k$-vector, is also alternating. In fact, this map is the "most general" alternating operator defined on $V^k;$ given any other alternating operator $f\; :\; V^k\; \backslash rightarrow\; X,$ there exists a unique linear map $\backslash phi\; :\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; \backslash rightarrow\; X$ with $f\; =\; \backslash phi\; \backslash circ\; w.$ This universal property characterizes the space of alternating operators on $V^k$ and can serve as its definition.

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Alternating multilinear forms The above discussion specializes to the case when , the base field. In this case an alternating multilinear function

$f\; :\; V^k\; \backslash to\; K$

is called an alternating multilinear form. The set of all Alternating map is a vector space, as the sum of two such maps, or the product of such a map with a scalar, is again alternating. By the universal property of the exterior power, the space of alternating forms of degree $k$ on $V$ is naturally isomorphic with the dual vector space . If $V$ is finite-dimensional, then the latter is to . In particular, if $V$ is $n$-dimensional, the dimension of the space of alternating maps from $V^k$ to $K$ is the binomial coefficient .

Under such identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose and are two anti-symmetric maps. As in the case of of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. Depending on the choice of identification of elements of exterior power with multilinear forms, the exterior product is defined as

$\backslash omega\; \backslash wedge\; \backslash eta\; =\; \backslash operatorname\{Alt\}(\backslash omega\; \backslash otimes\; \backslash eta)$

or as

$$

\omega \dot{\wedge} \eta = \frac{(k+m)!}{k!\,m!}\operatorname{Alt}(\omega \otimes \eta), where, if the characteristic of the base field $K$ is 0, the alternation Alt of a multilinear map is defined to be the average of the sign-adjusted values over all the of its variables:

$$

\operatorname{Alt}(\omega)(x_1,\ldots,x_k) = \frac{1}{k!}\sum_{\sigma \in S_k}\operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)}).

When the field $K$ has finite characteristic, an equivalent version of the second expression without any factorials or any constants is well-defined:

$$

{\omega \dot{\wedge} \eta(x_1,\ldots,x_{k+m})} = \sum_{\sigma \in \mathrm{Sh}_{k,m}} \operatorname{sgn}(\sigma)\, \omega(x_{\sigma(1)}, \ldots, x_{\sigma(k)})\, \eta(x_{\sigma(k+1)}, \ldots, x_{\sigma(k+m)}), where here is the subset of shuffles: σ of the set such that , and . As this might look very specific and fine tuned, an equivalent raw version is to sum in the above formula over permutations in left cosets of .

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Interior product Suppose that $V$ is finite-dimensional. If $V^*$ denotes the dual space to the vector space , then for each , it is possible to define an antiderivation on the algebra ,

$$

\iota_\alpha : {\textstyle\bigwedge}^{\!k}(V) \rightarrow {\textstyle\bigwedge}^{\!k-1}(V) .

This derivation is called the interior product with , or sometimes the insertion operator, or contraction by .

Suppose that . Then $w$ is a multilinear mapping of $V^*$ to , so it is defined by its values on the -fold Cartesian product . If u₁, u₂, ..., u _k−1 are $k\; -\; 1$ elements of , then define

$$

(\iota_\alpha w)(u_1,u_2,\ldots,u_{k-1}) = w(\alpha,u_1,u_2,\ldots, u_{k-1}).

Additionally, let $\backslash iota\_\backslash alpha\; f\; =\; 0$ whenever $f$ is a pure scalar (i.e., belonging to ).

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Axiomatic characterization and properties The interior product satisfies the following properties:

1. For each and each (where by convention $\backslash Lambda^\{-1\}(V)=\backslash \{0\backslash \}$),
2. : $\backslash iota\_\backslash alpha\; :\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; \backslash rightarrow\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k-1\}(V)\; .$
3. If $v$ is an element of $V$ (), then is the dual pairing between elements of $V$ and elements of .
4. For each , $\backslash iota\_\backslash alpha$ is a graded derivation of degree −1:
5. : $\backslash iota\_\backslash alpha\; (a\; \backslash wedge\; b)$

= (\iota_\alpha a) \wedge b + (-1)^{\deg a}a \wedge (\iota_\alpha b).

These three properties are sufficient to characterize the interior product as well as define it in the general infinite-dimensional case.

Further properties of the interior product include:

• $\backslash iota\_\backslash alpha\backslash circ\; \backslash iota\_\backslash alpha\; =\; 0.$
• $\backslash iota\_\backslash alpha\backslash circ\; \backslash iota\_\backslash beta\; =\; -\backslash iota\_\backslash beta\backslash circ\; \backslash iota\_\backslash alpha.$

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Hodge duality Suppose that $V$ has finite dimension . Then the interior product induces a canonical isomorphism of vector spaces

$$

{\textstyle\bigwedge}^{\!k}(V^*) \otimes {\textstyle\bigwedge}^{\!n}(V) \to {\textstyle\bigwedge}^{\!n-k}(V) by the recursive definition

$\backslash iota\_\{\backslash alpha\; \backslash wedge\; \backslash beta\}\; =\; \backslash iota\_\backslash beta\; \backslash circ\; \backslash iota\_\backslash alpha.$

In the geometrical setting, a non-zero element of the top exterior power $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !n\}(V)$ (which is a one-dimensional vector space) is sometimes called a volume form (or orientation form, although this term may sometimes lead to ambiguity). The name orientation form comes from the fact that a choice of preferred top element determines an orientation of the whole exterior algebra, since it is tantamount to fixing an ordered basis of the vector space. Relative to the preferred volume form , the isomorphism is given explicitly by

$$

{\textstyle\bigwedge}^{\!k}(V^*) \to {\textstyle\bigwedge}^{\!n-k}(V)

\alpha \mapsto \iota_\alpha \sigma .

If, in addition to a volume form, the vector space V is equipped with an inner product identifying $V$ with , then the resulting isomorphism is called the Hodge star operator, which maps an element to its Hodge dual:

$\backslash star\; :\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; \backslash rightarrow\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !n-k\}(V)\; .$

The composition of $\backslash star$ with itself maps $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; \backslash to\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ and is always a scalar multiple of the identity map. In most applications, the volume form is compatible with the inner product in the sense that it is an exterior product of an orthonormal basis of . In this case,

$\backslash star\; \backslash circ\; \backslash star\; :\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; \backslash to\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)\; =\; (-1)^\{k(n-k)\; +\; q\}\backslash mathrm\{id\}$

where id is the identity mapping, and the inner product has metric signature — p pluses and q minuses.

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Inner product For a finite-dimensional space, an inner product (or a pseudo-Euclidean inner product) on defines an isomorphism of $V$ with , and so also an isomorphism of $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V)$ with . The pairing between these two spaces also takes the form of an inner product. On decomposable -vectors,

$$

\left\langle v_1 \wedge \cdots \wedge v_k, w_1 \wedge \cdots \wedge w_k\right\rangle = \det\bigl(\langle v_i,w_j\rangle\bigr),

the determinant of the matrix of inner products. In the special case , the inner product is the square norm of the k-vector, given by the determinant of the Gramian matrix . This is then extended bilinearly (or sesquilinearly in the complex case) to a non-degenerate inner product on $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}(V).$ If e _i, , form an orthonormal basis of , then the vectors of the form

constitute an orthonormal basis for , a statement equivalent to the Cauchy–Binet formula.

With respect to the inner product, exterior multiplication and the interior product are mutually adjoint. Specifically, for , , and ,

$$

\langle x \wedge \mathbf{v}, \mathbf{w}\rangle = \langle \mathbf{v}, \iota_{x^\flat}\mathbf{w}\rangle where is the musical isomorphism, the linear functional defined by

$x^\backslash flat(y)\; =\; \backslash langle\; x,\; y\backslash rangle$

for all . This property completely characterizes the inner product on the exterior algebra.

Indeed, more generally for , , and , iteration of the above adjoint properties gives

$$

\langle \mathbf{x} \wedge \mathbf{v}, \mathbf{w}\rangle = \langle \mathbf{v}, \iota_{\mathbf{x}^\flat}\mathbf{w}\rangle where now $\backslash mathbf\{x\}^\backslash flat\; \backslash in\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !l\}\backslash left(V^*\backslash right)\; \backslash simeq\; \backslash bigl(\{\backslash textstyle\backslash bigwedge\}^\{\backslash !l\}(V)\backslash bigr)^*$ is the dual -vector defined by

$$

\mathbf{x}^\flat(\mathbf{y}) = \langle \mathbf{x}, \mathbf{y}\rangle for all .

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Bialgebra structure There is a correspondence between the graded dual of the graded algebra $\{\backslash textstyle\backslash bigwedge\}(V)$ and alternating multilinear forms on . The exterior algebra (as well as the symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopf algebra structure, from the tensor algebra. See the article on for a detailed treatment of the topic.

The exterior product of multilinear forms defined above is dual to a coproduct defined on , giving the structure of a coalgebra. The coproduct is a linear function , which is given by

$\backslash Delta(v)\; =\; 1\; \backslash otimes\; v\; +\; v\; \backslash otimes\; 1$

on elements . The symbol $1$ stands for the unit element of the field . Recall that , so that the above really does lie in . This definition of the coproduct is lifted to the full space $\{\backslash textstyle\backslash bigwedge\}(V)$ by (linear) homomorphism. The correct form of this homomorphism is not what one might naively write, but has to be the one carefully defined in the coalgebra article. In this case, one obtains

$\backslash Delta(v\; \backslash wedge\; w)\; =\; 1\; \backslash otimes\; (v\; \backslash wedge\; w)\; +\; v\; \backslash otimes\; w\; -\; w\; \backslash otimes\; v\; +\; (v\; \backslash wedge\; w)\; \backslash otimes\; 1\; .$

Expanding this out in detail, one obtains the following expression on decomposable elements:

$$

\Delta(x_1 \wedge \cdots \wedge x_k) = \sum_{p=0}^k \; \sum_{\sigma \in Sh(p,k-p)} \; \operatorname{sgn}(\sigma) (x_{\sigma(1)} \wedge \cdots \wedge x_{\sigma(p)}) \otimes (x_{\sigma(p+1)} \wedge \cdots \wedge x_{\sigma(k)}). where the second summation is taken over all -shuffles. By convention, one takes that Sh( k,0) and Sh(0, k) equals {id: {1, ..., k} → {1, ..., k}}. It is also convenient to take the pure wedge products $v\_\{\backslash sigma(1)\}\backslash wedge\backslash dots\backslash wedge\; v\_\{\backslash sigma(p)\}$ and $v\_\{\backslash sigma(p+1)\}\backslash wedge\backslash dots\backslash wedge\; v\_\{\backslash sigma(k)\}$ to equal 1 for p = 0 and p = k, respectively (the empty product in $\{\backslash textstyle\backslash bigwedge\}(V)$). The shuffle follows directly from the first axiom of a co-algebra: the relative order of the elements $x\_k$ is preserved in the riffle shuffle: the riffle shuffle merely splits the ordered sequence into two ordered sequences, one on the left, and one on the right.

Observe that the coproduct preserves the grading of the algebra. Extending to the full space $\{\backslash textstyle\backslash bigwedge\}(V),$ one has

$$

\Delta : {\textstyle\bigwedge}^k(V) \to \bigoplus_{p=0}^k {\textstyle\bigwedge}^p(V) \otimes {\textstyle\bigwedge}^{k-p}(V)

The tensor symbol ⊗ used in this section should be understood with some caution: it is not the same tensor symbol as the one being used in the definition of the alternating product. Intuitively, it is perhaps easiest to think it as just another, but different, tensor product: it is still (bi-)linear, as tensor products should be, but it is the product that is appropriate for the definition of a bialgebra, that is, for creating the object . Any lingering doubt can be shaken by pondering the equalities and , which follow from the definition of the coalgebra, as opposed to naive manipulations involving the tensor and wedge symbols. This distinction is developed in greater detail in the article on . Here, there is much less of a problem, in that the alternating product $\backslash wedge$ clearly corresponds to multiplication in the exterior algebra, leaving the symbol $\backslash otimes$ free for use in the definition of the bialgebra. In practice, this presents no particular problem, as long as one avoids the fatal trap of replacing alternating sums of $\backslash otimes$ by the wedge symbol, with one exception. One can construct an alternating product from , with the understanding that it works in a different space. Immediately below, an example is given: the alternating product for the dual space can be given in terms of the coproduct. The construction of the bialgebra here parallels the construction in the tensor algebra article almost exactly, except for the need to correctly track the alternating signs for the exterior algebra.

In terms of the coproduct, the exterior product on the dual space is just the graded dual of the coproduct:

$$

(\alpha \wedge \beta)(x_1 \wedge \cdots \wedge x_k) = (\alpha \otimes \beta)\left(\Delta(x_1 \wedge \cdots \wedge x_k)\right)

where the tensor product on the right-hand side is of multilinear linear maps (extended by zero on elements of incompatible homogeneous degree: more precisely, , where $\backslash varepsilon$ is the counit, as defined presently).

The counit is the homomorphism $\backslash varepsilon\; :\; \{\backslash textstyle\backslash bigwedge\}(V)\; \backslash to\; K$ that returns the 0-graded component of its argument. The coproduct and counit, along with the exterior product, define the structure of a bialgebra on the exterior algebra.

With an antipode defined on homogeneous elements by , the exterior algebra is furthermore a Hopf algebra.Indeed, the exterior algebra of is the enveloping algebra of the abelian Lie superalgebra structure on .

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Functoriality Suppose that $V$ and $W$ are a pair of vector spaces and $f\; :\; V\; \backslash to\; W$ is a linear map. Then, by the universal property, there exists a unique homomorphism of graded algebras

$\{\backslash textstyle\backslash bigwedge\}(f)\; :\; \{\backslash textstyle\backslash bigwedge\}(V)\backslash rightarrow\; \{\backslash textstyle\backslash bigwedge\}(W)$

such that

$$

{\textstyle\bigwedge}(f)\left|_

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